An Introduction to Behavioral Economics

The causes of the financial crisis and the associated credit crunch are highly complex, and indeed controversial, but undoubtedly one factor played a big part: ambiguity-aversion. People, and institutions, do not like situations where outcomes are uncertain and they cannot estimate probabilities of these outcomes. The state of the financial system and credit crunch was such an unprecedented event in recent times that people had no past history on which to draw in order to estimate Bayesian probabilities. Their reaction was to refuse to lend to borrowers whose ability to repay was uncertain, and refuse to buy securities whose value was uncertain.

Behavioral economics, more specifically neuroeconomics, is helpful in understanding ambiguity-aversion. Studies by John Coates (Coates and Herbert, 2008), a neuroscientist and former derivatives trader, indicate that uncertainty among financial traders causes the adrenal cortex in their brains to increase the output of cortisol, a stress-related hormone. In most environments this is advantageous, aiding the body’s healing processes with an anti-inflammatory response. However, repeated stress over weeks and months can produce so much cortisol that the brain focuses excessively on negative memories and perceives threats where they do not exist. Maybe the credit crunch was ultimately caused by raging hormones?

5.1 Background

Expected utility theory

The standard economic model of decision-making under risk is EUT. This has been widely accepted and applied as both a descriptive model of economic behavior and as a normative model of rational choice; this means that it assumes that rational people would want to obey the axioms of the theory, and that they do actually obey them in general. The basic EUT model was described in mathematical terms in expression (1.1) in the first chapter. In disaggregated form it is repeated here:

We are now concerned with examining (1), (3) and (4), omitting a discussion of (2) until Chapter 7.

The concept of utility was discussed in the last chapter, and we have seen that we can think of it in terms of subjective value. In this case we are using the term in a psychological sense, although we have also seen that it can be used in a more technical neurological sense. Decision-making under risk can be considered as a process of choosing between different prospects or gambles. A prospect consists of a number of possible outcomes along with their associated probabilities. Thus any theory of decision-making under risk will take into account both the consequences of choices and their associated probabilities. In keeping with the rest of this text, situations will often be explained in words or in terms of examples first, before using mathematical notation, even though the latter is often both more concise and more precise, in terms of avoiding ambiguity. As explained in the Preface, this sequence should help the reader to relate abstract terms to real concepts more easily. This is particularly important in the initial stages of analysis, until the reader becomes familiar with mathematical notation. A simple example of a situation involving decision-making under risk is given below, where an individual is required to choose between two alternative courses of action, resulting in two prospects:

Prospect A:	50% chance to win 100	50% chance to win nothing
Prospect B:	Certainty of winning 45

It will be convenient to denote prospects with bold case letters. In general terms a prospect can be described mathematically as follows:

q = (x₁, p₁; …; x_n, p_n)

where x_i represents outcomes and p_i represents associated probabilities. Riskless prospects yielding a certain outcome x are denoted by (x). Thus prospect A above can be expressed: q = (100, 0.5; 0, 0.5), but for simplicity null outcomes can be omitted, thus reducing prospect A to (100, 0.5). Prospect B can be expressed: r = (45).

The axioms underlying EUT were originally developed by von Neumann and Morgenstern (1947), and are related to the axioms of preference described in the previous chapter:

1 Completeness

This requires that for all q, r:

Either or or both.

2 Transitivity

If we take any three prospects, q, r, s:

if and , then

Sometimes the two above axioms are combined together and referred to as the ordering axiom.

3 Continuity

This principle guarantees that preferences can be represented by some function that attaches a real value to every prospect. In formal terms this can be expressed as follows:

For all prospects, q, r, s where and , there exists some probability p such that there is indifference between the middle ranked prospect r and the prospect (q, p; s, 1–p). The latter is referred to as a compound prospect, since its component outcomes q and s are themselves prospects. We will see that such prospects are important in experimental studies, since they often lead to violations of EUT.

The majority of economists have assumed that the existence of some preference function involving the above axioms is the starting point for constructing any satisfactory model of decision-making under risk. Essentially the assumption amounts to consumers having well-defined preferences, while imposing minimal restrictions on the precise form of those preferences.

A further axiom of EUT is the independence axiom; this imposes quite strong restrictions on the precise form of preferences. It is described below, and examples will be given in the following subsection related to anomalies in EUT.

4 Independence

This axiom relates to the cancellation principle described in Chapter 3, that any state of the world that results in the same outcome regardless of one’s choice can be cancelled or ignored. Kahneman and Tversky refer to this axiom as the substitution axiom in their 1979 paper. Let us illustrate the independence axiom with a simple numerical example first, before giving the formal representation. If prospect q = ($3000) is preferred to prospect r = ($4000, 0.8), then prospect q = ($3000, 0.25) is preferred to prospect r = ($4000, 0.2). The reader should note that the last two prospects have 25% of the probabilities of the first two prospects. The independence axiom can be generalized in the following formal representation:

For all prospects, q, r, s:

(5.1)

It can be seen that there is a common component of the compound prospects (s, 1 – p); according to the cancellation principle this component can be ignored in the comparison. In the example above s = 0 and p = 0.25.

EUT provides a simple model for combining probabilities and consequences into a single measure of value that has various appealing qualities. One particularly important property is commonly referred to as the monotonicity principle. Although we have discussed monotonicity earlier, in the context of indifference curves, in EUT the term implies that objective improvements to a prospect, meaning increasing some of its payoffs while holding others constant, should make it at least as attractive if not more so than before. It is thus related to the principle of dominance described in the last chapter, but in this situation we say that the dominance is stochastic since we are comparing not just outcomes but also probabilities. Again an example will aid understanding before giving a formal representation; this one comes from Tversky and Kahneman (1986, p. 263):

Consider the following pair of lotteries, described by the percentage of marbles of different colors in each box and the amount of money you win or lose depending on the color of a randomly drawn marble. Which lottery do you prefer?

The reader can verify that option B will be preferred to option A because it stochastically dominates it. For white, red and yellow marbles the situation is the same, but for both green and blue marbles the outcomes are more favorable, with the probabilities remaining the same.

It has been generally held by economists that the monotonicity principle is fundamental to any satisfactory theory of consumer preference, both descriptive and normative. Therefore its formal representation as an axiom is necessary:

5 Monotonicity

Let x₁, x₂,…, x_n be outcomes ordered from worst (x₁) to best (x_n). A prospect q = (p_q₁,…, p_q_n) stochastically dominates another prospect r = (p_r₁,…, p_r_n) if for all i = 1,…,n:

(5.2)

with a strict inequality for at least one i.

EUT also is based on the Expectation principle, that the overall utility of a prospect is the expected utility of its outcomes. In mathematical notation:

U(x₁, p₁; …; x_n, p_n) = p₁u(x₁) + ... + p_nu(x_n)

Based on the five axioms above, and the expectation principle, EUT states that consumers will behave in such a way that they will maximize the following preference function:

V(q) = Σp_i · u(x_i)

(5.3)

Where q is any prospect, and u(.) is a utility function defined on the set of consequences (x₁, x₂,…, x_n). We have seen a simple example of this in Chapter 1, where a student is deciding whether to drink coffee or beer before attending a lecture. However, in that example we ignored two factors that now need to be discussed. These concerned asset integration (since we only considered gains) and the nature of the utility function (we measured all gains in terms of utility rather than money, so no conversion was necessary). Thus, in addition to the five axioms above, two further assumptions are commonly made in EUT:

(i) Asset integration: a prospect is acceptable if and only if the utility resulting from integrating the prospect with one’s assets exceeds the utility of those assets alone. Thus it is final states that matter, not gains or losses. In mathematical notation:

(x₁, p₁; …; x_n, p_n) is acceptable at asset position w iff

U(w + x₁, p₁; …; w + x_n, p_n) > u(w)

(ii) Risk aversion: a person is said to be risk-averse if he prefers the certain prospect (x) to any risky prospect with expected value x. In EUT risk-aversion is caused by the concavity of the utility function. This characteristic is in turn caused by the law of diminishing marginal utility. Concavity and risk-aversion are best explained by means of a graphical example. A graph of the EUT utility function is shown in Figure 5.1.

If a person starts at any point of wealth x₁ then any gain in wealth will produce a relatively small gain in utility (u₂ – u₁) compared with the loss of utility (u₁ – u₃) associated with an equal loss in wealth. Nobody would ever want to make a fair bet, meaning one where the expected value of the prospect is zero, for example, betting on the toss of a coin.

It can be seen in Figure 5.1 that this utility function is monotonically increasing, meaning that u is increasing throughout the range of x; in mathematical terms: du/dx or u′ > 0. However, the slope of the utility function decreases, meaning that the second derivative of u with respect to x is negative. In mathematical notation: u″ < 0. The most common type of mathematical function that conforms to these characteristics of risk-aversion is a power function of the form u = x^b, where b < 1. With this type of function risk-seeking occurs if b > 1, and the utility function is convex. When b = 1, the power function becomes a linear function, implying risk neutrality.

Figure 5.1 EUT utility function

Evolutionary foundation

What can we say about EUT in terms of its foundations in evolutionary biology? At first sight it might seem that there was a sound evolutionary foundation for EUT. If individuals have to choose among a number of prospects related to expected offspring, evolution through natural selection would favor the choice of the prospect that leads to the highest expected offspring, assuming that the economic and biological risk is independent across all individuals. If, as discussed in Chapter 3, expected offspring translates into expected utility, then evolutionary theory would support EUT.

However, as Robson (2002) argues, this foundation is heavily dependent on the underlying assumption that all risks are independent, and this assumption is not reasonable in an evolutionary setting. In this setting there is an important difference between idiosyncratic risk and aggregate risk. Idiosyncratic risk refers to situations where decisions have purely selfish effects, such as the risk of heart disease or driving a car. Aggregate risk refers to situations where outcomes are shared by others, like the risk of catching ‘mad cow’ disease or being involved in a plane crash. Other things being equal, gambles involving idiosyncratic risk are preferable from an evolutionary standpoint to gambles involving aggregate risk. This may explain why people tend to be averse to the aggregate risk, for example, regarding air travel as more dangerous than travel by car.

There is a very important implication of this evolutionary distinction between idiosyncratic risk and aggregate risk that has already been mentioned in the previous chapter and will be further explored in this chapter: what matters biologically is not absolute success in producing offspring, but success relative to other types (Robson, 2002). Translated into economic terms, this means that utility is reference-dependent. Thus by relaxing the assumption relating to all risks being independent, we can understand why evolutionary theory can explain anomalies in EUT. We now need to examine the nature of these anomalies in more detail.

Anomalies in EUT

A number of anomalies have been observed relating to the axioms and assumptions of EUT, arising both from laboratory experiments and from field data. Some of these observations predate Kahneman and Tversky’s classic paper on prospect theory of 1979, some relate specifically to the paper, and others have come to light since then. At this point we will move on to discuss the last category of observations first; this will give the reader a general feel for the problems of EUT, since all the observations come from field data. We will then examine two examples predating prospect theory; both date back to Allais (1953), although both are also discussed in the 1979 prospect theory paper. Finally, the specific results from Kahneman and Tversky (KT) and other studies are discussed in more detail in later sections, examining the components of prospect theory and adopting a more rigorous analysis.

Many of these anomalies have been described, categorized and analyzed by Camerer (2000). A useful summary is given in Table 5.1, which is adapted from Camerer. This table names and describes the phenomena involved, classifies the anomalies according to domain, and indicates which elements of prospect theory are relevant in terms of explaining the anomalies. These anomalies are discussed in more detail in later sections of the chapter, and also in the following chapter on mental accounting, after the elements of prospect theory have been explained.

Let us now examine a more detailed example of an anomaly in EUT, this time from experimental data. This is sometimes referred to as the Allais paradox, and dates back to 1953. This is illustrated in the payoff matrix in Table 5.2. Each row represents an act involving a prospect, while each column represents a ‘state of the world’, with the associated probabilities at the top of each column. The values in the matrix represent payoffs to each act (in $, for example) given a certain state of the world. Subjects are first presented with a choice between options A and B. The independence axiom implies that since these two acts have the same consequence (a payoff of $500) in the third state of the world, the third state should be irrelevant to that choice. The same argument applies when subjects are then presented with a choice between options C and D (the common consequence being a payoff of 0). It should now be seen that, when the third state of the world is ignored, the choice between A and B is identical to the choice between C and D, that is a choice between (500, 0.11) and (2500, 0.1).

Table 5.1 Phenomena inconsistent with EUT

Therefore, according to the independence axiom, if A is preferred to B, C should be preferred to D, and vice versa. However, there is evidence from numerous studies that many people faced with similar pairs of choices choose A over B, but D over C, violating the independence axiom. This phenomenon is an example of what is called the common consequence effect.

Table 5.2 The Allais paradox

Another anomaly in EUT is described in the KT paper, again relating to the independence or substitution axiom. This situation is shown in the payoff matrix in Table 5.3. It can be seen that the choices between A and B and between C and D involve the same payoffs, and also the same relative probabilities, with the probability of the lower payoff being twice the probability of the higher payoff. Once again the independence axiom implies that if A is preferred to B then C should be preferred to D, and vice versa. The KT paper found a very contrasting result, with only 14% of their subjects preferring A to B, but 73% preferring C to D. This is an example of a phenomenon called the common ratio effect, since the ratio of the probabilities in each option is the same in both choices.

Table 5.3 Same payoffs but different probabilities of winning

Choice 1	A	(6000, 0.45)
Choice 1	B	(3000, 0.90)
Choice 2	C	(6000, 0.001)
Choice 2	D	(3000, 0.002)

These preliminary examples should give the reader a flavour of the anomalies related to EUT. Furthermore, the observed departures from the theory were systematic, meaning that they were in a predictable direction, rather than being random errors. Now we can turn our attention to various attempts that have been made since the late 1970s to account for these violations of EUT in terms of proposing a more satisfactory theory.

It is useful here to use the broad classification proposed by Starmer (2000), who distinguishes between conventional and non-conventional theories. The former accepted the first three axioms of completeness, transitivity and continuity, but were prepared to allow violations of the independence axiom, since these violations had been widely observed since the work of Allais. However, these conventional theories proposed that preferences should still be ‘well-behaved’; in particular this characteristic involves maintaining monotonicity or dominance, and also the principle of invariance described in the last chapter. Non-conventional theories do not insist on preferences being well-behaved in these ways. We will consider the conventional theories first, since these represent both the earlier departure and the least modification of EUT.

5.2 Conventional approaches to modifying EUT

There are a large number of different theories that have been proposed in this category, and it is not intended to review all of them here. Instead the leading contenders are described in terms of their main features, advantages and disadvantages. It is not possible to conduct such a review on an entirely chronological basis, since many theories have been developed over some period of time by several writers. Instead, theories are presented largely in the order that corresponds to the extent of their departure from EUT, with those theories that depart least being presented first.

Weighted utility theory

One of the earliest extensions of EUT was termed weighted utility theory (Chew and MacCrimmon, 1979). The preference function is represented as:

V(q) = [Σp_i · g(x_i) · u(x_i)] / [Σp_i · g(x_i)]

(5.4)

Where u(.) and g(.) are two different functions assigning nonzero weights to all outcomes. This model incorporates EUT as a special case when the weights assigned by g(.) are identical for every outcome. The model has been axiomatized by various economists, including Chew and MacCrimmon (1979), Chew (1983) and Fishburn (1983). These all involve a weaker form of the independence axiom, for example:

A similar model, which generalized this approach, was proposed by Machina (1982). In behavioral terms these theories proposed that people became more risk-averse as the prospects they face improve. The main advantage of such theories was that they explained the violations of independence in the common consequence and common ratio cases, but they lacked intuitive appeal because there was no psychological foundation. They were empirically rather than theoretically grounded.

Disappointment theory

Later theories had a better psychological foundation. An example is the disappointment theory developed by Bell (1985) and Loomes and Sugden (1986). In the latter version the preference function is represented as follows:

V(q) = Σp_i·[ u(x_i) + D (u(x_i) – U)]

(5.5)

where u(x_i) refers to a measure of ‘basic’ utility, meaning the utility of x_i considered in isolation from the other outcomes of q, and U is a measure of the prior expectation of the utility of the prospect. It is assumed in the model that if the outcome of a prospect is worse than expected, meaning if u(x_i) < U, a sense of disappointment or regret will be experienced. On the other hand, if an outcome is better than expected, there will be a sense of elation. When D(·) 0, the model reduces to EUT. However, the extension to the EUT model is intended to capture the psychological intuition that people are disappointment-averse; this entails the disappointment function D(·) being concave in the negative region and convex in the positive region. Disappointment theory is closely related to regret theory; in this case the decision-maker is trying to minimize a regret function, which represents the difference between the outcome yielded by a given choice and the best outcome that could have been achieved in that state of nature.

Betweenness models

Theories of this type have been proposed by Gul (1991) and Neilson (1992). Again they involve a weakened form of independence. Betweenness can be described as follows:

if then for all p < 1.

In behavioral terms this implies that any probability mixture of two lotteries will be ranked between them in terms of preference and, given continuity, an individual will be indifferent to randomization among equally valued prospects.

Non-betweenness models

Other models do not impose restrictions as strong as betweenness. Quadratic utility theory, proposed by Chew, Epstein and Segal (1991) is based on a weakened form of betweenness called mixture symmetry. This can be represented as follows:

If q ~ r then (q, p; r, 1 – p) ~ (q,1 – p; r, p)

The lottery-dependent utility model of Becker and Sarin (1987) has even weaker restrictions, assuming nothing regarding independence (although still assuming completeness, transitivity, continuity and monotonicity).

Decision-weighting theories

All the theories described so far assign subjective weights, or utilities, to outcomes. The value of any prospect is then determined by a function that combines these subjective weights with objective probabilities. Decision-weighting theories involve the use of probability transformation functions which convert objective probabilities into subjective decision weights. Betweenness does not generally hold in these cases.

These theories are empirically grounded in that there is much evidence that people tend to systematically underestimate probability in some situations and overestimate in others. For example, Pidgeon et al. (1992) found that people underestimate the probability of dying from common causes, like heart disease and cancer, and overestimate the probability of dying from rare causes, like in an airline accident. This evidence is reviewed in more detail in the next section on prospect theory. The effects of this phenomenon can be captured by incorporating decision weights in the preference function. An early version of this kind of model was proposed by Edwards (1954, 1955, 1961, 1962), being called a subjectively weighted utility (SWU) model. This model used subjective probabilities but objective outcomes, meaning that outcomes were entered into the model in a ‘raw’ form with u(x_i) x_i. The resulting preference function is given by:

V(q) w_i · x_i

(5.6)

A similar model, better known as the subjective expected value (SEU) model, was axiomatized by Savage in 1954. This model proposed that people used Bayesian methods in estimating subjective probabilities.

A later version, developed by Handa (1977), employed a probability weighting function π (p_i) which transforms the individual probabilities of outcomes into weights. π (·) is assumed to be increasing with π (0) 0 and π (1) 1. Variations of this model were proposed that allowed nonlinear transformations of probabilities of both probabilities and outcomes; both probabilities and outcomes are measured subjectively. Starmer (2000) refers to such forms as simple decision weighted utility models, and they are also sometimes referred to as ‘stripped prospect theory’, since they essentially use the second stage of the prospect theory process, but omit the editing rules of the first stage (explained in the next section).

The corresponding preference function is shown below:

V(q) = Σ π (p_i) · u (x_i)

(5.7)

Since nonlinear transformations of probabilities did not satisfy the monotonicity principle as far as preferences were concerned these models were generally not taken seriously by most economists. For example, Machina (1983) argued that any such theory will be: ‘in the author’s view at least, unacceptable as a descriptive or analytical model of behavior’ (p. 97).

Rank-dependent EUT

Rank-dependent utility (RDU) models were first developed by Quiggin (1982, 1993) in response to the problem described above. They proposed decision-weighting with more sophisticated probability transformations designed to ensure monotonicity in the preference function. In this type of model the weight attached to outcomes depends not only on the true probability of the outcome but also on its ranking relative to the other outcomes of the prospect. It is more complex to describe mathematically than previous models, but the starting point is to rank the outcomes x₁, x₂,..,x_n from worst (x₁) to best (x_n). The model then proposes the preference function:

V(q) = Σw_i · u (x_i)

(5.8)

where the weights are given by the weighting function:

w_i = π (p_i +…+ p_n) – π (p_i₊₁ +…+ p_n)

The interpretation of this function is that π (p_i +…+ p_n) is a subjective weight attached to the probability of getting an outcome of x_i or better, while π (p_i₊₁ +…+ p_n) is a weight attached to the probability of getting an outcome of better than x_i. In this theory therefore π (·) is a transformation on cumulative probabilities. A variation on this model is the rank-and-sign dependent utility (RSDU) model of Luce and Fishburn (1991, 1995), which involves an asymmetric utility function to take account of loss-aversion.

Rank-dependent models have become popular among economists because they have both empirical and theoretical appeal. Empirically they take into account the psychological tendency to overestimate and underestimate probabilities related to particularly good or bad outcomes. Theoretically the appeal has been the preservation of monotonicity.

The form of π (·) is critical in determining the predictions of the model. For example, convexity of π (·) implies a pessimistic outlook, in that it attaches relatively high weights to lower ranking outcomes compared with higher ranking ones. A favorite form, employed by Quiggin (1982), involves a more complex inverted S-shape, illustrated in Figure 5.2.

This function has π (p) p when p p*; it is concave below p* and convex above it. These forms of weighting function can be obtained using a variety of mathematical forms, some using one parameter and some using two. It goes outside the scope of this book to discuss these in detail, but the single-parameter model of Tversky and Kahneman (1992) is described in the next section.

As with other conventional extensions of EUT, rank-dependent EUT relies on a weakened form of the independence axiom called co-monotonic independence. This imposes the restriction that preferences between prospects will be unaffected by substitution of common consequences so long as these substitutions do not affect the rank order of the outcomes in either prospect. Other variations of the rank-dependent model have proposed similar axioms, like ordinal independence (Green and Jullien, 1988).

Figure 5.2 Rank-dependent probability weighting function with inverted S-shape

Conclusions

These more complex extensions to EUT have without doubt explained some of the observed violations, in particular those relating to independence. Over the last 50 years a very large body of studies has built up evidence that can discriminate between the theories described above in terms of their predictive ability. In particular, studies by Conlisk (1989), Camerer (1992), Harless (1992), Gigliotti and Sopher (1993) and Wu and Gonzalez (1996) tend to support the decision-weighting models in favor of other conventional models. Furthermore, those models involving an inverted S-shaped weighting function tend to have better empirical fit than those using other functional forms of the weighting function (Lattimore, Baker and Witte, 1992; Tversky and Kahneman, 1992; Camerer and Ho, 1994; Abdellaoui, 1998; Gonzalez and Wu, 1999). These more sophisticated models show significant predictive improvement over EUT.

However, even these models are still unable to explain widely observed violations of monotonicity, as well as violations of transitivity and invariance. A well-known example is the Ellsberg paradox. The examples given earlier in Tables 5.2 and 5.3, relating to the Allais paradox, involve situations where objective probabilities are known, in other words risky situations. However, frequently people face situations where objective probabilities cannot be calculated; these are referred to as situations involving uncertainty or ambiguity. There is a well-known paradox here also, first investigated by Ellsberg (1961). Suppose that there are two urns, 1 and 2. Urn 2 contains 100 balls, 50 red and 50 blue. Urn 1 also contains 100 balls, again a mix of red and blue, but the subject does not know the proportion of each. Thus choosing from Urn 1 involves a situation of ambiguity, whereas choosing from Urn 2 involves risk. Subjects are asked to make two successive choices. Both choices consist of two gambles, each of which involves a possible payment of $100, depending on the color of the ball drawn at random from the relevant urn. These choices are illustrated in Table 5.4.

Table 5.4 The Ellsberg paradox

Choice 1	A1 – ball is drawn from Urn 1	$100 if red, $0 if blue
Choice 1	A2 – ball is drawn from Urn 2	$100 if red, $0 if blue
Choice 2	B1 – ball is drawn from Urn 1	$100 if blue, $0 if red
Choice 2	B2 – ball is drawn from Urn 2	$100 if blue, $0 if red

Ellsberg’s experiments showed that A2 is generally preferred to A1, while B2 is preferred to B1. These choices are not only inconsistent with EUT, but also with SEU. The choice of A2 implies a subjective probability that less than 50% of the balls in Urn 1 are red, while the choice of B2 implies the opposite, that more than 50% of the balls in Urn 1 are red.

The general conclusion to be drawn from the experiments is that people do not like situations where they are uncertain about the probability distributions of a gamble, i.e. situations of ambiguity. This phenomenon of ambiguity-aversion has been commonly found, not just in experiments, but also in the field, as demonstrated in the introduction to the chapter in relation to financial trading. Furthermore, there is neuroeconomic evidence that provides a basis for the phenomenon. A study by Hsu et al. (2005) involved asking subjects to make choices under conditions of both risk and uncertainty. Under conditions of uncertainty there is more activation of the VMPFC and the amygdala (associated with a primitive fear response), while there is less activation of the NAcc, which relates to expectation of reward. Thus it appears that people value uncertain outcomes less than when the outcomes are risky, as their brains are searching for missing information.

This finding of common occurrence in both experimental and field studies applies to all the violations mentioned here, and includes the various anomalies described by Camerer (2000), illustrated in Table 5.1. The nature of these violations is described in the next section, which is concerned with the exposition of prospect theory; this is classified by Starmer (2000) as a non-conventional theory.

5.3 Prospect theory

The first question at this stage is: what constitutes the difference between a conventional and a non-conventional theory? All the theories described so far, including EUT, have essentially been models of preference maximization, assuming that agents behave as if optimizing some underlying preference function. As we have noted in the last two chapters, the form of the function makes no claim regarding underlying psychological mechanisms or processes. On the other hand, as we have also seen, behavioral models try to model the psychological processes that lead to choice, and Starmer (2000) refers to these as procedural theories. Prominent features of these models include the existence of bounded rationality and the consequent use of decision heuristics. Bounded rationality implies that the agent has both imperfect information in a complex and dynamic decision environment, and limited computational ability; the agent’s objectives may also be imperfectly defined. Thus the concept of optimization becomes more complex, with constraints of time, computational resources, and often conflicting objectives. In such situations the use of heuristics becomes necessary; these are computational shortcuts that simplify decision-making procedures. Since the late 1970s a variety of these non-conventional theories have been developed, but without any doubt prospect theory has been by far the most influential of these. Thus this section concentrates on a detailed discussion of this particular theory. In the concluding section of the chapter some of the other procedural theories are discussed and compared.

Prospect Theory (PT) was originally developed in the KT paper of 1979, and then extended in a later paper by the same authors in 1992, being renamed cumulative prospect theory. We shall see, however, that most of the elements of the theory have important precedents, in particular the work of Markowitz (1952) and Allais (1953). PT models choice as a two-phase process: the first phase involves editing, while the second involves evaluation. The use of an editing phase is the most obvious distinguishing characteristic of PT from any of the theories discussed in the previous section. The second feature that distinguishes PT from these theories is that outcomes are measured as gains or losses relative to some reference point. These features, and the other features of PT are now discussed.

Editing

This phase consists of a preliminary analysis of the offered prospects, which has the objective of yielding a simpler representation of these prospects, in turn facilitating the evaluation process. Certain heuristic rules and operations may be applied, not necessarily consciously, to organize, reformulate and narrow down the options to be considered in the next phase. These operations include coding, combination, segregation, cancellation, simplification, and the detection of dominance.

1 Coding – Empirical evidence suggests that people normally perceive outcomes as gains or losses relative to some reference point, rather than as final states of wealth or welfare. This aspect is discussed in more detail in the next section under Evaluation.

2 Combination – Prospects can sometimes be simplified by combining the probabilities associated with identical outcomes. For example the prospect (100, 0.30; 100, 0.30) will be reduced to (100, 0.60) and evaluated as such. The operation of combining is also referred to as coalescing.

3 Segregation – Some prospects contain a riskless component that can be segregated from the risky component. For example, the prospect (100, 0.70; 150, 0.30) can be segregated into a sure gain of 100 and the risky prospect (50, 0.30). Likewise, the prospect (200, 0.8; 300, 0.2) can be segregated into a sure loss of 200 and the risky prospect (100, 0.2).

4 Cancellation – This aspect was described earlier in relation to the independence axiom. When different prospects share certain identical components these components may be discarded or ignored. For example, consider a two-stage game where there is a probability of 0.75 of ending the game without winning anything and a probability of 0.25 of moving onto the second stage. In the second stage there may be a choice between (4000, 0.80) and (3000). The player must make the choice at the start of the game, before the outcome of the first stage is known. In this case the evidence in the KT study suggests that there is an isolation effect, meaning that people ignore the first stage, whose outcomes are shared by both prospects, and consider the choice as being between a riskless gain of 3000 and the risky prospect (4000, 0.80). The implications of this effect, in terms of inconsistency in decision-making, are considered later, in the section on decision-weighting.

5 Simplification – Prospects may be simplified by rounding either outcomes or probabilities. For example, the prospect (99, 0.51) is likely to be coded as an even chance of winning 100. In particular, outcomes that are extremely improbable are likely to be ignored, meaning that the probabilities are rounded down to 0. This part of the editing phase is often performed first, and the sequence of the editing operations is important, because it can affect the final list to be evaluated. As we shall see, the sequence and method of editing can depend on the structure of the problem.

6 Detection of dominance – Some prospects may dominate others, meaning that they may have elements in common, but other elements involve outcomes or probabilities that are always preferable. Consider the two prospects (200, 0.3; 99, 0.51) and (200, 0.4; 101, 0.49). Assuming that the second component of each prospect is first of all rounded to (100, 0.5), then the second prospect dominates the first, with the outcome of the first component being the same, but its probability being greater.

As has already been stated with regard to the isolation effect, there are various aspects of the editing process that can cause anomalies such as inconsistencies of preference, which will be discussed later. The KT editing heuristic has therefore been criticised on various grounds. For example, Quiggin (1982) has argued that the editing process is redundant if the preference function is appropriately specified. He has also argued that the final stage of detecting dominance, while it may induce monotonicity, has the side-effect that it admits violations of transitivity. Quiggin refers to this as an ‘undesirable result’. Starmer (2005) argues that this comment, and other criticisms of PT, is motivated by ‘a pre-commitment to preference theories which satisfy normatively appealing criteria such as transitivity and monotonicity’ (p. 282). He further argues that such a pre-commitment is misplaced in the light of direct empirical evidence. This issue will be discussed further later when more examples have been examined.

Evaluation

Once the editing phase is complete, the decision-maker must evaluate each of the edited prospects, and is assumed to choose the prospect with the highest value. According to PT, the overall value of an edited prospect, denoted V, is expressed in terms of two scales, v and π. The first scale, v, assigns to each outcome x a number, v(x), which reflects the subjective value of that outcome. The second scale, π, associates with each probability p a decision weight π(p), which reflects the impact of p on the overall value of the prospect.

The first scale entails an explanation of reference points, loss-aversion and diminishing marginal sensitivity, while the second scale entails an explanation of decision-weighting, or weighted probability functions. These four aspects of PT are discussed at length in the next four sections, since they are core concepts to which repeated reference is made throughout the remainder of the text. There is one other element that is often ascribed to PT, concerning the framing of decisions, but this aspect is discussed in the next chapter, in the context of mental accounting.

A mathematical exposition of the basics of the KT model can now be given, which is essentially taken directly from the 1979 paper. As far as the first scale is concerned, the EUT utility function u(x) = x^b is replaced by the following value function:

(5.9)

There are four parameters in this model:

r = reference point

α = coefficent of diminishing marginal sensitivity for gains

β = coefficent of diminishing marginal sensitivity for losses

λ = coefficient of loss-aversion

The second scale, for decision-weighting, involves an inverted S-shaped curve, similar to that shown in Figure 5.2, with the following form:

(5.10)

There is now a fifth parameter, γ, which determines the curvature of the function. All of these parameters are explained in the following sections of the chapter, with numerical examples.

The original KT model was concerned with simple prospects of the form (x, p; y, q), which have at most two nonzero outcomes. In such a prospect, one receives x with probability p, y with probability q, and nothing with probability 1 – p – q, where p + q ≤ 1. A prospect is strictly positive if its outcomes are all positive, i.e. if x, y > 0 and p + q = 1; it is strictly negative if its outcomes are all negative. A prospect is regular if it is neither strictly positive nor strictly negative.

The basic equation of the theory describes the manner in which v and are combined to determine the overall value of regular prospects.

If (x, p; y, q) is a regular prospect (i.e. either p + q < 1, or x ≥ 0 ≥ y, or x ≤ 0 ≤ y), then

V(x, p; y, q) = π(p) v(x) + π(q) v(y)

(5.11)

Where v(0) = 0, π(0) = 0, π(1) = 1. As in utility theory, V is defined on prospects, while v is defined on outcomes. The two scales coincide for sure prospects, where V(x, 1.0) = V(x) = v(x). Equation (5.11) generalizes EUT by relaxing the expectation principle described earlier.

As a simple example, take the situation of tossing a coin where the outcome of heads results in a gain of $20 while the outcome of tails results in a loss of $10. We can now express the utility of this regular prospect below:

V(20, 0.5; –10, 0.5) = π(0.5) v(20) + π(0.5) v(–10)

The evaluation of strictly positive and strictly negative prospects follows a different rule. In the editing phase, as described earlier, such prospects are segregated into two components: (i) the riskless component, i.e. the minimum gain or loss which is certain to be obtained or paid; (ii) the risky component, i.e. the additional gain or loss which is actually at stake. The evaluation of such prospects is described in the next equation.

If p q = 1 and either x > y > 0 or x < y < 0, then

V(x, p; y, q) v(y) + π(p)[v(x) – v(y)]

(5.12)

That is, the value of a strictly positive or strictly negative prospect equals the value of the riskless component plus the value-difference between the outcomes, multiplied by the weight associated with the more extreme outcome. For example:

V(400, 0.25; 100. 0.75) = v(100) + π(0.25)[v(400) – v(100)]

The essential feature of equation (5.12) is that a decision weight is applied to the value-difference v(x) v(y), which represents the risky component of the prospect, but not to v(y), which represents the riskless component.

The following four sections discuss the main elements of the KT model of PT, including the additions of cumulative PT, which relate to the last of the elements. In each case PT is compared with other theories, and the relevant empirical evidence from various laboratory and field studies is examined.

5.4 Reference points

Nature

In PT outcomes are defined relative to a reference point, which serves as a zero point of the value scale. Thus the variable v measures the value of deviations from that reference point, i.e. gains and losses. As Kahneman and Tversky (1979) say:

This assumption is compatible with basic principles of perception and judgement. Our perceptual apparatus is attuned to the evaluation of changes or differences rather than to the evaluation of absolute magnitudes. When we respond to attributes such as brightness, loudness, or temperature, the past and present context of experience defines an adaptation level, or reference point, and stimuli are perceived in relation to this reference point (p. 277).

It should be noted that this element of PT was not an innovation of the KT model. It has a considerably longer history, involving, in particular, aspects of the work of Markowitz (1952) and Helson (1964), although we will see later that the Markovitz model differs in other important respects from the KT model. The concept of reference points is indeed part of folklore in some respects. For example, readers who are familiar with the children’s story A Squash and a Squeeze will recognize the wisdom of the old man in advising the woman who complains that her house is too small. After cramming all her animals into the house, and then clearing them all out again, she finds that her house now seems large.

It is often assumed in analysis that the relevant reference point for evaluating gains and losses is the current status of wealth or welfare, but this need not be the case. In particular, the relevant reference point may be the expected status rather than the current status, an example being given in the discussion of anomalies. We will return to this aspect of the determination of the reference point later.

As with the other three central components of PT, it is now useful to consider the psychological and neurological foundations of the concept, and then to explain its application by means of a number of examples, discussing how the concept helps to explain certain anomalies in the EUT.

Psychological foundation

Evolutionary psychologists attempt to go beyond describing mental attributes and try to explain their origins and functions in terms of adaptations. It can be seen that the psychological concept of a reference point is related to the broader biological mechanisms of homeostasis and allostasis. Both of these have fundamental functions that appear to have evolved as adaptations. Homeostasis is a well-known biological principle, whereby various systems in the body have an optimal set point, and deviations from this point trigger negative feedback processes that attempt to restore it. Examples are body temperature, the level of blood sugar and electrolyte balance. The term allostasis was introduced by Sterling and Eyer (1988) to refer to a different type of feedback system whereby a variable is maintained within a healthy range, but at the same time is allowed to vary in response to environmental demands. Heart rate, blood pressure and hormone levels are variables in this category. Thus, when we exercise, both heart rate and blood pressure are allowed to rise in order to optimize performance. Wilson et al. (2000) suggest that happiness is also a variable in this category; this issue is discussed further in Chapter 11.

A simple physical or biological illustration of the phenomenon of reference points is the experiment where a person places one hand in cold water and the other in hot water for a certain time, before placing both hands in the same container of lukewarm water. The subject experiences the strange sensation of one hand now feeling warm (the one that was in cold water), while the other hand feels cool (the one that was in hot water). It appears that the visible evidence from the eyes that both hands should feel the same is unable to override the separate reference points of previous temperature in the brain. We shall see that these fundamental evolved adaptation mechanisms underlie many of the anomalies in the standard model.

Neuroscientific foundation

We have already seen in Chapter 3 that there is strong evidence for the existence of reference dependence in terms of the phenomenon of the dopaminergic reward prediction error (DRPE or RPE), which is based on the difference between anticipated and obtained reward. In experimental neuroscientific studies risk is normally introduced into the decision-making situation by asking subjects to perform a series of gambling tasks, and using fMRI. There are complications in neuroscientific analysis because of the possibility of confounds, which is discussed in more detail later in relation to decision-weighting. Two studies, De Martino et al. (2006) and Windmann et al. (2006), have examined the differential responses of subjects to punishments and rewards in gambling tasks, and both support the existence of framing effects and reference dependence, with differences in OFC activity in particular. Caplin and Dean (2009) use an axiomatic approach that eliminates the confound problem and also find evidence for reference dependence based on expectations mediated by dopamine release.

There is another analytical problem here, however, described in Chapter 2, relating to the correlation–causation situation. As Fox and Poldrack (2009) point out, at the present it is difficult to determine whether the observed results reflect the neural causes or neural effects of reference dependence. Further research with lesion patients may help to clarify this situation, as again explained in Chapter 2, since it can usually be concluded that the neural deficit is the cause of any differences in behavior.

Empirical evidence

It is now time to analyze the effects of the reference point phenomenon on decision-making under risk, applying PT to various real-life situations. At this stage the examples are limited to situations where the reference point phenomenon can be isolated from other elements of PT. Further examples, including those given in Table 5.1, involve a combination of elements, and a discussion of these situations is therefore deferred until later in the chapter.

One of the best known anomalies in EUT is a phenomenon sometimes referred to as the ‘happiness treadmill’, which was mentioned in the previous chapter. The average income in the United States has increased by more than 40% in real terms since 1972, yet in spite of having far greater income and wealth Americans regularly report that they are no happier than previously. Similar findings have occurred in other countries. Furthermore, the phenomenon does not appear to be caused by the unreliability of self-reporting data; if other indicators of happiness or unhappiness are examined, such as suicide rates and the incidence of depression, we see the same story. Suicide rates are at least as high in rich countries, like the US, Japan, Sweden and Finland, as they are in poor countries. Similarly, within the same country, suicide rates tend to be at least as high among affluent groups as they are among poor groups.

There is some evidence that reference incomes may also play an important role in the gender gap in incomes, at least in some professions. Rizzo and Zeckhauser (2007) use data from the Young Physicians’ Survey and find that males set higher reference incomes and respond more strongly to reference incomes compared with females, and show that this explains the gender gap in earnings and earnings growth rates.

Even in situations like winning the lottery, where a large and sustained increase in happiness would be expected, at least according to EUT, empirical evidence indicates that winners report average satisfaction levels no higher than that of the general population within as little time as a year (Brickman, Coates and Janoff-Bulman 1978).

There is a happier side to the story with this phenomenon, and that is that it works in both directions. People who have suffered some kind of major personal tragedy, such as the loss of a loved one or serious injury, also tend to recover quickly in terms of reported happiness level. As indicated in the previous chapter, both of these regressions to normal levels of satisfaction tend to be unexpected by those involved, certainly as far as their rapidity is concerned. It is notable that half of prison suicides occur on the first day of imprisonment. Inmates generally adapt quickly to their new environment, in spite of their fearful expectations.

Expectations have an important role to play as far as the reference point phenomenon is concerned. When people expect a pay rise of 10%, for example, and then they are awarded just 5%, they tend to be disappointed. Their reference point in this case is not their current pay level but their expected pay level; thus they code and evaluate the pay award as a loss, not as a gain. A study by Abeler et al. (2011) indicates that work effort is significantly affected by expectations of rewards. In a series of experiments they find that, when expectations are high, subjects work longer and earn more money than if expectations are low.

Reference points are also strongly influenced by the status of others. We may be delighted with a pay rise of 5%, until we find out that a colleague has been awarded 10%, when we react with fury. In this case the new information shifts the reference point, turning what was initially coded as a gain into a loss. Again, EUT is unable to explain this phenomenon, which will be discussed in more detail in Chapter 10 in connection with fairness and social utility functions.

Another situation where the reference point may not correspond to the current level of wealth is where a person has not yet adapted to the current asset position. The KT (1979) paper gives the example:

Imagine a person who is involved in a business venture, has already lost 2000 and is now facing a choice between a sure gain of 1000 and an even chance to win 2000 or nothing. If he has not yet adapted to his losses, he is likely to code the problem a choice between (−2000, 0.50) and (−1000) rather than as a choice between (2000, 0.50) and (1000) (p. 286).

As will be explained later in the chapter, the phenomena of loss-aversion and diminishing marginal sensitivity cause the person who has not yet adapted to be more likely to take a risk than the person who has adapted to the recent loss.

Shifts in reference points may also arise when a person formulates his decision problem in terms of final assets, as proposed in EUT, rather than in terms of gains and losses. This causes the reference point to be set to zero wealth, and we will see later that this tends to discourage risk-seeking, except in the case of gambling with low probabilities, for example entering lotteries.

5.5 Loss-aversion

Nature

In the words of KT (Kahneman and Tversky, 1979):

A salient characteristic of attitudes to changes in welfare is that losses loom larger than gains. The aggravation that one experiences in losing a sum of money appears to be greater than the pleasure associated with gaining the same amount (p. 279).

For example, most people would not bet money on the toss of a coin, on the basis that a heads outcome gives a specific gain, while a tails outcome gives an equal loss. In mathematical terms, people find symmetric bets of the form (x, 0.50; x, 0.50) unattractive. The phenomenon can be expressed in more general mathematical terms as follows:

v(x) < – v(–x) where x >0

(5.13)

Again, this element of PT is not an innovation; it is discussed, for example, by Galanter and Pliner (1974).

Expression (5.13) can also be used to derive a measure of loss-aversion, often denoted by the coefficient λ, where λ is the ratio –v(–x)/v(x). For example, in Tversky and Kahneman’s 1992 study it was found that the median value of for their college student subjects was 2.25. However, there is no universally agreed measure of loss-aversion. Whereas in the above situation is measured over a range of x values, it can also be measured as –v(–$1)/v($1). Another measure is to take the ratio of the slopes of the value function, v′(–x)/v′(x), in the loss and gain regions; the meaning of this will become clearer after reading the next section related to the value function.

As with the previous element of PT, it is useful to consider the psychological foundation of the phenomenon first, before moving on to examine a more detailed analysis of examples, indicating how these explain anomalies in EUT.

Psychological foundation

Evolutionary psychologists have speculated on the origins of the phenomenon, in terms of its adaptationary usefulness. Pinker (1997) has proposed that, whereas gains can improve our prospects of survival and reproduction, significant losses can take us completely ‘out of the game’. For example, an extra gallon of water can make us feel more comfortable crossing a desert; a loss of a gallon of water may have fatal consequences. While such conjectures regarding the origins of the phenomenon are by necessity in the nature of ‘just-so’ stories, there is no doubt of the existence of the asymmetry in real life, as will be shown when we examine empirical evidence.

A recent theory relating to the psychology of loss-aversion involves the concept of regulatory focus. Instead of explaining behavior in terms of the distinction between ‘hot’ and ‘cold’ systems, this theory examines the fundamental motivational aspects of behavior, and strategies related to these aspects. Regulatory focus theory proposes the coexistence of two motivational systems, the promotion system and the prevention system. These each serve fundamentally important but different needs. The promotion system is concerned with nurturance needs related to advancement, aspirations and accomplishment, and is marked by a sensitivity to gains versus non-gains. People with promotion focus are more sensitive to positive than to negative changes from neutrality or the status quo, i.e. some reference point. In contrast, the prevention system relates to duties, responsibilities and security, and is sensitive to losses versus non-losses. Prevention-focused people are more sensitive to negative than to positive shifts from the status quo (Higgins, 2007). Thus, prevention-focused persons should be more concerned about falling below the previous status quo or reference point, a negative change or loss, than should promotion-focused persons. However, the attitude to risk of such persons will depend on the ability to restore the status quo. If this is more likely to be achieved by playing it safe, then they will be risk-averse. However, if restoring the status quo can only be achieved by a high-risk strategy, then prevention-focused persons will be risk-seeking (Scholer et al., 2010). An example of this second situation is the ‘end-the-day effect’, discussed in the section on empirical evidence.

Neuroscientific foundation

We have already seen in the previous chapter that the main neural evidence for loss-aversion comes from studies that indicate that gains and losses are encoded in different ways in different brain regions. However, there is not just a simple two-system model in operation here. When risk is involved, it appears that the neural activity evoked by potential gains and losses is only partially overlapping with that evoked by actual gains and losses.

A study by Tom et al. (2007) introduced risk into the decision-making situation by imaging subjects during a gamble acceptability task series, in which subjects decided whether to accept or reject mixed gambles offering a 50% chance of gain and 50% chance of loss. The size of gain and loss were varied across different trials, with gains ranging from $10 to $40 and losses from $5 to $20. The loss-aversion coefficient was measured to have a median of 1.93. The study reported that a network of regions, including ventral and dorsal striatum, ventromedial and ventrolateral PFC, ACC and dopaminergic midbrain regions, showed increasing activity as potential gain increased. The loss-aversion asymmetry is shown by the fact that many of these regions showed not just decreasing activity, but a greater decrease in activity for potential losses compared with the increase in activity for potential gains. A more recent study by Caplin et al. (2010) indicates another type of asymmetry. It suggests that, even in the NAcc, which is activated by dopamine with both gains and losses, the signal is different in each case. For gains the signal has a shorter time lag and is less intense than for losses.

Empirical evidence

Over the last 30 years there have been a large number of studies relating to loss-aversion. As with the discussion of reference points, it is helpful to start with a phenomenon where loss-aversion can be isolated from other elements of PT. An example is the observation of asymmetric price elasticities of demand for consumer goods. Price elasticities indicate price sensitivity, measuring the percentage change in quantity demanded divided by the percentage change in price. Loss-averse consumers dislike price increases more than they like the gains from price cuts, and will reduce purchases more when prices rise than they will increase purchases when prices fall. Therefore loss-aversion implies that price elasticities will be asymmetric, with demand being more elastic in response to price rises than in response to price reductions. Such asymmetric responses have indeed been found in the case of eggs (Putler, 1992) and orange juice (Hardie, Johnson and Fader, 1993). In terms of measurement, the study by Hardie, Johnson and Fader found the loss-aversion coefficient to be around 2.4 for orange juice.

One of the best known manifestations of loss-aversion involves differences between prices potential buyers are willing to pay (WTP) for goods and prices potential sellers are willing to accept (WTA) for the same goods, often referred to as the endowment effect. Endowment effects are discussed in Section 5.7, since this is an area where PT has attracted significant criticism.

An anomaly in finance, particularly in stock markets, is a phenomenon known as the ‘disposition effect’ (Shefrin and Statman, 1985), where investors tend to hold on to stocks that have lost value (relative to their purchase price) too long, while being eager to sell stocks that have risen in price. This involves both loss-aversion and reference points, with the purchase price acting as a reference point in this case. EUT is unable to explain this phenomenon easily, since according to this aspect of the standard model people should buy or sell based on expectations of the future price, not the past price. Furthermore, tax laws encourage selling losers rather than winners, in order to reduce capital gains tax liability. Although investors sometimes claim that they hold on to losers because they expect them to ‘bounce back’, a study by Odean (1998) indicated that unsold losers only yielded a return of 5% in the following year, compared with a return of 11.6% for winners that were sold later. Genesove and Meyer (2001) have reported a similar disposition effect in the housing market; owners appear to be unwilling to sell their properties for less than they paid for them and, therefore, tend to hold on to them too long before selling when the market goes into a downturn. However, a more recent study by Barberis and Xiong (2006) questions the ability of prospect theory to explain the disposition effect in stock markets. They test the relationship between stock price movements and traders’ realization of gains and losses and find an opposite result to that predicted by prospect theory. The problem here is that there is a difference between frequency of portfolio evaluation (maybe twice a year) and frequency of checking portfolio values (maybe once a month), and the predictions of PT depend on assumptions regarding whether investors realize utility only at a point of evaluation or every time they check values.

Another anomaly of EUT that can be explained by PT concerns the ‘end-of-the-day effect’ observed in racetrack betting. Bettors tend to shift their bets toward longshots, and away from favorites, later in the racing day (McGlothlin, 1956; Ali, 1977). In this case again both loss-aversion and reference points are involved. Because the track stacks the odds against bettors in order to make a profit, by the last race of the day most of them are suffering losses. It appears that most bettors also use zero daily profit as a reference point; they are, therefore, willing to bet on longshots in the last race, since a small bet can generate a sufficiently large profit to break even for the day. Some studies indicate that this effect is so large that conservatively betting on the favorite to show (a first, second or third place finish) in the last race is profitable, even allowing for the track’s biasing of the odds. It is important to note that EUT cannot account for this phenomenon if bettors integrate their wealth, meaning that they regard gains and losses from the last race on one day as being in the same category as gains and losses on the next outing.

An excellent recent study of loss-aversion relates to professional golf (Pope and Schweitzer, 2011). There are a number of reasons why this study is significant: (1) it relates to highly experienced players, who should have taken advantage of any learning effects; (2) it involves high stakes in a competitive environment, so there is no lack of incentive as far as optimizing behavior is concerned; (3) it involves a situation where there is a natural reference point, par, for each hole played, so that a bogey or any score above par would be considered a loss. This study is discussed in more detail in Case 5.3, but the conclusion is that even the best professional golfers exhibit loss-aversion.

Further anomalies involving loss-aversion, combined with other effects, are discussed in more detail in the other case studies at the end of the chapter, and also in the first two case studies in the next chapter. These involve endowment effects, insensitivity to bad income news, the equity premium puzzle and the downward-sloping labor supply curve.

In general it can be concluded that there is robust evidence supporting the existence of loss-aversion in both experimental and field studies, with different kinds of subjects and goods. Many of these studies have also measured the loss-aversion coefficient, and the results have been summarized by Abdellaoui, Bleichrodt and Paraschiv (2007). The estimates vary between 1.43 and 4.8, but it should be noted that the values are not strictly comparable; not only do they utilize different measures of loss-aversion described earlier, but they make different assumptions regarding the form or shape of the value functions and probability weighting functions described in the next two sections.

There have also been a number of recent studies investigating and reviewing the factors determining loss-aversion and its boundaries (Ariely, Huber and Wertenbroch, 2005; Camerer, 2005; Novemsky and Kahneman, 2005a, 2005b). It is appropriate to discuss these more fully in the context of mental accounting in the next chapter.

5.6 Shape of the utility function

Nature

There is an ongoing debate in the literature regarding this issue. We will consider four main possibilities here: (1) the traditional concave function of the standard model; (2) the Friedman–Savage (FS) function; (3) the Markowitz (M) function; and (4) the Prospect Theory (PT) function.

1 The standard model utility function

As already discussed, this has a concave shape, caused by the law of diminishing marginal utility. The implication of this is that there is risk-aversion at all levels of wealth.

2 The Friedman–Savage utility function

Friedman and Savage (1948) observed that the traditional concave function failed to explain various widely observed phenomena, such as gambling. They proposed a function that had two concave regions, with a convex region between them, in order to explain these anomalies. This is shown in Figure 5.3.

Figure 5.3 Friedman–Savage utility function

Figure 5.4 Markowitz utility function

Although the FS utility function does explain some anomalies of the standard model function, it still fails to explain various empirical observations. For example, it predicts that in the middle region of wealth people will be willing to make large symmetric bets, for example betting $10,000 on the toss of a coin. In reality people do not like such bets, as was pointed out by Markowitz (1952).

3 The Markowitz utility function

Markowitz proposed various amendments in order to remedy the failings of other functions to explain empirical data. He anticipated the work of Kahneman and Tversky by including both reference points and loss-aversion in his analysis. His utility function was S-shaped in the regions of both gain and loss. However, as can be seen in Figure 5.4, in the middle region of small gains and losses between points A and B, the function has a reversed S-shape. The implications of this shape of function are that people tend to be risk-seeking for small gains (explaining most gambling), and risk-averse for small losses (explaining why many people take out insurance). However, people would be risk-averse for large gains and risk-seeking for large losses.

It should be noted that this graph also takes into account the other two elements of Markowitz, that are also features of PT: reference points (measuring outcomes in terms of gains and losses); and loss-aversion (the function is steeper in the loss domain than in the gain domain).

4 The Prospect Theory (PT) utility function

In PT risk-aversion may be caused by two factors. One factor is the nature of the decision-weighting factor (π), discussed in the next section. The other factor is the phenomenon of diminishing marginal sensitivity, which determines the shape of the function v(x). KT (1979) proposed a utility function that featured diminishing marginal sensitivity in the domains of both gains and losses. The marginal value of both gains and losses generally decreases with their magnitude (Galanter and Pliner, 1974); this is essentially an aspect of the well-known feature of the standard model, the law of diminishing returns.

As we have seen earlier, the PT value function was parameterized by Kahneman and Tversky (1992) as a power function:

(5.9) repeated

where α, β > 0 measure the curvature of the value function for gains and losses respectively, and λ is the coefficient of loss-aversion.

The relationship is shown graphically in Figure 5.5.

It can be seen that in PT the value function for changes of wealth is normally concave above the reference point and usually convex below it. In mathematical terms this can be expressed as follows:

v″(x) < 0 for x > 0 and v″(x) > 0 for x < 0

(5.14)

This type of function implies that diminishing marginal sensitivity generally causes risk-aversion in the domain of gains and risk-seeking in the domain of losses. For example, when faced with the prospects of (200, 0.5) and (100) people generally choose the latter, since the gain of 200 does not usually have twice as much utility as the gain of 100. In mathematical terms, v(200) < 2v(100). However, if we reverse this situation in order to consider people’s attitudes to losses we find that people generally prefer the prospect (–200, 0.5) to (–100). They are prepared to gamble in this situation, since the loss of 200 does not usually have twice as much disutility as the loss of 100. In mathematical terms, –v(200) < –2v(100). KT named this phenomenon the reflection effect, meaning that the preference between negative prospects is the mirror image of the preference between positive prospects.

Figure 5.5 PT utility function

Psychological foundation

In the words of KT (1979):

Many sensory and perceptual dimensions share the property that the psychological response is a concave function of the magnitude of physical change. For example, it is easier to discriminate between a change of 3° and a change of 6° in room temperature, than it is to discriminate between a change of 13° and a change of 16°. We propose that this principle applies in particular to the evaluation of monetary changes (p. 278).

As with the other elements of the KT model, evolutionary psychologists like to go beyond the descriptive statement above and speculate on how such a mental adaptation can have evolved. It appears that in many situations it is relative changes rather than absolute changes that are the key to survival and reproduction. For example, when one is hungry, a kilogram of meat is extremely useful (compared with zero or a very small amount); 10 kilograms is not much more useful, unless there are a lot of mouths to feed, since the additional amount could not be stored in our ancestral environment.

However, there does appear to be one exception to diminishing marginal sensitivity, in the loss region. As was mentioned in the discussion of loss-aversion, a really large loss can have fatal consequences. The implication of this is that people may be risk-averse for very large losses, while being risk-seekers in the case of smaller losses. Examples of such situations are given in the discussion of anomalies later.

Neuroscientific foundation

There are a number of problems in attempting to establish a neuroscientific basis for diminishing marginal sensitivity in the utility function. First, as we have seen in the previous chapter, there are different kinds of utility, and the difference between decision and experienced utility is an important one in terms of neural correlates. Anticipatory utility is also relevant. When subjects receive an immediate reward following a decision it is impossible to disentangle the combination of different utilities involved, since at present the fMRI technique lacks sufficient temporal resolution to do this. One approach is to design studies where the relevant reward is delayed and temporally separated from the decision, as in buying a stock or a lottery ticket.

Second, there are problems with confounds, since in practice it is difficult to separate the effects of the shape of the value function from the effects of the shape of the decision-weighting function. As a simple example of this, people may appear to give a relatively low value to receiving a large gain, but this may be due to the assignment of a low decision weight to the receipt of the gain rather than a low utility. This is explained in more detail in the next section.

One consequence of these problems, and others, is that different studies have produced some conflicting results. An example here is the role of the amygdala in responding to stimuli of different valences. It was originally believed that the amygdala was activated only when negative stimuli were given, resulting in a fear or anxiety response. However, later studies have indicated that there is activation in the amygdala in some cases when a positive stimulus involving some kind of reward is administered. At this point, therefore, there is still much that is unknown regarding neural response systems to rewards and punishments of different magnitudes, and it can be concluded that it is not yet clear how neural activity relates to the S-shaped value function proposed by PT.

Empirical evidence

We will start by comparing EUT and PT as far as the shape of utility functions is concerned. Diminishing marginal sensitivity is also a feature of EUT as well as PT, but when combined with reference points and loss-aversion it has different implications in PT compared with EUT.

Let us consider the reflection effect. As an empirical demonstration of this effect, the KT paper examined the attitudes of a sample of 95 respondents to positive and negative prospects and found the following results: when choosing between the prospects (4000, 0.80) and (3000), 80% preferred the second prospect (a sure gain), but when choosing between the prospects (–4000, 0.80) and (–3000), 92% preferred the first prospect, taking a risk of making a greater loss. The reflection effect had also been observed by other researchers before the KT study, for example Markowitz (1952) and A.C. Williams (1966).

EUT regards diminishing marginal utility and the concavity of the utility function as being the cause of risk-aversion. It tends to explain risk-aversion in terms of expected values and variances. However, when the prospects of the losses in the previous example, ( 4000, 0.8) and ( 3000), are compared, the second choice has both higher expected value and lower variance and, therefore, should be preferred according to EUT. As seen earlier, the empirical evidence of the KT study contradicts this prediction, with 92% of their sample preferring the first prospect.

EUT also has problems explaining various attitudes towards insurance. It may seem initially that taking out insurance implies risk-aversion, as implied by EUT. However, by assuming a concave utility function throughout the domain of the level of assets, EUT implies that risk-aversion is universal. This is contradicted by the fact that many people prefer insurance policies that offer limited coverage with low or zero deductible over comparable policies that offer higher maximal coverage with higher deductibles. Thus taking out insurance may be risk-averse compared with not taking out any insurance at all, but some policies, which may be popular, may be risk-seeking compared with others.

There is one phenomenon related to insurance which may appear to be an anomaly for both PT and EUT. This concerns attitudes to probabilistic insurance. This type of policy involves the purchaser paying a fraction of the premium of full insurance, but only gives the probability of the same fraction of paying out if an accident occurs. It appears that such a policy involves more risk than standard insurance. Empirical evidence from the KT study indicates that such insurance would not be popular, which appears to contradict the predictions of the PT model. This apparent anomaly in PT, along with certain tendencies for risk-seeking in the gain domain mentioned earlier, can only be explained after a discussion of decision-weighting in the next section.

So far we have compared the EUT and PT utility functions in terms of empirical evidence. However, it can also be seen that there are important differences between the PT function and the Markowitz (M) function. Markowitz noted the presence of risk-seeking in preferences among positive as well as negative prospects, and, as we have seen, he proposed a utility function that has convex and concave regions in both the gain and loss domains. The Markowitz function has received some empirical support from a study by Jullien and Salanié (1997), which related to racetrack betting. This study found that the utility function for small amounts of money was convex. Another study by Levy and Levy (2002) has also claimed to contradict the PT function and support the Markowitz model.

These studies raise some important issues:

1 How does the PT model explain activities like gambling? This widely observed empirical phenomenon appears to imply risk-seeking in the gain domain.

2 How does the PT model explain why people take out insurance? This is also a common activity, and seems to imply risk-aversion in the loss domain.

It is only possible to discuss the Jullien and Salanié and Levy and Levy studies, and examine the apparent anomalies in the PT model, after a discussion of the remaining element of the model, decision-weighting.

5.7 Decision-weighting

Nature

As with some of the previous elements of PT, which feature in other theories prior to the original KT paper, decision-weighting is not a unique element of PT. Various conventional theories incorporate decision-weighting, notably rank-dependent EUT. This is also the area where there are some substantive differences between the original 1979 paper (henceforth referred to as original prospect theory, or OPT) and the revised 1992 paper, which introduces the term ‘cumulative prospect theory’, or CPT. The latter version is more complex, but more satisfactory in a number of ways. In particular, it is more general, applying to situations involving uncertainty as well as those involving risk, and it is also better supported empirically, fitting a wide number of studies by different researchers in different countries. However, in order to better understand the development of the theory, we will discuss the original version first before extending it.

As with the other elements of the KT model, it is appropriate to start with a quotation from the 1979 paper:

In prospect theory, the value of each outcome is multiplied by a decision weight. Decision weights are inferred from choices between prospects much as subjective probabilities are inferred from preferences in the Ramsey-Savage approach. However, decision weights are not probabilities: they do not obey the probability axioms and they should not be interpreted as measures of degree or belief (p. 280).

There are actually two reasons why decision weights may be different from objective probabilities, and it is important to distinguish between them, even though they may have the same biasing effect, as we will see. These reasons relate to estimation and weighting. The first aspect relates to situations where objective probabilities are unknown, while the second relates to situations where these probabilities are known but do not necessarily reflect decision preferences according to EUT.

1 Estimation of probabilities

People are often lousy at estimating probabilities of events occurring, especially rare ones. They overestimate the probability of dying in plane crashes, or in pregnancy, or suffering from violent crime. An example of overestimating low probabilities, concerning playing the lottery, was given in Chapter 4. The California lottery, one of the biggest in the world, requires matching six numbers between 1 and 51 in order to win the main prize. The odds against doing this are over 18 million to one. In other words, if one played this lottery twice a week, one could expect to win about every 175,000 years. It was found by Kahneman, Slovic and Tversky (1982) that people overestimated the odds of winning by over 1000%.

Another example of situations where people are often bad at estimating probabilities is where conditional probabilities are involved, as we have also seen in Chapter 4. A simple example is where, after several consecutive coin tosses turning up heads, people are inclined to think that tails is more likely on the next toss (in this case the objective probability should be known but appears to be rejected). A more complex example involving conditional probabilities is given by Casscells, Schoenberger and Grayboys (1978), and relates to a situation where a person takes a medical test for a disease, like HIV, where there is a very low probability (in most circumstances) of having the disease, say one in a thousand. However, there is a chance of a false prediction; the test may only be 95% accurate. Under these circumstances people tend to ignore the rarity of the phenomenon (disease) in the population and wildly overestimate the probability of actually being sick. Even the majority of Harvard Medical School doctors failed to get the right answer. For every thousand patients tested, one will be actually sick while there will be 50 false positives. Thus there is only a one in fifty-one chance of a positive result meaning that the patient is actually sick.

2 Weighting of probabilities

The 1979 paper by Kahneman and Tversky concentrates on the discussion of those decision problems where objective probabilities, in terms of stated values of p, are both known and adopted by respondents. In such situations decision weights can be expressed as a function of stated probabilities: π(p) f(p). These decision weights measure the impact of events on the desirability of prospects, and not the perceived likelihood of these events, which was discussed above in the context of estimation. As an illustrative example, consider the tossing of a fair coin, where there is an objective probability of 0.50 of winning 100 and the same probability of winning nothing. In this case it is usually observed empirically that π(0.50) < 0.50, meaning that there is risk-aversion.

There are a number of important characteristics of the weighting function that were observed by KT. Most obviously, π is an increasing function of p, with: π(0) 0 and π(1) 1; this means that impossible events are ignored and the scale is normalized so that π(p) is the ratio of the weight associated with the probability p to the weight associated with a certain event. In addition there are three important characteristics of π which violate the normal probability axioms in EUT: subadditivity, subcertainty and subproportionality. These are now discussed in turn.

The characteristic of subaddivity relates to situations where p is small. For example the KT paper found that the prospect (6000, 0.001) was preferred to (3000, 0.002) by 73% of the respondents. This contravenes the normal risk-aversion for gains and diminishing marginal sensitivity described in earlier elements of PT, and can only be explained by a weighting function involving subadditivity.

In the above example: π(0.001)v(6000) > π(0.002)v(3000)

Since v(3000) > 0.5 v(6000) because of diminishing marginal sensitivity (concavity of v)

π(0.001)v(6000) > π(0.002)v(3000) > π(0.002)0.5v(6000)

Cancelling v(6000) from both sides of the first and last terms,

π(0.001) > 0.5π(0.002)

π(0.5 × 0.002) > 0.5π(0.002)

In general terms the subadditivity principle can be expressed as follows:

π(rp) > rπ(p) for 0 < r < 1

(5.15)

The overweighting of probabilities illustrated above was also observed by KT in the domain of losses. They observed that 70% of their respondents preferred the prospect (–3000, 0.002) to the prospect (–6000, 0.001), which shows the reflected preferences of the previous example. However, they did not find that this principle applied to larger probabilities, for example where p = 0.90. The significance of these findings is discussed in the section relating to anomalies.

The second principle of subcertainty can be illustrated by a couple of examples coming from studies by Allais (1953). Allais noted that people tend to overweight outcomes that are considered certain, relative to outcomes that are merely probable. He found that 82% of his respondents preferred the prospect (2400) to the prospect (2500, 0.33; 2400, 0.66). Yet 83% of his respondents preferred (2500, 0.33) to (2400, 0.34).

Thus in the first case: v(2400) > π(0.66)v(2400) + π(0.33)v(2500)

While in the second case: π(0.33)v(2500) > π(0.34)v(2400)

Thus v(2400) > π(0.66)v(2400) + π(0.34)v(2400)

Dividing by v(2400),

1 > π(0.66) + π(0.34)

In general terms the subcertainty principle can be expressed as:

π(.p) (1 p) < 1

(5.16)

One main implication of subcertainty is that preferences are generally less sensitive to variations in probability than EUT would suggest. This is illustrated in Figure 5.6, where: π(p) is less steep than the 45° line.

The discontinuities of the function at low and high probabilities reflect the phenomenon that there is a limit to how small a decision weight can be attached to an event, if it is given any weight at all. The implication is that events with very low probabilities are ignored, or given a weight of zero, and there is then a discrete quantum jump to a minimum decision weight that is applied to events that are regarded as just sufficiently likely for them to be worth taking into consideration. A similar effect occurs at the upper end of the probability spectrum, where there is a discrete jump between certainty and uncertainty.

A final characteristic of decision-weighting functions is that they involve subproportionality. This means that they violate the independence or substitution axiom of EUT. For example, the KT study found that 80% of respondents preferred the certainty of (3000) to the uncertain prospect (4000, 0.8), but when these outcomes had their probabilities reduced by a common factor of a quarter, the situation was reversed. Thus 65% of their respondents preferred the prospect (4000, 0.2) to the prospect (3000, 0.25).

Figure 5.6 A typical PT weighting function (1979 version)

The independence axiom was stated in formal terms near the beginning of the chapter. In more simple terms this principle states that if a prospect A (x, p) is preferred to prospect B (y, q) then it follows that any probability mixture (A, r) must be preferred to the mixture (B, r). In the example above r = ¼. In contrast the subproportionality principle of PT states the following:

(5.17)

This means that, for a fixed ratio of probabilities, the ratio of the corresponding decision weights is closer to unity when the probabilities are low than when they are high. In more simple terms, people judge probabilities that are the same compared in relative terms (1 to 0.8 and 0.25 to 0.2) to be more similar when probabilities are small (0.25 is judged more similar to 0.2 than 1 is to 0.8).

This completes the description of the original version of decision-weighting in the KT 1979 paper. However, certain empirical anomalies were later observed, described in the final subsection, and the revised ‘cumulative prospect theory’ was developed in the 1992 paper. The essential difference in general terms is that the principle of diminishing marginal sensitivity is now applied to weighting functions as well as the utility function. In the words of Tversky and Kahneman (1992):

Diminishing sensitivity entails that the impact of a given change in probability diminishes with its distance from the boundary. For example, an increase of 0.1 in the probability of winning a given prize has more impact when it changes the probability of winning from 0.9 to 1.0 or from 0 to 0.1, than when it changes the probability of winning from 0.3 to 0.4 or from 0.6 to 0.7. Diminishing sensitivity, therefore, gives rise to a weighting function that is concave near 0 and convex near 1 (p. 303).

In cumulative prospect theory, sometimes called second-generation PT, the probability weighting function is used in a rank-dependent way. Decision weights are assigned cumulatively, starting with the largest gain and working downwards, and a mirror-image method applies to assigning weights to losses. A numerical example will help explain this. Consider a lottery that gives monetary gains of $0, $5 and $10 with respective probabilities 0.5, 0.3 and 0.2. The weight for the largest gain, $10, is determined by a direct transformation of the relevant probability, giving a weight of w(0.2). The decision weight for the next largest gain, $5, is defined as w(0.2 0.3) w(0.2). It should be noted that the sum of the weights for the $5 and $10 gains is w(0.5), which is the sum of the transformation of the corresponding probabilities. The advantage of this innovation in second-generation prospect theory is that it maintains stochastic dominance, unlike original prospect theory.

The weighting function, for both gains and losses, therefore has an inverted S-shape, shown in Figure 5.7. This is again similar to the weighting function used in many conventional rank-dependent EUT models. Instead of using the symbol π to denote decision weights, the symbol w(p) is used, and the general shape of the function is described by the following mathematical form, the same as (5.10):

(5.18)

The parameter γ determining the curvature of the function may be different for losses compared with gains. As Tversky and Kahneman point out, this form has several useful features: it has only one parameter, γ, thus maintaining parsimony; it is versatile, in accommodating weighting functions with both convex and concave regions; and it does not require w(0.5) = 0.5 as a point of symmetry for the curve. Related to this last feature, the most important advantage of the form is that it fits empirical data well, as we will see shortly.

The practical implications of these modifications to the weighting function are that, instead of having a simple twofold attitude to risk, involving risk-aversion for gains and risk-seeking for losses, there is now a more complex fourfold pattern: risk-aversion for gains and risk-seeking for losses of high probability; risk-seeking for gains and risk-aversion for losses of low probability. As we will see in the last part of this section, this refined model fits better with empirical observations.

There have been other attempts to parameterize the decision-weighting function, involving more complex two-parameter models, for example, Lattimore, Baker and Witte (1992) and Prelec (1998). These produce a similar inverse-S-shaped function to that shown in Figure 5.7, and are difficult to distinguish from the KT version in analyzing empirical data.

Figure 5.7 A typical PT weighting function (1992 version)

Psychological foundation

There are various puzzles regarding the attitudes to risk described above. First, there is the issue concerning why we are bad at estimating probabilities, particularly those of rare events. Second, there is the issue regarding why we are usually risk-averse when gains are at stake, but risk-seeking when losses are involved. Finally, there is the issue of exceptions: Why are we sometimes risk-seeking for gains, particularly when probabilities are low? Evolutionary psychologists have ventured a number of theories relating to these issues, and there is some interesting evidence both from other species and from neuroscientific studies.

As far as the first issue is concerned, it appears that we are bad at estimating probabilities for events that have no resemblance to those that have occurred in our evolutionary past. Betting on the lottery is obviously in this category. Complex problems involving conditional probabilities would also often fall into this category of unfamiliarity. On the other hand, events that were a high risk in our evolutionary past, such as death in pregnancy or through violent assault, tend to be overestimated in importance in our current environment (Slovic, Fischhoff and Lichtenstein, 1982; Glassner, 1999).

As an interesting corollary, there is evidence that some animals are extremely good at estimating probabilities in situations that directly affect their survival. One might not expect woodpeckers to be exceptionally good mathematicians, but in one respect they seem able to solve a problem that would stump many skilled humans (Lima, 1984). In a laboratory experiment, woodpeckers were presented with two kinds of artificial trees, both of which had 24 holes. In one group they were all empty, while in the other six of the holes contained food. The problem facing the woodpeckers was similar to the one facing oil wildcatters: to determine how many holes to try before moving on. If they left too soon they would be deserting a tree that may still contain food, but if they stayed too long they missed out on opportunities elsewhere. Using sophisticated mathematics it can be calculated that the woodpeckers would maximize their food intake by leaving a tree after encountering six empty holes. In the study it was found that the average number of holes tried by the woodpeckers was 6.3, remarkably close to perfect. Furthermore, when the number of empty holes was varied in the experiment, the woodpeckers changed their behavior accordingly.

What is the implication of this? It is not that woodpeckers are better mathematicians than humans. However, the process of natural selection has honed the instincts of woodpeckers over millions of years. Those ancestral woodpeckers who were better at solving this problem, through some neurological superiority, were more likely to pass on their genes, with the same capacity, to future generations, Over time competition between woodpeckers would ensure that only the most successful would survive and breed, so that today’s woodpeckers have adapted extremely well to solving their perennial problem. We shall see that this process of natural selection can lead to the building of extremely effective mechanisms for solving problems, particularly in the domain of behavioral game theory.

However, returning to human behavior, there is at least one other important factor that is relevant in terms of explaining our poor estimation of probabilities. Our senses of perception have become attuned through natural selection to become highly selective, filtering out ‘noise’ and trivia, and external events have to compete for our attention. We tend to give more weight to events that attract our attention for whatever reason. In modern times the media play an important role here. Because events like plane and train accidents receive much media coverage, they attract our attention more than car accidents, and this affects our estimation of probabilities, causing us to overestimate them.

Let us now consider the second issue mentioned above. What explains the general pattern of risk-aversion for gains? Once again animal studies are instructive. In general animals often appear to be risk-averse, for example, in competition for mates. Such competition rarely results in fatal injuries. Often there are displays and shows of strength to begin with; if this fails to discourage one of the contenders, the situation may escalate to some form of sparring. This preliminary ‘sizing up’ usually results in one or other contender backing down, except when the rivals are very evenly matched. This rarity of intra-species lethal combat was once explained by biologists in terms of group selection, meaning that the phenomenon was good for the species as a whole. However, this group selection explanation has been long discredited, at least in this context (G.C. Williams, 1966), with biologists now favoring individual selection, at the level of the ‘selfish gene’, to use another expression from Dawkins (1976). In simple terms, the aphorism ‘those who fight and run away live to fight another day’ is appropriate.

This leads us to the third main issue. There are obviously situations where humans are risk-seeking in the region of gains. What can account for this? Again it is necessary to consider animal studies and neurological research, as well as human studies. A study of macaque monkeys by Platt and McCoy (2005) has demonstrated that, like humans, they are fond of a gamble. An experiment indicated that they preferred an unpredictable reward of fruit juice to a reward of a certain amount, where the expected values of both prospects were the same. The experiment also showed that the monkeys still preferred to gamble even when the unpredictable prospect delivered a series of miserly portions. Platt’s conclusion was: ‘… it seemed as if these monkeys got a high out of getting a big reward that obliterated any memory of all the losses that they would experience following that big reward’. It is also notable that the gambling behavior was mirrored by neuronal activity in the brain region associated with the processing of rewards.

In simple terms, there are situations where it does pay most species to take risks, even life-threatening ones. Were it not for this propensity to take such risks the human race would never have ventured forth to populate the whole planet, and would have stayed concentrated in Africa, where homo sapiens originated about a hundred thousand years ago. There are still a few tribal societies in existence today where life is very much as it was then. The Yanomamö in South America are such a tribe, surviving by hunting and small-scale farming. Violence is a way of life for this tiribe, with a quarter of the men dying from this cause. The killers in turn are often killed by relatives of their victims. The question therefore arises: why do Yanomamö men risk killing each other? An extensive, long-term study by the anthropologist Chagnon (1988) has revealed that those who kill and survive end up having more wives and babies. The study compared 137 Unokais (men who had killed at least one other man) and 243 non-Unokais in the tribe. The Unokais had on average 1.63 wives and 4.91 children, while the non-Unokais averaged only 0.63 wives and 1.59 children.

Therefore taking risks can obviously pay dividends. However, it appears that there is no particular strategy regarding risk that is optimal for all individuals within a population in all situations. We have already seen in the Scholer et al. (2010) study discussed in the section on loss-aversion that some people display a motivational system that appears to be prevention-focused, while other people display a motivational system that appears to be promotion-focused. Prevention-focused people are loss-averse, but may display either risk-aversion or risk-seeking, depending on the potential for restoring a particular status quo. Thus attitudes toward risk depend both on an individual’s motivational focus and on the particular environmental situation with which they are faced.

Neuroscientific foundation

One conclusion that can be drawn from the above discussion is that our brains require mechanisms built into them over the course of evolution that provide rewards for taking risks. These mechanisms are biochemical in nature, and the most important one involves the neurotransmitter dopamine. Evidence suggests that some people have a variation of the dopamine D4 receptor gene, sometimes referred to as the ‘novelty seeking’ gene. This variation can lead to a number of variations in behavior patterns (Benjamin et al., 1996). Studies have shown how such variations affect migration (Chen et al., 1999) and sexual behavior (Hamer, 1998). A discussion of how far such a gene might spread throughout a population is deferred until the chapter on behavioral game theory, since it involves an explanation of the concept of an evolutionarily stable strategy (ESS). At this stage it just needs to be noted that such a gene bestows certain advantages on its possessor (the tendency to take more opportunities), but also certain disadvantages (the tendency to come to grief when taking such opportunities). This also helps to explain why different individuals have a different motivational focus and different attitudes to risk.

As far as probability weighting is concerned, studies attempting to identify neural correlates of distortions in this characteristic have a fairly recent origin. Paulus and Frank (2006) estimated a nonlinear probability weighting function using a gamble-certainty-equivalent paradigm. They reported that activity in the ACC was correlated with the nonlinearity parameter, with subjects who had more ACC activity for high versus low prospects tending also to have more linear or objective weighting of probabilities.

Another study by Hsu, Zhao and Camerer (2008) has also estimated nonlinearities in probability weighting, showing a significant correlation between behavioral non-linearity in gambling tasks and nonlinearity of striatal response. A study by Berns et al. (2007) examined probability-weighting distortions for aversive outcomes, such as the prospect of receiving an electric shock. These researchers reported that there was fairly wide-scale overweighting of low-probability aversive events recorded in a number of brain regions, including dorsal striatum, PFC, insula and ACC.

Again these studies raise the issue of cause and effect, as is so often the case with identifying neural correlates of behavior. However, as mentioned earlier, future studies involving subjects with brain lesions may clarify the position, if subjects with lesions in certain areas tend to have linear probability weighting.

Empirical evidence

Some of the anomalies in EUT have already been discussed in the previous sections. However, some further explanation of these phenomena is necessary, and a number of others can also be discussed. In particular we are now in a position to show that PT is in general better at explaining various real-life phenomena than EUT, or any of the conventional models extending EUT. As Camerer (2000) has observed in a review and comparison of PT with other theories, PT, and in particular cumulative PT, can explain not only the same observations that EUT can explain, but also the various anomalies that cannot be explained by EUT.

Let us begin by examining empirical evidence supporting the inverted S-shaped weighting functions for both gains and losses. Tversky and Kahneman (TK: 1992) performed a study with graduate students to reveal their preferences in terms of certainty equivalents (CEs) for a number of prospects. Table 5.5 gives a sample of results that were observed, showing the expected values (EVs), median CEs, and attitudes to risk for each prospect.

Thus with the first prospect the subjects were prepared to pay an average of $78 dollars to obtain an expected value of $95, showing risk-aversion. In general, if the CE > EV, this indicates risk-seeking for both gains and losses, while if the CE < EV, this indicates risk-aversion for both losses and gains.

These data can be transformed in order to draw a decision-weighting function. We do not have to take into account diminishing marginal sensitivity as far as the utility function is concerned in this case, since all the amounts of money involved are the same in this example, $100. Thus attitude to risk is affected only by diminishing marginal sensitivity in the decision-weighting function. The plotting of this function requires the calculation of the ratios of the certainty equivalents of each prospect (c) to the nonzero outcome (x). Thus for the first prospect in Table 5.5 this ratio c/x = 0.78. One can interpret this ratio as a subjective or weighted probability, in that a subject may know the objective probability of gaining $100 is 0.95, but because of risk-aversion they really perceive the probability as 0.78 in terms of decision-making. From the small sample of results, a decision-weighting function is drawn, in this case for gains, shown in Figure 5.8. It should be noted that the 45° line, where c/x = p, represents risk-neutrality; points above the line occur where c/x > p, implying risk-seeking, while points below the line occur where c/x < p, implying aversion. In the case of decision-weighting functions for losses, the situation is reversed: points above the line represent risk-aversion and points below the line represent risk-seeking. It should also be noted that the curve is not symmetrical, since w(0.5) = 0.36.

Table 5.5 Empirical results related to weighting function

Figure 5.8 Empirical decision-weighting function for TK data

Figure 5.8 relates only to a very small sample of observations, for the sake of simplicity. However, when applied to a much larger sample of the TK observations, the general shape of the function fits well, for both gains and losses. The graph confirms the predictions of PT in terms of risk-seeking for gains of low probability and risk-aversion for gains of high probability, with this pattern being reversed for losses. In the TK sample as a whole, 78% of the subjects were risk-seeking for gains of low probability, while 88% were risk-averse for gains of high probability; 80% were risk-averse for losses of low probability, while 87% were risk-seeking for losses of high probability.

The empirical observations of the 1992 TK study have been replicated in a number of other studies. A particularly notable study was carried out by Kachelmeier and Shehata in 1992, and was one factor leading to a revision of the original version of PT. This study was carried out in China, and due to the prevailing economic conditions there, the investigators were able to offer substantial rewards of up to three times normal monthly income. The main finding was that there was a marked overweighting of low probabilities; this resulted in pronounced risk-seeking for gains. In the highest payoff condition described above, and with a probability of winning of 0.05, certainty equivalents were on average three times larger than expected values.

Many other studies, in a variety of different fields, provide general support for cumulative PT when it is compared with EUT. Insurance is a good example of a field where appropriate comparisons can be made. According to EUT, people buy insurance at a cost greater than the expected monetary loss because they have a utility function that is concave throughout, making them risk-averse in the domain of losses. Thus they dislike large losses disproportionately compared to the small losses of paying insurance premiums. The problem with this explanation is that in many situations people do not buy insurance voluntarily. For example, car insurance is compulsory by law in many countries or states (and even then many people drive uninsured). This failure to buy insurance is easier to reconcile with PT, where the utility function is convex for losses. Furthermore, cumulative PT can also explain why people do sometimes buy insurance in terms of the overweighting of low probabilities, or risk-aversion for low probabilities of loss, rather than in terms of the large disutility of loss, as with EUT.

One aspect of insurance provides crucial evidence regarding the validity of the two theories, and this concerns probabilistic insurance, described earlier. According to EUT, if there is a small probability, r, that the policy will not pay out, then a person should be prepared to pay approximately (1 r) times as much as the full premium for probabilistic insurance. For example, if there is a 1% chance that the claim will not be paid in the event of an accident, then a person should be prepared to pay about 99% of the full premium. However, empirical evidence indicates that people have a strong dislike of probabilistic insurance. A study by Wakker, Thaler and Tversky (1997) showed that people were only willing to pay 80% of the full premium in the situation above. On the other hand, cumulative PT can explain this once again in terms of the overweighting of low probabilities; since probabilistic insurance does not reduce the possibility of loss to zero, such a prospect is unappealing. Thus, in the words of Camerer (2000):

Prospect theory can therefore explain why people buy full insurance and why they do not buy probabilistic insurance. Expected utility cannot do both.

Various aspects of playing the lottery have already been discussed. There is one further phenomenon here that adds support to the PT model. People like big prizes disproportionately. Although EUT can explain this with the additional assumption of a convex utility function, the overweighting of low probabilities in PT can also be relevant. Larger states or countries tend to find that their lotteries are more appealing and more widely played. One might expect that the larger prizes offered would be offset by the lower probability of winning, but this appears not to be the case.

As far as gambling on horse racing is concerned, there is further interesting evidence available. There is a significant bias toward betting on longshots rather than on favorites. This bias can be measured in terms of the proportion of total money bet on longshots and comparing this with the proportion of times such horses actually win. Studies by Thaler and Ziemba (1988) and Hausch and Ziemba (1995) indicate that longshot horses with 2% of money bet on them only win about 1% of the time. The overweighing of low probabilities appears to be a relevant factor here. However, if one takes account of the study by Jullien and Salanié (1997) mentioned earlier, the convexity of the utility function for some amounts of gain may also be relevant.

There is another side to the story above, and that is the aversion towards betting on heavy favorites. In this case the Jullien and Salanié (1997) study found that there was a highly nonlinear weighting function for losses, causing probabilities of losses to be strongly overweighted. For example, it was found that π(0.1) 0.45 and π(0.3) 0.65. Thus it seems that, while people like to gamble, they are disproportionately afraid of the small possibility of losing when they bet on a heavy favorite.

There is one final aspect to this racetrack betting situation which has broader implications, and this is the ‘gambler’s fallacy’, sometimes referred to as the ‘the law of averages’, which was discussed in the previous chapter. Longshots are frequently horses that have lost several races in a row. Gamblers often believe that they are, therefore, ‘due’ for a win in these circumstances. The phenomenon is the same as the situation where a coin comes up heads on several consecutive tosses and people believe that there is then a greater chance that the coin will come up tails on the next toss.

There is some recent evidence regarding gender bias as far as decision-weighting is concerned. Fehr-Duda et al. (2011) report that women who are in a good mood tend to be more optimistic than men in terms of overestimating low probabilities. It appears from this study that mood has no such influence on probability estimation by men. However, we have seen in the previous chapter that probability estimation can be influenced by the phenomenon of visceral fit, which is obviously related to mood. Again it appears that more research needs to be done in this area, to examine not only how different visceral factors affect probability estimation, but how these different factors may have different effects based on gender.

While this subsection reviews some of the evidence supporting PT in terms of probability weighting, Section 5.8 will provide a more extensive review of empirical evidence in general terms, comparing PT with EUT and other conventional theories.

5.8 Criticisms of prospect theory

Prospect theory has now been around for over three decades. During that time, not surprisingly considering its non-conventional nature and radical implications, it has attracted numerous criticisms. It is important also to distinguish between original prospect theory (OPT) and cumulative prospect theory (CPT), since different criticisms may apply in each case. Some of the criticisms have been theoretical, and some empirical. In the first case it has been claimed that the theory contains contradictions, is incomplete, or lacks clarification, whereas in the second case the claims are that empirical data violate the assumptions of the theory and that the theory makes incorrect predictions. It is not possible to examine all the criticisms of prospect theory in both its forms here, so our approach is to focus on the ones that have received the most attention in the literature.

In terms of theory we will discuss four main criticisms here: (1) the lack of normative status; (2) internal contradictions; (3) incompleteness; (4) and the determination of reference points. Empirical criticisms relate to: violations of the combination principle; violations of stochastic dominance; the failure to explain the Allais paradoxes; violations of gain–loss separability; the nature of the utility function; endowment effects; the discovered preference hypothesis and misconceptions; and the nature of framing effects. Many of these empirical criticisms have come from various studies by Birnbaum (2008); he summarizes 11 paradoxes where ‘prospect theories lead to self-contradiction or systematic false predictions’. It is not appropriate to examine all of these in detail here, but we shall discuss the most fundamental ones.

Once these criticisms have been discussed, it is then possible to consider alternative decision models and come to certain conclusions regarding the current state of decision theory.

Lack of normative status

The most fundamental, although not necessarily the most serious, criticism relates to the normative aspects of the theory. Kahneman and Tversky propose PT as a descriptive rather than a normative theory. The authors treat the EUT model as the normative model to be used as a benchmark, but in rejecting its usefulness as a descriptive model, they do not propose any norm or norms to replace it. While the editing phase of the model adds explanatory power to the model in the descriptive sense, by introducing elements of bounded rationality and decision heuristics it not only makes the model less parsimonious, but it also makes it indeterminate. Thus the model loses the simplicity and tractability of EUT and some conventional models which optimize a single variable.

Let us now consider the nature of this indeterminacy. In general terms this is caused by the fact that, like other procedural models, there are features of the model that are underdetermined by the theory, such as the order in which certain operations are performed in the editing phase, the location of reference points, and the shape of the probability weighting function. The issue of the determination of reference points is discussed later in the section. As far as the weighting function is concerned, the cumulative function in (5.9) is unlikely to be accurate in detail. In the words of the 1992 study:

We suspect that decision weights may be sensitive to the formulation of the prospects, as well as to the number, the spacing and the level of outcomes. In particular, there is some evidence to suggest that the curvature of the weighting function is more pronounced when the outcomes are widely spaced (p. 317).

The evidence referred to here comes from Camerer (1992).

However, the main problem related to normative status concerns violations of both monotonicity and transitivity. Examples of these are given later in the section. Many economists would question whether it is even possible to have theories of ‘preference’ that violate transitivity. Certainly it becomes more difficult to talk about people maximizing any preference function, although not impossible as we will see. We shall also see that most recent theories of decision-making adopt the same descriptive rather than normative approach, in order to account for the violations of monotonicity and transitivity which have been widely observed.

Internal contradictions

These criticisms come in particular from Birnbaum (2008), who also reports many studies that are claimed to violate prospect theory empirically which will be discussed later. Birnbaum draws attention to the editing rules, which he states are ‘imprecise, contradictory, and conflict with the equations of prospect theory. This means that OPT often has the misfortune (or luxury) of predicting opposite results, depending on what principles are invoked or the order in which they are applied’(p. 468). The reason why Birnbaum uses the term ‘luxury’ in the above context is that the contradictions mean that prospect theory can explain most empirical results ex post simply by applying the editing rules selectively. Unfortunately, the downside of this characteristic is that the theory is claimed to be difficult to use for prediction purposes on an ex ante basis. This has unfortunate consequences also as far as the explanation of the Allais paradoxes is concerned, as we shall see. To illustrate the problem, consider the example in Table 5.6.

Table 5.6 Combination and cancellation

Each gamble is represented by an urn containing 100 marbles that are identical except for their color. The urn for gamble A contains one red and one blue marble, each of which pays $100, and it has 98 white marbles that pay nothing. Urn B contains one red marble paying $100, two green marbles that pay $45, and 97 white marbles that pay nothing. A marble will be drawn at random from whichever urn the participant chooses, the color of the marble determining the prize.

In the initial choice gambles (choice 1), a person has to choose between A and B; there are three branches, or outcomes, in each case. A branch corresponds to a probability-consequence event, but in gamble A it can be seen that the first two branches are identical in this respect. Because of this feature (actually it is sufficient for them to have the same consequence or payoff), the combination principle results in gamble A being represented as A′, which would then be compared with B. Thus the combination principle results in the choice being represented as choice 2. However, it can also be seen that the first branch of A and B is identical, therefore according to the cancellation principle the choice between A and B can be reduced to a choice between A″ and B′, as shown in choice 3. People may not have the same attitudes towards choices 2 and 3, resulting in preference reversals and inconsistent results.

Incompleteness

OPT is often criticized as being incomplete because it only applies to gambles with no more than two nonzero consequences. CPT is more general than OPT for several reasons: (1) it applies to gambles with more than two nonzero consequences; (2) it removes the need for the editing rules of combination and dominance detection, which are automatically guaranteed by the representation of CPT; and (3) it allows different weighting functions for positive and negative outcomes. Another possible aspect of incompleteness of both forms of PT concerns reference points, discussed next.

The determination of reference points

Some economists view it as a weakness of PT that reference points are not determined endogenously. The determination of reference points is necessary in order to estimate the incidence and effects of loss-aversion; a good example of this problem relates to the endowment effect, discussed in Case 5.1. If subjects are given something for free in an experiment they are likely to value this object differently than if they had ‘earned’ it in some way (Cherry, Frykblom and Shogren, 2002). In practice, however, it is difficult to construct an experimental design where subjects can experience real losses without destroying the incentive to take part. The Cherry et al. study overcame this problem by allowing subjects to ‘earn’ wealth by taking part in a quiz and answering questions correctly. This study is discussed in more detail in Chapter 10 in the context of fairness games.

KT, and other supporters of PT, generally use either the existing situation as a reference point, or some anticipated or expected situation. However, more precision regarding the determination of reference points would be an aid to constructing better models of behavior. Certainly more research into how reference points are determined in different situations would be valuable, and this necessitates a detailed theory relating to how people form and adjust expectations. For example, it would be useful to know if different types of saver or investor have the same reference points. Furthermore, an understanding of the process of the dynamic adjustment of reference points over time would aid the analysis of various psychological phenomena; an example is the ‘writing off’ of sunk costs, discussed in more detail in the next chapter.

Violations of the combination principle

CPT (and many other decision theories like RDU) satisfy coalescing and transitivity, and therefore cannot explain ‘event-splitting’ effects. Starmer and Sugden (1993) found that preferences depend on how branches are split or coalesced (combined). Birnbaum (1999, 2004, 2007) and Humphrey (1998, 2000, 2001a, 2001b) have reported widespread and robust findings of event-splitting effects. In order to gain an understanding of this phenomenon and how it contradicts both OPT and CPT we need to give an example; the one in Table 5.7 below comes from Birnbaum (2004), and presents subjects with two choices, the first between A and B, and the second between A′ and B′. As in Table 5.6, in each choice a marble is drawn randomly from an urn, with the color of the marble determining the prize. The subjects have to choose which urn to draw from.

Table 5.7 Violations of the combination principle

It can be seen that A′is the same as A, with the last two outcomes combined. Likewise, B′ is the same as B, with the first two outcomes combined. Thus, if a person obeys the combination principle, then they should make the same choice between A and B as between A′ and B′. However, Birnbaum (2004) reported that 63% of subjects chose B over A, and 80% chose A′ over B′, a highly significant result.

Violations of stochastic dominance

CPT and similar theories must satisfy stochastic dominance, as explained earlier. It seems intuitive that people would not choose an option that was obviously stochastically dominated by another option. In order to understand the issue better we will repeat the example given earlier in the chapter from Tversky and Kahneman (1986, p. 242):

It is clear in this example that Option B is preferable to, or dominates, Option A, as we saw earlier. However, there are situations where stochastic dominance is not so obvious. Consider the following example, from the same study by Tversky and Kahneman (1986, p. 242), which is a slightly modified version of the above problem.

In this version Option C is basically the same as Option A, but combines blue and yellow marbles into the same category because they both result in a loss of $15. Similarly, Option D is basically the same as Option B, but combines red and green marbles, since they both result in a win of $45. However, the framing of the options makes it more difficult to detect the dominance of D over C. Kahneman and Tversky found that 58% of subjects preferred the dominated option C. Thus, although the authors of CPT were aware of such a violation, CPT itself is not able to explain it.

Birnbaum and Navarrete (1998, p. 61) also tested for violations of stochastic dominance by asking subjects to compare the following gambles:

It should be noted that the representation of the gambles is in a different form from the PT representation, since the focus of Birnbaum’s theory is on trees with branches, rather than on prospects. This is explained in the next section in relation to configural weights models. It is easier to see which of the gambles A and B is dominant by showing a ‘root gamble’, G, which involves 90 red marbles winning $96 and 10 white marbles winning $12. We can now use the principles of combination, cancellation and transitivity to determine dominance. A is dominant over G, and G is dominant over B, therefore A is dominant over B. However, Birnbaum and Navarrete (1998) found that 73% of their student subjects violated dominance in this example and chose B. This result was confirmed in various other similar choices in this study, and was also repeated in Birnbaum, Patton and Lott (1999), with again 73% of another undergraduate student sample violating dominance in various similar problems. Birnbaum (2006) claims that by 2006 he has completed 41 studies with 11,405 participants testing stochastic dominance in various formats, and reports that violation is a very robust finding.

Failure to explain the Allais paradoxes

Different theories explain the Allais paradoxes in different ways. We have already seen that PT was originally constructed in order to explain these paradoxes in terms of editing principles. In order to test the implications of PT, Birnbaum (2007) devised some tests which dissected the Allais paradoxes, by presenting them in gambles of different forms, some using the combination principle, some using the cancellation principle, as illustrated earlier in Table 5.6. Based on empirical data from 200 participants, Birnbaum concludes: ‘neither OPT nor CPT with or without their editing principles of cancellation and combination can account for the dissection of the Allais paradoxes’ (Birnbaum, 2008).

The nature of the utility function

Another criticism, relating to the utility function, has been raised by Levy and Levy (2002), who claimed to have found evidence contradicting an aspect of PT in a series of experiments which compared the PT and Markowitz utility functions. The Levy and Levy (LL) study argued that the original KT data did not provide a reliable indicator of the shape of utility functions because it always asked subjects to compare prospects that were both either positive or negative. In reality, the LL study claimed, most prospects are mixed, involving situations where there is a possibility of either gain or loss, for example in investing on the stock market. Their study included a number of experiments, asking respondents to choose between such mixed prospects. The main objective was to test whether the data supported the PT model, with the S-shaped utility function, or the M model, with the reversed S-shaped function throughout most of the range. The study used a total of 260 subjects, consisting of business students and faculty from a number of institutions, along with a number of professional practitioners. One of the tasks in the experiments will serve as an example of the methodology. The subjects were asked to consider that they had invested $10,000 in stock and were evaluating the possible returns, choosing between the following two mixed prospects:

Prospect F: (–3000, 0.5; 4500, 0.5) Prospect G: (–6000, 0.25; 3000, 0.75)

Both prospects are not only mixed, involving the possibility of either gain or loss, but their expected values are the same, 750, and the same for both the gain and loss components, –1500 and 2250. According to the PT model, people are risk-averse in the domain of gains; therefore, they should prefer the gain of 3000 with a probability of 0.75 in prospect G to a gain of 4500 with the probability of 0.5 in prospect F. Similarly, the PT model proposes that people are risk-seeking in the domain of losses; thus they should prefer a loss of 6000 with a probability of 0.25 in prospect G to a loss of 3000 with a probability of 0.5 in prospect F. Therefore prospect G is dominant over prospect F according to the PT model, while the situation is reversed according the M model. The LL study found that 71% of their subjects preferred F while only 27% preferred G. They interpreted this finding as showing strong evidence against the PT model, and, combined with the results of other tasks in their experiments, they concluded that the M model was better supported.

However, the LL study has been criticized in a paper by Wakker (2003). Wakker claims that the data of the LL study can still be used to support the PT model, since the LL study ignored the element of decision-weighting. The LL study justified this on the grounds that the probabilities involved in their experiments were always at least 0.25 in magnitude, and that probabilities in this range should involve a linear weighting function. Wakker disputes this, claiming that nonlinearities in this range can have a significant distorting effect, enough to make the results compatible with PT. When we consider the finding of Jullien and Salanié (1997) reported above, with losses involving π(0.3) = 0.65, Wakker’s conclusion seems to have some justification.

Violations of gain–loss separability

The gambles described in Tables 5.6 and 5.7 have been positive gambles. Mixed gambles, termed regular prospects in PT, were described earlier, and are evaluated according to expression (5.9), meaning that the gains and losses are valued separately and then added together. This assumes the feature of gain–loss separability. In simple (non-mathematical) terms this means that if you prefer the good part of B to the good part of A, and if you prefer the bad part of B to the bad part of A, then you should prefer B to A. Various empirical studies by Wu and Markle (2008), and Birnbaum and Bahra (2007) have contradicted this feature. Wu and Markle (2008) report a reversal between preferences for mixed gamble and the associated gain and loss gambles such that mixed gamble A is preferred to mixed gamble B, but the gain and loss portions of B are preferred to the gain and loss portions of A. The implication of this is that the argument given in PT for the kinked utility function described in (5.12) and illustrated in Figure 5.5 is false. This does not necessarily mean that the utility function is not kinked, but the existence of a kink must rest on a different premise from that given in both OPT and CPT.

The discovered preference hypothesis and misconceptions

There are some studies that combine a number of objections. The discovered preference hypothesis (DPH) developed by Plott (1996) proposes that people’s preferences are not necessarily revealed in their decisions. They have to be discovered through a process of information gathering, deliberation and trial-and-error learning. Subjects must, therefore, have adequate opportunities and incentives for discovery, and it is claimed that studies lacking these factors are unreliable. Plott argues that most studies that support the endowment effect are in this category, lacking the necessary elements of experimental design that ensure reliability. Binmore (1999) makes a similar claim.

Plott and Zeiler (2005, 2007) pursue this issue further, by performing experiments to test whether subject misconceptions, rather than PT preferences, can account for the gap between willingness to pay (WTP) and willingness to accept (WTA) that PT refers to as the endowment effect. It should be noted that the methodology in this study was different from that in List (2004) and many other studies, because it did not focus on willingness to trade as such, but on the concepts of WTA and WTP, which entail various problems in terms of analysis. Plott and Zeiler (PZ hereafter) draw attention in particular to the concept of ‘subject misconceptions’, and point out that this is not operationally defined or quantified. It is in effect a compound effect of several effects that PZ identify: misunderstanding an optimal response; learning effects; lack of subjects’ attention because of inadequate incentives to give an optimal response; and giving a strategic response. Since these problems apply to much research in experimental economics it is worth giving some explanation regarding each of them and discussing the PZ approach to solving these problems. This approach incorporates the following four elements.

1 Use of an incentive compatible elicitation device

When subjects are asked to state a WTP or WTP they are essentially in a kind of auction scenario, and this is not the same buying/selling scenario as the normal marketplace. This lack of familiarity may cause subjects to misunderstand how to give an optimal response. Therefore an important principle in the PZ study was to use an incentive compatible elicitation device. A common technique used in experimental economics in such situations to elicit valid responses is the Becker– DeGroot–Marschand (BDM) mechanism. This mechanism pits each buyer and seller against a random bid, which determines the price paid by the buyer and that received by the seller. All sellers stating bids lower than the random bid sell the good, and all buyers stating bids higher than the random bid buy the good. Sellers bidding higher than the random bid and buyers bidding lower do not transact. The purpose of this mechanism is to elicit bids that reflect the true value to each party. The optimal response is to state a bid equal to the subject’s true value.

2 Training

Since it is not obvious to subjects that stating one’s true value is an optimal response, especially given that the random bid is determined by a lottery, PZ took time in their study to fully explain the mechanism using numerical examples. An illustration here will clarify the situation. Say a seller’s true value is $6, but they overbid, stating $7, maybe under the misapprehension that this may cause the buyer to bid higher, as in many real-life situations. This is an example of a strategic response, where one party to a transaction takes into account the behavior and reactions of the other party. If the random bid is $6.50 they will not transact, and there is an opportunity cost of $0.50, because they are forgoing a transaction that would give them a consumer surplus of $0.50. Now take the situation where a buyer’s true value is $6. They may underbid, stating $5, maybe under the misapprehension that this may cause the seller to come down in price, again as in real-life situations. If the random bid is $5.50, they will not transact, and will again forgo $0.50. The training was therefore designed to ensure that subjects understood the nature of the BDM mechanism, and therefore stated bids that represented their true values.

3 Practice rounds

This procedure allows subjects to ‘learn though using the mechanism while still educating themselves about its properties’. Subjects can also ask questions and the experimenter can check that subjects are understanding the nature of the task.

4 Anonymity

Anonymity in decisions and payouts is important because otherwise subjects may again be inclined to make strategic responses, either to impress other subjects or to impress the experimenter.

PZ (2005) found that, while they could replicate the WTA–WTP gap in the study of Kahneman, Knetsch and Thaler (1990) using an experimental procedure lacking in controls, when they implemented the full set of controls described above the gap was not observed. PZ concluded that this ability to ‘turn the gap on and off’ constituted a rejection of the PT interpretation of the gap as being an endowment effect, in favor of the theory of subjects’ misconceptions as being the cause.

The PZ study is certainly a valuable and informative one in many ways, but its conclusion has one main weakness. This is that the methodology is ‘all-or-nothing’, in the sense that either there is very little experimental control, or various controls are combined together. The result is that a number of effects are confounded together in the ‘subjects’ misconceptions’ category. Plott and Zeiler admit this, giving five possible interpretations of these misconceptions (some of which they reject) in their conclusion. In particular, in the summary of their 2007 paper, they explicitly state that their objections to endowment effect theory do not challenge prospect theory in general. However, like Birnbaum, they claim that a kink in the utility function is not necessary to explain empirical findings, in this case WTA–WTP asymmetries.

Further research is needed, using various degrees of control in the experimental design, to establish whether the switching off of the WTA–WTP gap is mainly due to misunderstanding the optimal response, learning effects, giving some kind of strategic response, or misinterpreting the intentions of the experiment or experimenter. It may be that learning effects, through practice rounds, could be the main factor; this would support the findings of List (2004) reported earlier.

It has been argued that the best type of experimental design to ensure that the requirements of the DPH are met is a single-task individual-choice design (Cubitt, Starmer and Sugden, 2001). Such a design can ensure that subjects get an opportunity to practice a single task repeatedly, with the requisite learning effect, and it can also ensure simplicity and transparency, which are difficult to achieve in market-based studies, where tasks are more complex and involve interactions with others. However, when Cubitt, Starmer and Sugden reviewed the results of nine different experiments involving such a design, they found that the results still violated the independence axiom for consistent choices in Allais-type situations, discussed earlier. Another study by Loomes, Starmer and Sugden (2003) also questioned the interpretation of the disappearance of the WTA–WTP gap under market experience. These researchers note that:

… even after repeated trading, individuals’ valuations of given lotteries remain subject to a high degree of stochastic variation, arguably reflecting many subjects’ continuing uncertainty about what these lotteries are really worth to them (p. 166).

Having mentioned these results and conclusions, we can now consider an associated problem with the PZ conclusion. If subjects did not have a clear understanding of the experiment they may have been inclined to state certain values which did not reflect their true values, perhaps still giving a strategic response. Although Plott and Zeiler think this unlikely, it would mean that their results do not provide evidence rejecting the occurrence of the endowment effect in the real world outside the laboratory. Recent field studies confirm the existence of the endowment effect under most real-world conditions, although they also report that the economic environment does play a role in affecting people’s perceived reference states and consequently on their valuations (Köszegi and Rabin, 2006; Knetsch and Wong, 2009).

The nature of framing effects

Inconsistent results have been reported as far as the ability of PT to explain framing effects. Different types of framing effect have been demonstrated by Levin, Schneider and Gaeth (1998): standard risky choice, attribute framing and goal framing. This study claimed that PT probably best explains the first type of effect, but not the other two. It also doubted that PT could interpret the empirical evidence of risky choices in different contexts. A similar conclusion was reached by Wang and Johnston (1995), who indicated that framing effects are context-dependent, rather than being a generalized phenomenon. Other evidence suggests that a framing effect depends on task, content and context variables inherent in the choice problem (Wang, 1996; Fagley and Miller, 1997).

A further body of research criticizes the original approach of Tversky and Kahneman (1981) and their illustration of framing effects in a situation sometimes referred to as the ‘Asian disease’ problem. People are informed about a disease that threatens 600 citizens and asked to choose between two undesirable options (Tversky and Kahneman, 1981). In the ‘positive frame’ people are given a choice between (A) saving 200 lives with certainty, or (B) a one-third chance of saving all 600 with a two-thirds chance of saving nobody. Most people prefer A to B here. In the ‘negative frame’ people are asked to choose between (C) 400 people dying with certainty, or (D) a two-thirds chance of 600 dying and a one-third chance of nobody dying. In this case most people prefer D to C, in spite of the fact that A and C are identical results or ‘prospects’ and B and D are identical results. As well as illustrating framing effects and preference reversal, this example also illustrates loss-aversion (saving lives is seen as a gain, while dying is seen as a loss).

Several studies have argued that this approach actually confounded two different effects: a framing effect and a reflection effect (Arkes, 1991; Kühberger, 1995; Levin, Schneider and Gaeth, 1998; Chang, Yen and Duh, 2002). This distinction now needs to be explained in some detail in order to understand the implications.

A framing effect depends on whether the problem is framed in a positive or negative frame, which depends on the negation ‘not’. A reflection effect depends on the domain of the problem, meaning whether it relates to a gain or loss. To illustrate this difference, statement A: ‘200 people will be saved’ represents both a positive frame and a positive domain, whereas statement C: ‘400 people will die’ involves both a negative frame and a negative domain. It is therefore argued that, because frame and domain correlate perfectly in the TK treatment of the Asian disease problem, it is impossible to disentangle the framing and reflection effects. On the other hand, it can be claimed that the statement, ‘400 people will not be saved’, although identical in meaning with statement A, involves a negative frame but a positive domain. Similarly, the statement, ‘200 people will not die’ is identical in meaning with statement C, but involves a positive frame with a negative domain. Thus, by restating A and C, it is possible to test prospect theory against other theories as far as explaining framing effects is concerned. This aspect is discussed in the next section.

5.9 Recent theories and conclusions

It should not be inferred from the discussion above that these have been the only criticisms of prospect theory; we have concentrated on these issues since they have attracted the most discussion in the literature. In response to the various theoretical and empirical issues and problems raised, a number of new theories or models of decision-making under risk and uncertainty have been proposed over the last two decades, some even predating CPT, which have claimed to surmount these problems and explain empirical findings in a more satisfactory way. It is impossible to survey all these models here, but six main ones will be considered: (1) third-generation prospect theory; (2) probabilistic mental models (PMM); (3) fuzzy trace theory (FTT); (4) the priority heuristic; (5) imprecision theory; and (6) configural weights models.

Third-generation prospect theory

It makes sense to discuss this first, since it builds on the elements of both the first (1979) and second (1992) versions of prospect theory. Schmidt, Starmer and Sugden (2008) proposed third-generation prospect theory (PT³) as a model that extends the predictions of prospect theory in an area in which the theory had previously been silent, the situation where reference points are uncertain. Therefore PT³ can be applied to situations where decision-makers are endowed with lotteries and have the opportunity to sell or exchange them, for example, where they buy insurance or sell stocks, unlike the original versions of prospect theory. Furthermore, PT³ was intended to explain two commonly observed anomalies in EUT, (1) discrepancies between WTA and WTP valuations of lotteries; and (2) preference reversals in gambles involving what are referred to as P-bets and $-bets. These concepts were originally developed by MacCrimmon and Smith (1986).

Some detailed explanation here will aid an understanding of both methodology and the relationship between uncertainty, preference reversals and transitivity. A P-bet offers a relatively large probability of a modest sum of money and a residual probability of zero. A $-bet offers a smaller probability of a considerably bigger prize and a larger chance of zero. Respondents are asked to place certainty equivalents on each bet and also to make a straight choice between the two bets. A common preference reversal observed here is that people place a higher money value on the $-bet, but choose the P-bet. As an example, consider the following two prospects:

$-bet: A (0.1, $140) certainty equivalent of $14

P-bet: B (0.8, $15) certainty equivalent of $12

A has the higher certainty equivalent, but people often still prefer B, the P-bet. If we now refer to prospect C as a sure bet of $13 (or any sum between $12 and $14), this would result in the following ordering of preferences:

which violates transitivity.

In order to generalize prospect theory to apply to situations with uncertain reference points, Schmidt, Starmer and Sugden (2008) propose two components: (1) a definition of ‘gain’ or ‘loss’ relative to stochastic reference points, known as reference acts; and (2) a rank-dependent method of assigning decision weights to any act f, relative to any reference act h.

In its most general form PT³ incorporates the approach of Sugden (2003) in terms of using reference-dependent subjective expected utility (RDSEU), and proposes a value function of the form:

V (f, h) = Σ_i v(f[s_i ], h[s_i]) W(s_i ; f, h)

Where W(s_i ; f, h) is the decision weight assigned to state s_i when f is being evaluated from h. PT³ also uses parameterizations relating to utility functions and probability weighting functions whose validity has already been established by previous versions of prospect theory. Schmidt, Starmer and Sugden claim three main advantages of this modeling approach:

1 Generality – it is more general than previous versions of prospect theory, applying to three key aspects of preferences: attitudes to consequences, attitudes to probability, and attitudes to gain and loss.

2 Parsimony – the model is as simple as possible given its application above; only one parameter is involved for each of the three aspects.

3 Congruence with reality – when the model is applied to existing evidence it explains both the anomalies mentioned earlier, in that it predicts both WTA and WTP discrepancies for lottery valuations and the preference reversal with P and $ bets.

Regarding the last advantage, the explanation of preference reversal is significant, in that it is based on ‘the interaction of empirically plausible degrees of loss-aversion, diminishing sensitivity and probability weighting’ (p. 221) rather than being caused by the violation of procedural invariance, which had been the interpretation of many psychologists. Applying these three different factors to the example above, this means in simple terms that people value the $-bet more than the P-bet because of overweighting of low probabilities, they value the sure bet more than the P-bet because of loss-aversion, and they prefer the P-bet to the $-bet because of diminishing marginal sensitivity.

Schmidt, Starmer and Sugden do not propose that PT³ accounts for all preference reversals of the P and $ bet type; they acknowledge that violations of procedural invariance are important, in that different methods of eliciting information can influence decisions, but they claim that, given its psychological plausibility and the way their model fits the evidence, PT³ does play a significant role in the explanation of preference reversal.

Probabilistic mental models

According to the theory of probabilistic mental models (PMM) (Gigerenzer, Hoffrage and Kleinbolting, 1991), people first attempt to construct a local mental model (LMM) of the task given to them, and then utilize it to solve the problem using long-term memory and elementary logical operations. If, as in any complex problem, this process is not possible, then a PMM is constructed using probabilistic information generated from long-term memory. Thus PMM theory suggests that a decision-maker solves a problem by applying inductive inference, meaning that they put the specific decision task into a larger context. The theory explains framing effects in terms of the inferences people make when presented with incomplete information.

We can compare PMM with PT by examining how each views the Asian disease problem. Prospect theory explains the Asian disease problem by using different reference points for different comparisons. Statements A and B are worded in terms of people being saved, both involving a perceived positive domain (the actual domain is negative since people are still dying); thus in the domain of gains people are risk-averse and prefer option A to B. On the other hand, statements C and D are expressed in terms of people dying, involving a negative perceived domain, or losses. In this domain people are risk-seeking, and therefore prefer option D to C.

By contrast, according to PMM, when people edit the statement ‘200 people will be saved’ they may infer that maybe over time more than 200 will be saved. On the other hand, the statement ‘400 people will die’ may be edited so that it is inferred that maybe more than 400 people will eventually die. Thus when statements A and C are expressed differently, with negative frame and positive domain and vice versa, it is possible to test PMM theory against PT. For example, when asked to compare A': 400 people will not be saved’ with B: 1/3 chance that 600 will be saved, and 2/3 chance that 0 will be saved, PT predicts that A' will be favored, while PMM theory predicts that B will be preferred, interpreting A' to mean that maybe more than 400 people will not be saved.

This is the approach taken by Chang, Yen and Duh (2002), who test prospect theory against two competing models: probabilistic mental models and fuzzy-trace theory (Reyna and Brainerd, 1991), discussed next.

Fuzzy-trace theory

Fuzzy-trace theory (FTT) proposes that people prefer to reason using simplified representations of information, i.e. the gist of the information, rather than using exact details. For example, both numerical outcomes and probabilities are represented dichotomously; this means that the Asian disease options can be simplified as follows:

Statement A:	Some people will be saved.
Statement B:	Some people will be saved or nobody will be saved.
Statement C:	Some people will die.
Statement D:	Nobody will die or some people will die.

In choosing between A and B, the first part of the statement is common to both options, thus the choice centres on the difference ‘nobody will be saved’, and A is preferred to B. Similarly, in choosing between C and D the difference is ‘nobody will die’, and D is preferred to C. Therefore, given the original four options FTT makes the same predictions as PT. However, option A′ now becomes: ‘some people will not be saved’, which cannot be compared directly with B. Likewise, option C′ becomes ‘some people will not die’, which cannot be compared directly with D. Under these circumstances, according to FTT, people are forced to think in more detail about the problem, calculating expected values, and choosing according to their attitudes to risk.

The study by Chang, Yen and Duh (2002) attempts to test the different theories against each other by performing two experiments. In both cases the experiments are expressed in terms of an investment decision problem, but in the first case the options are presented in the same way as the original Asian disease problem, with A being compared to B and C being compared to D. The results confirm all three theories and it is impossible to test for differences between them. However, in the second experiment A′ is compared to B and C′ is compared to D. In this case all three theories make different predictions. Chang, Yen and Duh find that FTT explains the results best, since there is no significant difference between responses according to whether the frame is positive or negative. Thus they conclude that there is no framing effect in the situation where domain and frame are different, confirming the study of Stone, Yates and Parker (1994). They find further evidence in favor of FTT in the comments of subjects relating to the scenario. In the first experiment only 18% of the subjects (undergraduate business students) mentioned the calculation of expected values in their comments. By contrast, in the second experiment they find that 35% of the subjects refer to the calculation of expected values.

However, there is one main shortcoming of the Chang, Yen and Duh study, which is admitted by the authors. This is that the subjects are not asked to indicate their perceived problem domain or problem frame. For example, the study assumes that option A (400 people will not be saved) involves a perceived problem domain of gain. This assumption is certainly questionable, since it can be argued that the reference point used here may be that all people will be saved, and that therefore A′ involves a perceived loss. If this is indeed the case then the predictions of PT are confirmed. More research needs to be done in this area to ascertain how people perceive problem domains.

There are important policy implications of framing effects, in particular in the type of accounting situation described by Chang, Yen and Duh. Over the last few years there have been numerous accounting scandals in both the US and Europe involving the reporting of financial information to both shareholders and auditors. If framing effects are better understood, this may enable government legislators and standard setters, like the International Accounting Standards Board, to better determine both the kind of information and presentation of information so as to prevent fraud and deception.

The priority heuristic

In its philosophy this model is similar to the two above, being a later development of them, and can be regarded as part of the ‘fast and frugal heuristics’ research program (Gigerenzer and Goldstein, 1996; Gigerenzer, Todd and The ABC Research Group, 1999; Gigerenzer, 2004). The priority heuristic originated with Brandstätter, Gigerenzer and Hertwig (2006), who proposed it as a simple sequential cognitive process in contrast to more complicated models involving nonlinear transformations of utilities and probabilities on top of EUT. The priority heuristic is a three-step process, consisting of a priority rule, a stopping rule, and a decision rule as follows:

Priority rule	go through reasons in the order: minimum gain, probability of minimum gain, maximum gain, probability of maximum gain.
Stopping rule	stop examination if the minimum gains differ by 1/10 (or more) of the maximum gain; otherwise, stop examination if probabilities differ by 1/10 (or more) of the probability scale.
Decision rule	choose the gamble with the more attractive gain (probability). The term attractive refers to the gamble with the higher (minimum or maximum) gain and the lower probability of the minimum gain.

Brandstätter, Gigerenzer and Hertwig (2006) report that their empirical findings show that the priority heuristic can explain the Allais paradox and the same four-fold pattern of risk attitudes as CPT. Furthermore, they claim that it can explain certain observed intransitivities which cause preference reversals that cannot be explained by CPT. In view of its simplicity, the principle of Occam’s razor would tend also to favor its application.

It has been noted by its authors that the priority heuristic is designed to apply to difficult decisions where trade-offs are inevitable. Brandstätter, Gigerenzer and Hertwig (2008) extend their approach by advocating what they refer to as an Adaptive Toolbox Model of risky choice. This employs the priority heuristic as a second stage of a decision-making process, where the first stage involves seeking a no-conflict solution. This in turn means searching for dominance, and if that fails, applying a similarity heuristic. This first stage is similar to the approach of Rubinstein (1988, 2003), which is explained in Chapter 8 in the context of intertemporal choice. Only if the first stage is unable to detect a no-conflict solution does the second stage involving the priority heuristic apply.

There have been various criticisms of the priority heuristic and the fast and frugal heuristics program. At the empirical level, Birnbaum (2008) claims that the priority heuristic fails to explain various observed results. There has also been criticism at the theoretical level. The model is proposed as a process model rather than an ‘as-if’ model, but it has been claimed that there are a number of components of fast and frugal heuristics that involve questionable assumptions, making the program in general psychologically implausible (Dougherty, Franco-Watkins and Thomas, 2008).

Imprecision theory

The basic premise of this theory is that many choices that people make involve a degree of uncertainty or imprecision. This idea has a long history, but it has only been since the mid-1990s that models of imprecision, often referred to as stochastic decision models since they involve a random error, began to be formulated. These models essentially took some form of deterministic model, like EUT, as their base, and then added a stochastic component. The main purpose was to explain certain violations involving preference reversal that could not be explained by purely deterministic models. There is a common finding in empirical studies that subjects make preference reversals even when faced with the same pairwise choice problem twice within the same experiment. Studies by Starmer and Sugden (1989), Camerer (1989), Hey and Orme (1994) and Ballinger and Wilcox (1997) indicate that between one-quarter and one-third of subjects switch preferences on repeated questions.

These violations again relate to the editing phase of the choice process, involving framing effects. An early model developed by Loomes and Sugden (1982, 1987) was called regret theory and was discussed earlier. This approach was explicitly designed to take into account both violations of monotonicity and transitivity. This theory has the further advantage that it posits a preference function that can be maximized, giving the model normative status. However, experiments by Starmer and Sugden (1998) suggest that regret theory does not explain all the observed violations. Loomes and Sugden (1995, 1998) also propose a random preference model, which proposes that people act on preferences based on a core theory, but the parameters to be applied in any context vary randomly, for example, the degree of risk-aversion. However, this and other stochastic models failed to account for more than a subset of the systematic deviations, i.e. preference reversals, commonly observed.

More recently Butler and Loomes (2007) have developed and tested a model that appears to account for a greater number of observed preference reversals. It involves a model originally developed by MacCrimmon and Smith (1986) based on P-bets and $-bets. As we have seen in the discussion of PT³, a P-bet offers a relatively large probability of a modest sum of money and a residual probability of zero. A $-bet offers a smaller probability of a considerably bigger prize and a larger chance of zero. Respondents are asked to place certainty equivalents on each bet and also to make a straight choice between the two bets. The preference reversal observed here is that people place a higher money value on the $-bet, but choose the P-bet, and this result violates transitivity, as we have already seen.

Butler and Loomes (2007) varied the standard procedure to allow for imprecision by allowing participants to respond in any of four ways, rather than just two. Thus instead of just offering the simple alternative: (1) prefer A; (2) prefer B, they offered four choices: (1) definitely prefer A; (2) think I prefer A but am not sure; (3) think I prefer B but am not sure; and (4) definitely prefer B. The study concluded:

in the absence of very precise ‘true’ preferences, respondents faced with equivalence tasks may be liable to pick one value from an imprecision interval, with their perception of the range of this interval, and their selection of a particular value from within it, both liable to be influenced by various ‘cues’ or ‘anchors’ (p. 293).

An important anchor turned out to be the starting point of the iterative procedure for determining certainty equivalents. This procedure, and the problems involved, is discussed in more detail in Chapter 7 in the section on methodology.

Another important conclusion by Butler and Loomes was that ‘all methods of eliciting those preferences, with or without incentives, are vulnerable to procedural effects of one kind or another’ (p. 294). They reject the idea of a ‘gold standard’ in terms of procedure.

Configural weights models

These represent a family of models rather than a single model. Three main versions have been developed in recent years: rank-affected multiplicative weights (RAM); transfer of attention exchange (TAX); and gains decomposition utility (GDU). It goes outside the scope of this text to describe these in detail here, so we will concentrate on describing the main common features, and how they compare and contrast with PT.

There are three common factors shared by configural weights models and the PT model: (1) both types of model are descriptive rather than normative; (2) both use reference-dependent utility or value functions; and (3) both are psychological models, so that they use ‘psychophysical’ utility and decision-weighting functions based on psychological theory.

The most fundamental difference is that in configural weights models people treat gambles as trees with branches rather than as prospects or probability distributions. Branches with lower consequences receive greater weight, as a consequence of risk-aversion; thus these models do not rely on nonlinear utility functions, although they do not rule them out. Weights depend in part on the rank of a branch with the set of total outcomes. Configural weights models generally violate the principle of combination or coalescing, and the principle of cancellation, used in PT.

Different models use different methods for assigning weights. In the RAM model, the weight of each branch of a gamble is the product of a function of the branch’s probability multiplied by a constant that depends on the rank and augmented sign of the branch’s consequence (Birnbaum, 1997). The augmented sign is positive for branches with lower consequences and negative for branches with higher consequences. A middle-ranked branch in a three-branch gamble has an augmented sign of zero. The decision situation is best illustrated by a simple binary gamble, like tossing a coin, shown in a tree diagram in Figure 5.9. The augmented sign for the lower consequence, tail, would be positive, thus increasing its weight, while the augmented sign for the higher consequence, head, would be negative, decreasing its weight.

The TAX model also represents the utility of a gamble as a weighted average of the utilities of the consequences, with the weights depending on probability and rank of the branches. The main difference is that in TAX the branch weights result from transfers of attention from branch to branch. This is how Birnbaum (2008) describes TAX:

Intuitively, a decision maker deliberates by attending to the possible consequences of an action. Those branches that are more probable deserve more attention, but branches leading to lower valued consequences also deserve greater attention if a person is risk-averse. In the TAX model, these shifts in attention are represented by weights transferred from branch to branch (p. 470).

Figure 5.9 TAX model

Again we can use Figure 5.9 for illustration.

Let us assume for simplicity that the utility function is linear and identical with money value. In the first diagram it can be seen that the expected value of the gamble is $50. Both branches have equal probabilities, so this has no effect on attention in this example; however, the lower valued consequence, tail, receives more attention, and it is assumed in the second diagram that a weight of 1/6 or 0.17 is transferred from the probability of a head to the probability of a tail, thus giving this consequence a decision weight of 0.67 compared to a weight of 0.33 for a head. A person would then value the gamble at $33. This example illustrates also how the TAX model accounts for risk-aversion, without resorting to a nonlinear utility function.

The GDU model takes the approach of decomposing multibranch gambles into a series of two-branch gambles. Thus a three-branch gamble is resolved in two stages: first, the chance to win the lowest consequence, and otherwise to win a binary gamble to win one of the two higher prizes. The binary gambles are represented by the RDU model described in Section 5.2.

The main advantage of configural weights models claimed by Birnbaum (2008) is that they explain many of the 11 paradoxes left unexplained by PT in either form. He furthermore claims that TAX in particular is able not only to explain all 11 paradoxes, but to predict them using parameters estimated from previous data, thus avoiding one of the methodological issues discussed in Chapter 2.

Conclusion

Perhaps at this stage the main conclusion that readers may have arrived at is that there are too many theories of decision-making under risk. This would echo the criticism of Fudenberg (2006) mentioned in Chapter 2, that there are too many theories in behavioral economics in general. On the face of it, we would agree with this conclusion; one unfortunate consequence is that it is easy for students to become confused, since there appear to be so many contradictions and conflicts. However, we believe that there are two important factors that need to be considered in this context that will help to guide the student of behavioral economics, and point the way to the future: (1) common factors; and (2) differentiation between phenomena and explanations for phenomena.

1 Common factors

It has already been seen in the discussion of configural weights models that these models share a number of common factors with prospect theory. This aspect of common ground can be generalized to most recent models of decision-making. However, it is easy to overlook this, since academics have an inevitable tendency to stress differences rather than similarities when they want to get papers published in journals. Maybe we should label this a ‘publishing bias’, which produces a ‘polarization effect’, so that the stances of different researchers appear to be more different than they really are. The most important factors which recent theories tend to share relate to having: a descriptive rather than normative approach; reference dependence; loss-aversion; and a psychological basis underlying the model. It is probably fair to say that any new models or extensions of existing ones will share these features. Reference dependence and loss-aversion appear to be fundamental psychological features, and we have seen that these features are also well supported by neuroeconomic studies. As we have also seen in Chapter 2, many researchers believe that progress can only be made by moving beyond ‘as-if’ models and utilizing process models that incorporate such psychological and neural features explicitly. However, at present there is a lack of clarity relating to the nature of some of these features, and this brings us on to the second factor.

2 Differentiation between phenomena and explanations for phenomena

We have already seen in the discussion of endowment effects that it is important to distinguish phenomena from explanations of these phenomena. There may often be several theories or explanations as to why a particular effect occurs. Thus it helps to avoid confusion if separate terms are used for phenomena and explanations. This differentiation is important with other fundamental concepts as well, such as risk-aversion and loss-aversion (Schmidt and Zank, 2005). We have also seen that there are different ways of accounting for risk-aversion. Prospect theory and many other models do this by using nonlinear utility functions. Birnbaum and Stegner (1979), on the other hand, propose that people place more weight on lower valued consequences because of asymmetric costs of over-and underestimating value. This was illustrated in Figure 5.9. A similar distinction applies to the concept of loss-aversion. PT and other theories often define loss-aversion in different ways, as we have seen, but they again attribute loss-aversion to the nature of the utility function, with a kink or asymmetry at the reference point. In contrast, configural weights models account for loss-aversion by a transfer of weights.

Of course, the different models obey or violate different principles, and as a result make different predictions. As has now been said several times, the ultimate test rests on empirical evidence relating to these predictions, provided that such evidence is correctly obtained, analyzed and interpreted. At this point such evidence is still indecisive, which is why there are so many models still being considered. Many studies have indicated the need for further research in order to determine with greater certainty which models have more real-world application and which do not. Some studies have been quite specific regarding what type of research is needed here. For example, Birnbaum (2008) states:

What has not yet been done is the assessment of the behavior of individuals over a large number of replications with a larger number of properties tested within the same person (p. 497).

The advantage of such studies is that it would become possible not only to determine appropriate models in general but also to see whether different models apply to different people. In the meantime Birnbaum suggests, and we would agree, that it is best to assume that everyone can be represented by the same model, while different people may have different parameters in that model. We can also conclude at this stage that the appropriate model for descriptive purposes will not in general be EUT, but some form of process model. Although it is not yet clear which model or models will turn out to be superior, it is at least clear that process theories like prospect theory and configural weights theories both explain and predict better than EUT.

As a final concluding point it should be noted that there are other anomalies in EUT, and most other theories and models, which are not easily explained by any additional assumptions that are consistent with the standard model. Some of these have been mentioned in Chapter 3. For example, one factor is that people appear to value control, even when this conveys no rational advantage. A study by Langer (1982) indicated that people who were allowed to choose their entries in a lottery valued their tickets more highly than those who were simply assigned entries at random. When researchers offered to buy the tickets back from the subjects they found a huge difference: those subjects who were assigned tickets were willing to sell them for an average of just under two dollars, while those who selected their own entries demanded more than eight dollars. Another anomaly that is discussed in Langer’s study involves the same factor of illusory control over events. When people played a game of chance against an opponent, the appearance of the opponent affected the amount that players were willing to bet. The game simply involved drawing a playing card, with the higher card winning. Half the bettors played against a well-dressed and confidently-acting opponent, while the other half played against opponents who acted in a bumbling manner and wore ill-fitting clothes. Of course, the chance of winning is a half in either situation, yet bettors were willing to bet 47% more when faced with opponents who appeared inferior.

This factor of control, however illusory, is likely to be one factor why people regard car travel as less risky than train or air travel. PT can accommodate this apparently irrational tendency in terms of the overweighting of certain probabilities, although it should be noted that in this case again the probabilities are not necessarily low. A different weighting function may be involved in these situations, in the same way that the Jullien and Salanié study (1997) found different weighting functions for losses compared with gains.

In more general terms, the strong desire to be in control appears to be an important reason why the majority of people still believe in the phenomenon of free will (Wegner, 2002). Again, evolutionary psychology has proposed an explanation for this emphasis on control. In our past our ancestors developed a very useful cause-imputing mental adaptation which enabled them to impute and analyze causes of events. It was often better from a survival point of view to impute an incorrect cause of an event than to consider the event as being causeless. For example, if one’s goods disappeared overnight, it might have been better for one’s future prospects to blame the wrong person for stealing the goods than to believe that the goods just vanished without a cause. A strong desire for accountability has always been a prominent feature of most criminal justice systems in all kinds of different societies. This aspect of social fairness and punishment is discussed in detail in Chapter 10.

It seems fitting to end this conclusion by commenting on the main criticism of PT discussed earlier – its lack of normative status. In the past there has been too much reliance on axioms like monotonicity and transitivity in spite of mounting empirical evidence of their violation. Models like PT have been rejected because they permit such violations. For example, Quiggin (1982) commented that the implication in prospect theory that certain choices may violate transitivity was ‘an undesirable result’.

Economics as a science needs to reject assumptions that are proved invalid, and in turn reject theories based on these assumptions which are incapable of accurate prediction. It should not be rejecting theories that disavow such assumptions and by doing so predict well. As Starmer (2000) states:

… there should be no prior supposition that the best models will be ones based on the principles of rational choice, no matter how appealing those may seem from a normative point of view (p. 363).

Therefore it is inappropriate to use normative criteria, as many economists have done, to evaluate a descriptive model. Prospect theory and other recent theories may be less neat and parsimonious than EUT and conventional extensions of EUT, but they are undoubtedly better predictors, and explain various paradoxes. In time normative versions of recent theories may be developed, once economists obtain a better understanding of phenomena like the learning process and reactions to incentives, but lack of normative status is not necessarily a weakness of any theory of decision-making.

5.10 Summary

Expected utility theory (EUT) rests on three main axioms: transitivity, continuity and independence. In addition assumptions are usually made regarding expectation, asset integration and risk-aversion.
In EUT risk-aversion is caused by the utility function being concave.
Conventional extensions to EUT relax the independence axiom in a number of ways, but still maintain monotonicity and transitivity.
Some conventional models incorporate probability weighting, thus including subjective evaluations of both outcomes and probabilities.
A prospect consists of a number of possible outcomes along with their associated probabilities.
Prospect theory states that decision-making under risk involves two phases: editing and evaluation.
The editing process involves coding, combination, segregation, cancellation, simplification and detection of dominance.
The evaluation of prospects in PT involves four main principles: reference points, loss-aversion, diminishing marginal sensitivity and decision-weighting.
Reference points are points denoted as zero on the value scale, with outcomes having values measured as deviations from this reference point, i.e. in terms of gains and losses.
Reference points are often the current level of assets or welfare, but may also involve expectations of the future. Sometimes people may not have yet adjusted to the current situation, so their reference point may relate to a past situation.

The biological basis of reference points is related to the processes of homeostasis and allostasis.

Reference points can explain the anomaly of EUT referred to as the ‘happiness treadmill’.

Loss-aversion means that the disutility from losses is greater than the utility from gains of the same size.

Loss-aversion can explain anomalies like the ‘disposition effect’ and the ‘end-of-the-day’ effect.

There is neuroscientific evidence for the psychological phenomena of reference dependence and loss-aversion.

Diminishing marginal sensitivity means that people become increasingly insensitive to larger gains and losses. Thus the value function is S-shaped, being concave in the region of gains and convex in the region of losses. This may cause risk-aversion for gains and risk-seeking for losses, depending on decision-weighting.

Decision-weighting means that outcomes are weighted not according to objective probabilities, as in EUT, but according to decision weights.

Decision weights may differ from objective probabilities for two reasons: people are often bad at estimating probabilities; and, even if such probabilities are known or stated, people may weight them subjectively.

People frequently overweight low probabilities.

Decision-weighting can explain many anomalies in EUT, such as gambling and insurance, in particular probabilistic insurance.

PT has been criticized on various grounds, theoretical and empirical: the lack of normative status; internal contradictions; incompleteness; and the determination of reference points. Empirical criticisms relate to: violations of the combination principle; violations of stochastic dominance; the failure to explain the Allais paradoxes; violations of gain–loss separability; the nature of the utility function; endowment effects; the discovered preference hypothesis and misconceptions; and the nature of framing effects.

There are various recent models of decision-making which claim to explain and predict better than prospect theory; the priority heuristic, imprecision theory, and configural weights models are prominent among these.

5.11 Review questions

1 John McEnroe has been quoted as saying ‘the older I get, the better I used to be’. Explain this is terms of prospect theory.

2 How does Pinker’s three-act tragedy relate to the standard model and to prospect theory?

3 How do EUT and PT differ in their views of risk-aversion and risk-seeking?

4 Use two numerical examples to explain the difference between risk-aversion and risk-seeking.

5 Explain the endowment effect in terms of the experiment with pens and mugs. What principles of prospect theory are relevant here?

6 Explain the ‘end-of-the-day effect’ in gambling in terms of prospect theory.

7 Give an example of an experiment where neuroeconomics has been helpful in explaining people’s behavior.

8 Why is the decision-weighting function important in PT?

9 Give three reasons from behavioral economics why people tend to prefer traveling by car than by plane.

10 Explain the difference between a descriptive and a normative theory. How do EUT and PT compare in these respects?

11 Use a numerical example to explain the principles of cancellation and dominance.

12 Explain the steps in the priority heuristic process.

13 Explain the nature and purpose of imprecision theory.

14 Describe the fundamental difference between PT and configural weights theories.

15 Show, using a numerical example, how the TAX model accounts for risk-aversion.

5.12 Review problems

1 EUT

A student is considering studying for a test tomorrow, but is tempted to go out and party with friends. He believes that the test could be either easy or hard, with equal probability, and estimates that he will achieve the marks shown below:

	Easy	Hard
Study	90	75
Party	75	55

The student believes his utility function is given by u = √x, where x is the number of marks achieved. He also estimates that studying involves a cost of two units of utility. Determine the course of action that maximizes the student’s utility.

2 Prospect Theory

The student now changes his beliefs regarding his utility function, since he realizes that he is affected by loss-aversion relative to a reference mark of 75.

His value function is now given by:

If the cost of studying is seven units on the v-scale, determine the student’s best course of action.

3 Prospect Theory

Before making the decision above the student views some previous tests and updates the estimated probabilities of the test being easy or hard accordingly. His revised estimates are: P(easy test) = 0.3; P(hard test) = 0.7. Do these revised probability estimates affect his choice of action?

5.13 Applications

Three case studies are included in this chapter, all of them relating to anomalies in EUT that have been observed in the field. The cases illustrate in particular the importance of reference points, and how they are determined in different circumstances, and the phenomenon of loss-aversion, along with its consequences.

Case 5.1 The endowment effect

According to the standard model, ownership or entitlement should not affect the value of goods. This assumption relates to the Coase theorem, which states that the allocation of resources will be independent of property rights. There are two main exceptions to the Coase theorem: (1) income effects may affect tastes; and (2) transactions costs may discourage trade. In addition to these exceptions, there are certain other situations where economists have proposed that value may be affected by ownership: (3) where ownership has conveyed experiential effects, causing people to value items they have owned for some time; and (4) where buyers and sellers need time to adjust to and learn market conditions, which may have recently changed.

Apart from the above exceptions, the standard model predicts that buyers and sellers should not on average demand different prices for the same good, i.e. the WTP of buyers should not differ significantly from the WTA of sellers. Stated in different terms, the standard model assumes that indifference curves are unaffected by ownership. However, many anomalies have been observed over the years. For example, a number of hypothetical surveys have shown that in the case of hunting and fishing rights the WTA of sellers has been between 2.6 and 16.5 times as large as the WTP of buyers. In a real exchange experiment, it was found that the ratio for deer hunting rights was 6.9 (Heberlein and Bishop, 1985). Another such experiment found that the ratio for lottery tickets was 4.0 (Knetsch and Sinden, 1984).

A particularly comprehensive and detailed study was performed by Kahneman, Knetsch and Thaler in 1990. One important objective of this study was to isolate any endowment effect from any of the other circumstances mentioned above that might cause discrepancies between WTP and WTA. For example, the researchers carried out a number of experiments with tokens first, to accustom the subjects to the situations. As expected, these induced-value experiments showed no difference between the WTP and WTA for tokens. However, when the experiments were repeated with consumer goods, using mugs and pens, significant differences appeared. Four trials were performed with the subjects (Cornell University students), in order to eliminate any learning effect over time, but it was found that there was very little difference between the trials. There were 44 subjects involved, divided into two equal groups, one with the property right to the good which they could sell, and the other without the property right initially, but in a position to bid for it. It was also stressed to the subjects that it was in their interest to state their true willingness to pay and accept in the questionnaires, because after the four trials one trial would be taken at random, the market-clearing price (MCP) would be calculated from the responses, and the relevant transactions would then take place. Thus, if the subjects with the property right indicated a WTA at or below the MCP they would then sell at this price, while subjects without the property right who indicated a WTP at or above the MCP would then buy at this price.

The following results were recorded:

Mugs – the median WTP soon settled to $2.25 after the first trial, while the median WTA was a constant $5.25 throughout all the trials. An average of 2.25 trades took place with each trial, compared with an expected 11 (50% of the 22 pairs of subjects would be expected to have the potential buyer value the good more than the seller).

Pens – the median WTP was a constant $1.25, while the median WTA varied between $1.75 and $2.50. An average of 4.5 trades took place per trial, compared with the expected 11.

The authors of the study came to the following conclusions:

1 There was evidence contradicting the standard model – people’s preferences do depend on entitlements.

2 Indifference curves depend on the direction of trade – an indifference curve showing acceptable trades in one direction may cross another indifference curve showing acceptable exchanges in the opposite direction.

3 Endowment effects reduce the gains from trade – the volume of trade will be lower than predicted by the standard model. This is not because of inefficiencies like transaction costs, but because there are less mutually advantageous trades available.

4 Endowment effects will be different for different goods – they are unlikely to exist at all for money tokens, or for goods that are purchased explicitly for the purpose of resale, or for goods where perfect substitutes are available at a lower price. The effects are likely to be strongest ‘when owners are faced with an opportunity to sell an item purchased for use that is not easily replaceable’. Examples given are tickets to a sold-out event, hunting licenses in limited supply, works of art, and a pleasant view.

5 Endowment effects can also apply to firms and other organizations – for example, firms may be reluctant to divest themselves of divisions, plants or products, and they may be saddled with higher wage levels than newer competitors.

There has been considerable laboratory evidence from numerous studies over many years that supports the existence of endowment effects in the traditional sense. These effects relate to the situation where an owner of a good, or seller, places a higher value on it than a non-owner, or buyer. According to PT this phenomenon arises through a combination of reference points and loss-aversion. The owner or seller’s reference point involves possessing the item, while the buyer’s reference point does not involve possession; the seller’s loss in a transaction is greater than the buyer’s gain in the transaction.

However, Plott and Zeiler (2005, 2007) point out that the term is now used to refer to two different phenomena, only one of which refers to endowment in the strict sense. Moreover, we have already seen in Chapter 3 that the term ‘endowment effect’ is problematical because it is now used to refer to a theory or explanation for a phenomenon, rather than the phenomenon itself (PZ, 2007). Therefore there are two main kinds of criticism of the ‘endowment effect’: (1) the effect does not really exist, when it is tested for under proper controlled conditions; and (2) the effect may exist, but it is caused by factors other than those proposed by prospect theory. We will examine each criticism in turn.

Many studies refer to differences between willingness to pay (WTP) and willingness to accept (WTA) as an endowment effect. This does not necessarily involve endowment, since sellers may not have been endowed with the good originally PZ (2005, 2007). Note that sellers may have come into possession of a good for many reasons, such as earning income to buy it, rather than it being gifted to them like ‘manna from Heaven’. Thus there is not just a single phenomenon occurring. Evidence suggests that WTA by sellers may depend significantly on how they came into possession of the good.

Further complications are raised by other possibilities. For example, the influence of the experimenter is an important factor. Subjects may interpret the choice of the good endowed by the experimenter as an indicator of relative quality. Furthermore, social preferences may be relevant if the endowed person regards the good as a gift from the experimenter, and therefore does not want to reject it for that reason.

Another relevant factor concerns learning effects. If the seller has had time to thoroughly inspect the good to fully learn its value, this increases WTA. Some economists have expressed the belief that the endowment effect is merely the result of a mistake made by inexperienced consumers and through time these consumers will learn ‘better’ behavior that conforms to the neoclassical standard model (Knez, Smith and Williams, 1985; Brookshire and Coursey, 1987; Coursey, Hovis and Schulze, 1987; Shogren et al., 1994). Some of these researchers have also reported empirical findings that do not support the endowment effect hypothesis. Most recently List (2004) has conducted a large-scale study involving more than 375 subjects who actively participated in a well-functioning marketplace. The purpose of the study was to test the predictions of prospect theory in terms of the endowment effect against the predictions of the standard model. All subjects actively traded sportscards and memorabilia. In the experiment they were endowed with either a candy bar or a mug of similar market value, and asked whether they would like to trade. List found that both inexperienced and experienced consumers did not trade as much as predicted by the standard model, revealing an endowment effect as people tended to value the good they were endowed with more than the other product. However, for ‘intense’ consumers who traded in their usual market at least 12 times monthly, and for dealers, there was no reluctance to trade and therefore no evidence of an endowment effect. List’s conclusion was that experience in the market did indeed tend to eliminate the endowment effect, and that furthermore there was a transference of this experience, meaning that experience in the subjects’ normal market of sportscards and memorabilia transferred its effects to trading other goods. In a more recent study involving a field experiment that exogenously induces market experience, List (2011) finds confirming evidence that market experience alone can eliminate the endowment effect.

Let us now consider the second criticism of the endowment effect, which is that it is caused by factors other than loss-aversion related to reference points. Birnbaum (2008) explains the endowment effect in terms of a configural weights model. It has been proposed by Birnbaum and Stegner (1979) that exchange asymmetries between WTA and WTP could be explained in terms of asymmetric costs to buyer and seller. A buyer makes a worse or more costly error by overestimating value than by underestimating value, whereas for a seller the more costly error is to underestimate value rather than overestimate it. The result is that, in a configural weights model, buyers assign greater weight to lower estimates of value and sellers assign greater weight to higher estimates of value. Birnbaum claims that various empirical studies involving judgments of buying and selling prices of either ‘sure things’ of uncertain value, like used cars and stocks, or standard risky gambles, support this explanation of the endowment effect and are not consistent with the loss-aversion explanation (Birnbaum and Stegner, 1979; Birnbaum and Zimmermann, 1998).

The conclusion, at least based on existing research, is that not only is the endowment effect an ambiguous term but also that both its causes and the circumstances of its effects remain controversial issues in behavioral economics.

Questions

1 Explain why the term ‘endowment effect’ is ambiguous.

2 Explain how prospect theory can explain endowment effects.

3 Explain how configural weights theories explain endowment effects.

4 Explain, with the aid of a graph, how the endowment effect may cause indifference curves to cross, contrary to the standard model.

5 Wimbledon tickets are allocated by a lottery process. Given that there is a secondary market for such tickets, what implications does the endowment effect have in this situation?

6 We have seen that studies by List (2004) and PZ (2005) argue that the endowment effect is not present under various circumstances. What circumstances may eliminate the endowment effect?

Case 5.2 Insensitivity to bad income news

There are various versions of the standard model as far as the relationship between income and spending over time is concerned, for example the Friedman ‘permanent income hypothesis’. These models are used to predict how much consumers will spend now and how much they will save, depending on their current income, anticipations of future income, and their discount factors. However, all these models assume that consumers have separate utilities for consumption in each period and that they use discount factors that weight future consumption less than current consumption. Moreover, these models make many predictions that seem to be contradicted by empirical evidence. They generally predict that people should plan ahead by anticipating future income, estimating their average income over their lifetime, and then consuming a constant fraction of that total in any one year. Since most people earn increasing incomes over their lifetime, the standard model predicts that people will spend more than they earn when they are young, by borrowing, and will save when they are older. In fact, however, consumer spending tends to be close to a fixed proportion of current income and does not vary across the life cycle nearly as much as the standard model predicts. Also, consumption falls steeply after retirement, which should not be the case if people anticipate retirement and save enough for it.

There are a number of aspects of behavioral economics that are relevant in examining the relationship between income and spending. Some of these were discussed in Chapter 3, and some are discussed further in Chapters 7 and 8 on intertemporal choice. At this point we are concerned with applications of prospect theory. In particular, the concepts of reference points and loss-aversion are relevant.

It is therefore instructive to focus on a study by Shea (1995a). This yielded another result that contradicted the predictions of the standard model. Many groups of workers have their wages set in advance for the following year. According to the standard model, if next year’s wage is unexpectedly good, then the workers should spend more now, and if next year’s wage is unexpectedly low, the workers should reduce their spending now. Shea (1995b) examined the behavior of unionized teachers and found a ratchet effect: the teachers did spend more when their future wages were unexpectedly high, but they did not reduce spending when their future wages were unexpectedly low.

This ratchet effect was explained by a model developed by Bowman, Minehart and Rabin (BMR) in 1999. Whereas in the standard model consumers have separate utilities for consumption in each period, represented as u(c_t), the BMR model was a two-period consumption-savings model in which workers have reference dependent utility, u(c_t – r_t) where r_t represents a reference level of consumption for time period t. In the next time period the reference point is an average of their previous reference point and their previous level of consumption. Thus r_t = αr_t_-1 + (1 – α) c_t-1. This means that the pleasure workers get from consumption in any time period depends on how much they consumed in the previous time period through the effect of previous consumption on the current reference point. If workers consumed a lot in a previous time period their current reference point will be high, and they will be disappointed if their current consumption and standard of living fall, i.e. if c_t < r_t.

Furthermore, the BMR model proposes loss-aversion, meaning that the marginal utility of consuming just enough to reach the reference point is always larger than the marginal utility from exceeding it. There is also a reflection effect, so that if people are consuming below their reference point, the marginal utility of consumption rises as they get closer to it.

Given these features in the BMR model, we can see how it explains the ratchet effect observed in the behavior of the teachers in Shea’s study. If teachers are currently consuming at their reference point and then get bad news about future incomes, they may not reduce their current consumption at all for two reasons:

1 Loss-aversion implies that cutting current consumption will cause their consumption to fall below their reference point, which results in great displeasure.

2 Reflection effects imply that workers are willing to gamble that next year’s incomes may be better. They would prefer to gamble on the prospect that they will either consume far below their reference point or consume right at it than accept the certain prospect of consuming a relatively small amount below their reference point.

Questions

1 Explain what is meant by a ‘ratchet effect’ in the context of Shea’s study.

2 Explain in words the meaning of the expression: r_t = r_t ₁ + (1 – ) c_t ₁.

3 Explain the reflection effect described above in terms of a numerical example.

4 What implications do you think there might be for government policy arising from the Shea study and the BMR model?

Case 5.3 Loss-aversion in golf

We have seen that loss-aversion is a fundamental bias that is anomalous with the standard model. However, many economists believe that this and other types of bias tend to be eliminated in market situations when the stakes are high, competition is great, and traders are experienced. The study by Pope and Schweitzer (2011) is of seminal importance in this respect because it is a field study as opposed to an experiment, and by studying behavior of professional golfers in PGA tournaments, it satisfies the conditions of high stakes, competition, and experience. Furthermore, the study involves a situation where the reference point is highly salient, a par score for each hole, and thus it allows for a precise test of the existence of loss-aversion. Because the outcomes of a tournament, and the payoffs golfers receive, depend on their overall scores over 72 holes (4 rounds), the standard model predicts that golfers should have the same incentive to hole any putt, regardless of whether the putt is to make a par score, or to make a birdie (one stroke below par) or an eagle (two strokes below par). On the other hand, Prospect Theory predicts that golfers have more incentive to make a par putt than a below-par putt because of loss-aversion. Thus there is an opportunity here to test the two theories directly against each other in a market setting. In reality there are problems of confounds, which also need to be addressed. Therefore, this study is valuable for two reasons: (1) it is a good test of the existence of behavioral biases; and (2) it is a good example of how a field study can be designed and implemented to test theories.

The study examined putts, analyzing a sample of 2.5 million putts in total, using laser measurements of distance. Although the study considered all putts, the majority of the putts were either for par or for a birdie. If a player misses a par putt, then they may register this as a loss, whereas this would not occur for a birdie putt; making the latter putt would be registered as a gain. There is an alternative explanation: par putts are more likely to be second putts on the green than birdie putts, so players may focus more on holing these putts to avoid ‘three-putting’. However, this is still a loss-aversion explanation, but using number of putts on the green as a reference point rather than par for the hole.

The means of testing the theories was to estimate a regression function with probability of holing a putt as a function of effort and a number of relevant putt characteristics, like distance from the hole. The central hypothesis was that, with a Prospect Theory value function incorporating loss-aversion, golfers would invest more effort in holing par, bogey and double-bogey putts than birdie or eagle putts. This method allows the comparison of the proportion of par putts made with the proportion of birdie putts. The main finding is that the birdie putts are made 2% less often than comparable par putts, thus supporting the loss-aversion theory. The term ‘comparable’ is important, because of the existence of a number of possible confounding factors. Most obviously, birdie putts are generally of longer distance, so distance has to be controlled for, and this was possible through accurate measurements of distance from the hole. Thus it was possible to compare putts on the same hole in the same tournament that were within one inch (2.5cm) of each other. In addition, the authors considered and controlled for a number of other confounds or contrasting explanations to loss-aversion:

1 Players attempting par putts may have learned something about the green conditions after having already attempted and failed a birdie putt. Learning may also occur by watching a partner putt. These learning effects were controlled for by using dummy variables in the regression analysis relating to number of putts already taken on the green by both the player and the partner.

2 Players attempting birdie putts may be starting from a more difficult position on the green, because of a longer approach shot. This effect was controlled for by dividing all the greens into different sections and using dummy variables for the different sections.

3 Player-specific factors may be relevant. Some players may be long drivers but poor putters. In this case they may be more likely to be in a position to attempt a birdie putt, but also less likely to hole it. These fixed effects were accounted for in the analysis.

4 Hole-specific factors may be relevant. Some holes may have easy fairways but difficult greens and vice versa. Thus with an easy fairway it might be more likely to have a birdie putt, but this putt may be more difficult. Again these fixed effects were accounted for.

5 Tournament-specific fixed factors may be relevant. This means that a player in a better position in the tournament may be more likely to be going for a birdie or eagle putt, and may also have a greater incentive to hole it. This effect was controlled for by using a dummy variable to take into account a player’s overall score in the tournament.

6 Players may be overconfident on birdie putts. A good drive or approach shot may result in the possibility of a birdie putt, and also induce a state of overconfidence. As we have seen, overconfidence can reduce performance. However, in observing golfing performance, the study finds that there is positive rather than negative autocorrelation between hole scores, meaning that a low score on one hole tends to be associated with a low score on the following hole. The authors infer from this result that there is also likely to be autocorrelation between quality of stroke on each hole, so that a good drive or approach shot is likely to be followed by a good putt, whereas overconfidence would produce the opposite result.

7 Players may be more nervous on birdie putts. This is likely to be related to position in the tournament, and as seen in (5) above, this was controlled for. Furthermore, nervousness should be more relevant in later rounds when there is more pressure to perform well, but in fact it was observed that the performance differential between par and birdie putts was reduced in later rounds.

Thus all these alternative explanations for the performance differential between par and birdie putts can be eliminated. As well as the general finding supporting the existence of loss-aversion, the study reports two more interesting findings. The second finding is that the difference between par putts holed and birdie putts holed is most marked in the first of the four tournament rounds, and that the effect is less than half as large in the fourth and final rounds. This is also in keeping with the theory of loss-aversion, combined with shifting reference points. The existence of par as a reference point is likely to be most salient in the first round. By the final round the scores of the other players are likely to become a more important reference point, in particular the score of the leader; thus after the first round scores are generally measured in terms of strokes behind the leader.

A final finding of importance concerns risk-aversion. According to PT people tend to be risk-averse for gains of relatively high probability and risk-seeking for losses of relatively high probability (professionals make both birdie and par putts with a frequency over 80%). The implication is that golfers should be more cautious in attempting birdie putts than in attempting par putts. A cautious putt errs on the side of being short, since a putt that is too long frequently involves a more difficult follow-up putt with a less predictable line and slope. Pope and Schweitzer (2011) find that birdie putts do indeed tend to fall short more often than par putts, thus supporting another aspect of PT.

Perhaps it is appropriate to leave the final word regarding loss-aversion to the most famous golfer of the last decade, Tiger Woods (Pope and Schweitzer, 2011):

Any time you make big par putts, I think it’s more important to make those than birdie putts. You don’t ever want to drop a shot. The psychological difference between dropping a shot and making a birdie, I just think it’s bigger to make a par putt.

Questions

1 Compare the predictions of the standard model with Prospect Theory as far as putting is concerned.

2 Explain the nature of reference points in a PGA tournament with four rounds.

3 Explain the nature of confounds and contrasting explanations in this study.

4 Explain how the study tests for risk-aversion.