Chapter 7

Social and Behavioral Applications

7.1. Introduction

IN this chapter, we discuss modeling examples with the social and behavioral aspect as the leading theme; emphasis has been put on social–behavioral problems which are generally unstructured and difficult to model.

7.2. Courts and Justice

We consider measures of justice and injustice.

Example . Mathematical Model of Justice

This model is a first approximation to assigning values to levels of injustice due to crime and to waiting for trial in a community. It takes into account the types of crimes, the intensity felt in the community about each crime, the time lapse between apprehension and prosecution, the length of sentences (and the deviations from what they should be), and the possibilities for convicting an innocent man. Some uses of this model may be (i) the establishment of a comparative index to be evaluated for a community over a given time period and to be compared with other communities and time periods, and (ii) a second index to be used for the best organization of a docket: the “optimal” sequencing of trials.

Let (γ j )j = 1…n be the set of all types of crime. Each γ j has a magnitude c j , (proportional to the sentence to be given), and a mean trial length l j and sentence length m j .

An individual crime i belonging to class τ may be assumed to have an average intensity 1/β i depending on the character of the criminal and/or the victim and upon the method of and motive for committing the crime (level of brutality, existence of a case for self-defense, amount stolen, etc.). This impact on a victim of the intensity of a crime is assumed to decrease over time at an exponential rate.

We assume

where I i (t ) is the level of injustice due to an unsolved crime i at time t ; t i is the time when the i th crime was committed.

The total injustice due to an unsolved crime is then

There are many other possible expressions; we give this as an illustration.

7.3. Academic Activities

As we change the type of institution being considered, we usually find little change in the type of problems encountered. Our first example relates to policy planning in a university.

Example . University Campus Planning Model [Oliver, Hopkins, and Armstrong]

The purpose of this model is to represent mathematically the relationship among various components of an academic institution. This will be helpful in answering a number of specific policy questions such as: What input flows of new students are required to meet specific output flows of degree winners? How sensitive are enrollment levels to teacher–student ratios?, etc. The formulation of the problem is analogous to an input–output production model for the institution. This may be represented by the linear transformation Y = AX , where X is a vector whose components correspond to student degrees of various types and levels; Y is a vector whose components include capital equipment, instructional staff, and also the input stream of new students; A is a matrix whose coefficients include technological requirements such as teacher–student ratio and the fraction of instructional staff derived from the students themselves.

The elements of X and Y are expressed in terms of inventories (e.g., student enrollments and faculty levels), rather than flows.

Let h be the vector of student arrivals into the educational system, whose components represent the students in different educational categories (e.g., lower division, upper division, master's, doctoral, etc.); v be the matrix (v ij ), 0 ≤ v ij ≤ 1, where v ij is the fraction of students who enter in the i th category and who subsequently reach the j th category; g be the vector of internal demand for educated students (number per unit time) whose components are students from different categories working as teaching assistants; f be the vector of external demand for educated students (number per unit time) whose components are students leaving the system with different educational levels; L be the vector of student inventories or enrollment levels of students (number); N be the vector of teaching staff inventories; A be the technological requirement matrix (dimensionless); and W be a diagonal matrix of average times required per student to obtain education.

We wish to express the instructional staffing levels in terms of the final demand for trained students.

The average enrollment level is given by

the average instructional level N is given by

we also have

Substituting (7.1 ) and (7.2 ) in (7.3 ) yields

Multiplying both sides of (7.5 ) by V gives

Finally, we combine (7.4 ) and (7.6 ) to give the required relation

Further, combining (7.1 ) and (7.6 ) gives

Thus the instructional staffing levels and the required student flows have been obtained.

7.4. Communication and News Transmissions

The same basic situation may be perceived in a variety of ways each of which leads to a different kind of model. Very briefly we give an example of a communication model.

Example . Diffusion Models of News Transmission

Diffusion models describe the diffusion of some object or idea through a population. The unit of adoption may vary from an individual to, for example, a city, or a nation.

A diffusion model takes the form of a differential equation. Both deterministic and stochastic versions have been formulated.

(i) Deterministic:

where n is the number of persons who have adopted a given object or idea and N is the total number of persons in the population under consideration.

(ii) Stochastic:

where p i is the probability that there are i persons who have adopted the given object or idea at time t , and N is the total number in population.

7.5. Population and Pollution

We now consider models in the area of population and pollution. In our first example we study emigration from the farm to the city.

Example 1 . The Relative Population of Cities from Farm Emigration

The general farm population has been emigrating to the cities; we assume that the level of mobility is sufficiently high and that the cost of transportation is negligible in comparison to the cost of the total move so that distance is no object.

There are m cities to which the farm population can go whose population sizes S i may be ranked as follows: S 1 > S 2 …> S m . The chance of choosing a city depends on its rank. The first city gets m times as many immigrants as the smallest. Thus, the chance of moving to S 1 are m times as great as those of moving to S m . An empirical observation is that the tendency of immigrants is to move so that each city gets about the same immigration in relation to its size as any other city. (The rank is used as the weight.) How does this effect the size of cities?

We wish to minimize

subject to

which is a given average population per city. The constraints can be written as

where C 1 is the total population of n cities.

We use Lagrange multipliers to obtain 2iS i + λ = 0.

Thus

is the required population of city i .

Our next model refers to the pollution of rivers.

Example 2 . River Pollution

In 1963 R. V. Thomann extended a previous model due to D. J. O'Connor by dividing a river estuary along its longitudinal axis into segments flowing into each other and studying the relationship between the amounts of dissolved oxygen (DO) and the biochemical oxygen demand (BOD) from segment to segment. Both of these affect life in the river and also affect the way in which wastes thrown into the river are converted into harmless materials.

Let L i be the mean concentration of BOD in the ith segment; C i be the mean concentration of DO in the ith segment; t be the time; V i be the volume of the i th segment; Q i be the net waterflow across the upstream boundary of the i th segment; ξ i be a dimensionless advection factor; E i be the turbulent exchange factor for the upstream boundary of the i th segment; d i be the BOD decay rate constant in the i th segment; J i be the rate of BOD loading to the i th segment from an external source (lb/day); r i be the reaeration rate of the i th segment (per day); C se be the saturation DO value (lb/ft3 ); P i be any other source or sink of the DO in segment i (lb/day).

They obtained 2n equations which described a mass balance of BOD and DO for each of the n segments of the estuary (assuming vertical and lateral homogeneity). These equations may be easily derived and are given by:

7.6. Economic Models

Mathematical models in economics are usually highly sophisticated. Unfortunately, they are often much less accurate in predicting what will occur than in describing what is occurring. Perhaps one reason for this is the fact that economics is a behavioral science and that in economics behavior dominates logic, rather than the reverse, as in queueing theory, for example. Nevertheless, economic models are vitally important and we present several of them.

Example 1 . Demand Analysis [Baumol]

A customer has a fixed budget which is to be spent on a number of commodities in such a way as to maximize his total utility. A model for this may be derived as follows:

Let the utility function be

and the budget constraint be

where Q i is the quantity of commodity i purchased, P i is the price of commodity i , and M is the total budget available.

Then, using a Lagrange multiplier,

and using ∂f /∂Q i as the marginal utility of commodity i , we have

i.e., the ratio of the marginal utilities of two commodities is the same as the ratio of their prices.

Since ∂f /∂Q i , ∂f /∂Q j will be functions of Q 1 …Q n , solutions of these equations will in general give quantities Q 1 …Q n for maximum utility.

Example 2 . Variation of Income [Shapiro]

National income consists of the sum of three components: (i) consumer expenditures, (ii) induced private investments, and (iii) government expenditures. Data is available at discrete time intervals denoted by the subscript t . Let

where Y t is the national income, C t is the consumer expenditures, I t is the private investment, and G t is the government expenditures.

The following assumptions are made relating the variables:

(1) consumer expenditures are proportional to the national income of the preceding period;

(2) private investment in a period is proportional to the increase in consumer expenditures for that period over the preceding period;

(3) government expenditure is the same in all periods.

The problem is now to analyze the behavior of national income under these conditions.

From (1),            C t = αY t – 1 .

From (2),            I t = β (C t – C t – 1 ).

From (3), since G t is constant, we may choose units such that G t = 1.

Combining equations we obtain

With values assigned to α , β and initial values to Y t – 2 , Y t – 1 , an oscillating curve of national income is produced. This curve is either gradually damped to a limit or undergoes increasing oscillations without limit.

Example 3 . The Harrod-Domar Growth Theory [Shapiro]

What growth rate in investment input is necessary to produce a given growth rate in output?

Let K be the capital stock, Y be the level of output, Y p be the potential level of output, and Y R be the actual level of output.

Define the average capital–output ratio as K /Y . The marginal capital output ratio ∆K /∆Y tells how much additional capital is necessary to provide a specified addition to the flow to output.

In addition, Y /K is the average productivity of capital and ∆Y P /∆K gives the ratio of increase in potential capacity output to the ∆K increase in capital stock.

It is assumed that

Since in any time period ∆K = I (investment),

It is also assumed that

where α is the marginal propensity to save.

Now at an equilibrium rate of growth

and hence

Further, ∆Y = σI = σ (αY ) in equilibrium, and so

and, therefore,

Thus the growth rate of investment and the growth rate of actual output are the same for equilibrium. In addition, the higher the propensity to save, α , the greater the required growth rate, and conversely. The higher the productivity of capital, σ , the greater the required growth rate and conversely.

Our next example is a very practical illustration of economic utilization of a scarce commodity.

Example 4 . Economic Utilization of Desalinated Water for an Irrigated Farm – A Linear Programming Model With Uncertain Cost Coefficients [Oak Ridge]

The problem is to determine the size of the farm, the crop pattern, and the schedule for storage and retrieval of water in order to make best use of the output of water in order to make best use of the output of a desalination plant.

Water requirements for the crop start at the planting date, rise with the increased evapotranspiration as foliage is developed, and begin to decline a month or so before the harvest. With high-efficiency sprinkler irrigation in a desert climate, cotton, sunflower, peanuts, soybeans, sorghum, and beans reach a peak water requirement of about 10 in. per month; wheat, potatoes, tomatoes, and citrus reach requirements of about 5 in. per month. Except for beans (three-month) and citrus (year-round), each of these crops has a growing season of about six months. Let j (1, 2,…, n ) designate a particular crop schedule suitable for one parcel of land; an example would be beans from June 1st to August 30th and then winter wheat from November 15th to the following May 15th. For parcel j , designate the water requirements in inches for each month i (i , 2,…, 12) as a ij . The annual requirement for parcel j is . (For the bean/wheat example, the January to December sequence, rounded to the nearest inch, might be: 2, 3, 5, 6, 1, 4, 10, 6, 0, 0, 1, 2 with a total of A = 40.) If X j is the number of acres in parcel j (planted according to schedule j ), then the water volume required in month i by the parcel is a ij X j acre-inches.

Many coastal locations have reservoirs suitable for underground storage of water. Let s i designate the volume (acre-inches) stored in month i , and let r i designate the volume retrieved. The total annual storage is , the total annual retrieval is

where f is the fraction of the water volume which is lost in storage. Let the desalination plant have a capability of delivering d acre-inches/month and D acre-inches/year. (Because of lost time for maintenance of the desalination plant, D < 12d .) The production constraint may be given as

Equations (7.9 ), (7.10 ), and (7.11 ) form a set of 14 constraints. Over the life of this project, income and expenses are not uniform from year to year. For example, certain expenses such as land reclamation do not occur each year, and for citrus crops there is no yield for the first five years of growth. In order to compare income versus expenditures, transactions are financially discounted to a base date. In other words, income and expenses are expressed in terms of their “present worth.” Let the present worth of the income associated with parcel j be designated as b j and the present worth of the expense as c j , both in $/acre. Income is the product of the yield of the crops on the parcel and their sales price. Expense is the cost for land development, construction of the irrigation system and crop storage facilities, materials such as seed, fertilizer, pesticides, and tractor fuel, and labor for operation and maintenance. In addition, there is a pumping expense of p $/acre-inch associated with retrieval of water from storage. To maximize net income, the objective is thus:

The cost of water, i.e., the cost of operating the desalination plant, is deliberately omitted. The cost of operating a dual-purpose plant built for producing both power and desalted water is assumed to remain constant whether or not water is produced. (If water is not produced, more waste heat is generated.) Under such a condition, there is no economic reason for the evaporator not to operate at full capacity. The problem is to make most advantageous use of the water when the plant output is considered fixed. (The cost of operating the evaporator enters the decision of whether or not to build the power plant/evaporator/farm complex but would not affect the design of the farm.)

The market, i.e., the sales price, for some crops is subject to considerable year-to-year fluctuation. It would be prudent, therefore, to solve the problem several times using different sets of sales prices, to indicate the expectation (probability) that each set of prices will occur, and then to decide which farm design is optimal.

Crop yield will respond to change in the quantity of water applied. On the basis of such data, alternative water schedules for particular crops can be tested.

In the model, crops grown sequentially on the same parcel of land were considered independent of crops grown simultaneously on adjacent parcels. Planting, cultivating, and harvesting activities, however, are staggered throughout the year, and joint use can be made of both labor and equipment. Recognition of such joint production economics would require revision of the objective function given by expression (7.12 ).

It should be noted that instead of maximization of net income, alternative objectives might be selected such as maximization of protein or caloric production.

7.7. Conflict Resolution

We have been particularly interested in models relating to conflict resolution and in the identification of problems in that area which are amenable to present modeling techniques. We have also been interested in developing new techniques for such problems; we discuss this further in the next chapter .

A fundamental question is that of measurement: How can we determine those aspects of human interaction which are quantifiable in a way which permits the construction of a model?

Measurement may enter conflict resolution in several clearly defined ways. One involves the use of numbers as probabilities; and this involves their use in assigning monetary values to commodities involved. In the latter case, a conflict problem may be analyzed in an economic context of benefits and losses.

Generally, the probability approach to conflict leads to solution in terms of moments (e.g., averages and variances) and in terms of asymptotic stability or the existence of steady-state solutions.

The economic approach also invokes concepts of equilibria and stability. Stability, as used here, connotes a balance of forces which could lead to a military standoff because neither side has more weapons than the other.

Both probabilistic and deterministic models are often based on data which allow determination of their parameters. Such models can be used for prediction and extrapolation.

Game theory is another method which may be used to analyze conflict and which requires the use of measurement. In game theory, the problem is to find a scheme of representing preferences which allows one to conduct an analysis consistent with definitions of stability.

There are two important areas of conflict and conflict resolution which are amenable to modeling. The first involves the analysis of armament and arms races, escalation of hostilities, and the conduct of wars. The second concerns negotiation and bargaining in the settlement of quarrels and hostilities. The main modeling approach used to analyze negotiations is game theory. However, conflicts and their escalation have been studied with a variety of models ranging from algebra and differential equations to control theory as well as game theory, particularly zero-sum games such as the Blotto Game and Submarine Duels. These models may be placed in two categories, descriptive models which analyze stability and normative models which provide means for finding an optimal policy. Most game theoretic analyses attempt to be normative but fall short because of the difficulty of measurement.

It should be evident that mathematical models in conflict resolution serve as an aid to better understanding of the problem rather than a means to obtain the answer to be implemented. The value of general models is actually to improve understanding rather than to give clear-cut numerical answers.

Example 1 . Measurement of Power: The Shapley Value [Luce and Raiffa]

Game theory has been analyzed in many different ways. A characteristic function v assigns a value v (C ) to every coalition C . (A coalition is a subset of the set N of all players.) A characteristic function is assumed to satisfy the following conditions:

 (i) v (Φ) = 0, where Φ is the empty set; and

(ii) v ( C 1 ∪ C 2 ) ≥ v ( C 1 ) + v ( C 2 ) for C 1 ∩ C 2 = Φ (i.e., superadditivity).

The Shapley value is an a priori assessment of the chances of a player in a characteristic function game. It is based on a system of axioms: (i) the efficiency axiom, (ii) the symmetry axiom, (iii) the dummy axiom, and (iv) the additivity axiom. The Shapley value can be described as an average over the marginal contribution of player i , v (C ) – v {C – (i )} to a coalition C with i in C .

We let |C | be the number of players in C and n be the total number of players. If we take a fixed coalition C , there must be (|C | – 1)!(n – |C |)! sequences of players in order in which player i contributes to C .

Now let L i denote the set of all C with i ∈C . The Shapley value for player i is defined as

This may be regarded as a measurement of the power of player i .

Application to Bargaining Situation [communicated by R. Selten]

Consider the following game of negotiations:

(1) There are five players called small ( S ).

(2) There are two players called big ( B ).

(3) The following coalitions formed among these players win the game:

(a) Two B players and at least one S player.

(b) One B player and at least three S players.

(c) Five S players.

The game follows the following cycle:

Step 0. The seven players negotiate among themselves until a winning coalition can be formed with each member of the coalition in agreement with respect to his share of the prize.

Step 1. The coalition is Registered. Ten-minute timer is started.

Step 2. Those players who are not members of the registered coalition try to break up the coalition by offering more attractive coalitions to the members of the registered coalition. All the players can negotiate during this step to protect their positions or to try to improve them. There are two possible results of this step:

(a) The registered coalition stands for 10 minutes. In this case the game is over and the prize is divided as agreed.

(b) The coalition is broken and a new one is formed. The game is then continued from step 1.

The value of this game to each of the players may be computed in a number of ways. One way is to use the Shapley value.

For characteristic functions with a small number of players a simple tabular method can be used in order to compute the Shapley value.

Computation of the Shapley-value for the foregoing seven-person game

This game has the following characteristic function:

v (i ) = 0	for i = 1…7;
v (C ) = 27	if C contains players 1, 2 and at least one other player;
	if C contains player 1 or player 2 and at least three of the players 3, 4, 5, 6, 7;
	if C contains the players 3, 4, 5, 6, 7;
v (C ) = 0	in all other cases.

TABLE 7.1. Computational table for the seven-person demonstration game:

Some conflicts are resolved by fighting; such situations may be regarded as zero-sum games. In that case each party would attempt to choose its strategy in the confrontation in such a way as to obtain the best return. A simple example follows.

Example 2 . Colonel Blotto

Colonel Blotto has four regiments with which he hopes to occupy two posts, but his enemy opposes him with three regiments. If either side has more regiments assigned to a post than the other, he wins by an amount equal to the number of regiments the enemy assigns plus one because a post is considered worth a regiment. The total amount gained or lost is the sum of the payoffs from both posts. If the forces are equal at a post, the payoff is zero at that post. What is the optimal strategy for allocating regiments for Colonel Blotto and for his enemy?

Colonel Blotto has five strategies and the enemy has four. Each strategy is described by an ordered pair of numbers indicating the number of regiments assigned to the first and to the second post respectively. The payoff matrix to Colonel Blotto is as follows:

This game has an optimal mixed strategy solution as follows: Blotto should play his respective strategies with probabilities (4/9, 4/9, 0, 0, 1/9); the probabilities for the enemy are (1/18, 1/18, 4/9, 4/9). The value of the game to Blotto is 14/9: if he plays his optimal strategy, he wins at least this number of regiments no matter what the enemy does. We now discuss arms races.

Example 3 . Arms Race Model [Richardson]

A very simple but effective model for arms races was given by Lewis Richardson (1881–1953) who developed a model to describe armament buildups between two countries. He assumed (i) that in an armament race between two countries, each country would attempt to increase its armament of the other, (ii) that economic factors impose constraints on armament that tend to diminish the rate of armament by an amount proportional to the size of the existing forces, and (iii) that a nation would build arms, motivated by ambition, grievances, and hostilities even if another nation posed no threat to it.

If the costs of the armament levels of the two sides are N 1 (t ) and N 2 (t ), respectively, where t represents time, then the foregoing three conditions may be expressed for each side

where k , a , g , l , b , and h are positive constants.

The constants in this model are sometimes referred to in the following way: k , l are called defense or reaction coefficients; a , b are called fatigue or expense coefficients; g , h are called grievance coefficients when positive, good-will coefficients when negative. They are usually assumed to be positive, as suggested in condition (iii) above.

In this model a balance of power is attained when a stable equilibrium at a constant level of expenditure is reached. Stability is obtained when kl < ab , that is, the product of the coefficients of reaction to the other side must be less than the product of the coefficients corresponding to the expense of armament. An unstable equilibrium occurs when ab < kl and indicates a runaway arms race.

Now let A and B denote the respective expenditure of two blocks for arms preparation and let A 0 and B 0 be the corresponding cooperative expenditures between the blocks. Put N 1 = A – A 0 , N 2 = B – B 0 , and assume for simplicity that k = l and a = b . If we substitute these quantities in Richardson's equations and add we obtain

which says that the rate of change of the combined armament expenditures of both blocks is proportional to the level of the combined expenditures. That is, we have a linear relation.

The last equation describes the arms race of 1909–1913 surprisingly well; Austria–Hungary and Germany were on one side, France and Russia on the other side.

Richardson collected data for this period and made a plot as shown in Fig. 7.1 .

FIG . 7.1

He drew the straight line suggested by the combined equation and obtained the indicated good fit of the model to the data giving the trend along the line with time. The graph indicates a positive proportionality factor or slope k – a . Thus k – a > 0 or k > a , indicating instability and a runaway race.

In general, it is not easy to obtain an accurate statistical measure of warlike preparations, but such measures are crucial for the credibility of the model. There are other studies in which data have been collected for the purpose of analyzing the stability of the arms race between the United States and the USSR in recent years.

7.8. Learning

A theory of learning is obviously central to the theory of behavior, for individual behavior is itself largely learned. Although there are several old and well-established theories of learning, the study of this subject has gathered momentum with the formulation of many new models. Much of the credit for this is due to Estes and to Bush and Mosteller, on whose work some of the following examples are based.

Example 1 . The Linear Model of Behavior

An individual is exposed to a certain stimulus which evokes one of a number of possible responses and as a result of this response, some event takes place. Over a number of trials, the stimuli become conditioned to certain responses; i.e., they tend to cause those responses by the individual.

Assume that an individual is constrained to two response classes with probabilities p and 1 – p , respectively. Then a linear recursive equation for p is given by

where n = number of trials, P n is the probability that the response of the first class is made on trial n , and α i , λ i are two parameters that depend upon the response actually made on trial n and upon the outcome of that trial. (1 – α i ) may be thought of as a rate parameter and λ i is the asymptote which would be approached if (7.13 ) were appropriate for every trial beyond a given point in the process. For each event i that could occur on trial n , there corresponds a similar equation.

Bush and Mosteller show that this linear equation can be derived from Estes’ theory of stimulus sampling and conditioning.

For more than two response classes, assume a probability vector p n for trial n and postulate

where τ i is a linear operator (a transition matrix).

In more general terms, let S be a stochastic operator that describes the effect of the event which has just occurred, i.e., p n + 1 = S p n .

Now suppose we wish to treat two or more of the response classes in an identical manner and so we combine them. We can represent this by applying a projection operator C to the probability vectors to obtain C p and CS p . The stochastic operator now becomes CS , i.e., (CS ) C p = C (S p ).

Consider again the two-response case

P n + 1 = α 1 P n + (1 – α 1 )λ 1    with probability p n ,

P n + 1 = α 2 P n + (1 – α 2 )λ 2    with probability 1 – p n .

A stable asymptotic distribution exists independently of p 0 except when λ 1 = 1 and λ 2 = 0. When this is the case, all probabilities are 1 or 0 in the limit. If f (p 0 , α 1 , α 2 ) is the probability that a “particle” beginning at p 0 is absorbed in the limit at 1, then

This functional equation has no closed analytic solution.

Various procedures for estimating the parameters of the model from experimental data have been used; it does not appear that any of the parameters are universal constants. The parameter values for a given experiment have not been determined independently of the data from that experiment. 1 – α i is usually observed in the range 0 ≤ 1 – α i , ≤ 0.2.

This model has been applied to a rote-learning experiment; i.e., a person memorizes a list of symbols. On each trial the subject is shown the list and then given a recall test. If recall occurs on trial n , the recall probability p n + 1 on the next trial is given by p n + 1 = αp n + (1 – α ). This model was found to fit the data well.

We now consider models of behavior viewed as an adaptive process and, in particular, look at behavior in terms of learning and motivation.

Example 2 . An Adaptive Behavior Model

Assume an individual performs a learning activity for some specified time over given periods (for example, x could be in hours/day). Let D , the level of difficulty, decrease exponentially with practice. Then dD /dt = aDx .

Further, assume that, at any given level of difficulty, practice is pleasurable up to a certain point and then becomes unpleasant.

Let x = (D ) be this point. Then motivation may be represented by the equation dx /dt = –b (x – ). Given initial values, D 0 , x 0 , at time t 0 , the time paths of D and x may be predicted.

This model has not been applied to empirical data, but it does have a number of the required properties and seems theoretically sound.

We now consider how to model an individual's change in opinions over time.

Example 3 . Attitude Change Model [Anderson]

At time t an individual may hold one of m opinions. The probability of holding the j th opinion at time t + 1 depends upon his opinion i at time t , denoted by P ij (t + 1), where

The matrix of transition probabilities P is given by

Therefore, given the individual's initial opinion and the matrix P , it is possible to calculate the probability of any sequence of opinions; it is also possible to find the limiting probabilities for each opinion.

The model was applied to a questionnaire study on voting intentions. P was estimated from data collected over a number of months and predictions were made for future opinion changes. The model did not predict well; Anderson attributes this to differences between vocal intent and actual behavior on the part of the voters.

There have been many other learning models; their greatest deficiency has been that they were empirically fitted to data.

Chapter 7—Problems

1. Derive a connection between group size and group performance in solving a problem. In what ways does group size aid in the solution? In what ways does it hinder the finding of a solution? How would you test your model?

2. How is information transmitted among the members of a group? What are the most effective means of transmitting information? Why? What types of models may be used to represent this communication? Can you ensure that all members of the group receive the information? In what time period?

3. How would you decide on the size of prison needed for a given state? [The university planning model will be helpful, but recall that prisoners stay for varying lengths of time.]

4. What forms of pollution exist in your town? How can they be handled? What affects whether or not they will be handled properly? Can you estimate the costs of doing so?

5. How can the courts in your area cut down the time it takes to bring someone to trial? What are the limiting factors? Formulate a simple model for the length of time it takes to bring a case to trial.

6. You have been asked to advise a resort on its tourist industry. How would you proceed? Remember to include not only hotel rooms and facilities, but also access roads, etc. Develop a model to show expected return for given levels of expenditure.

7. Suggest some measures for the grievance and good-will coefficients in Richardson's model of an arms race. How would you test these measures? Repeat for the defense and fatigue coefficients.

8. Formulate a model for group (or gang) formation. What assumptions do you need to make? What would be an optimal size for a group? Why? At what stage will groups tend to break into smaller groups? How would you test your model?

9. You have $ 20 to spend on dinner. Use the model on demand analysis to decide how you would allocate the money to maximize your satisfaction. Does the model work well? If not, why not?

10. You are teaching your friend how to write a computer program in some given language. How would you test his progress? Would any of the learning models in this chapter be of use? How would you test their validity?

References

Anderson, T. W., Probability models for analyzing time changes in attitudes, in Mathematical Thinking in the Social Sciences (ed. P. F. Lazarfeld), Glenco Free Press, 1954.

Baumol, William J., Economic Theory and Operations Analysis , Prentice-Hall, Englewood Cliffs, New Jersey, New York, 1961.

Bush, R. R. and C. Mosteller, Stochastic Models for Learning , Wiley, New York, 1955.

Defleur, Melvin, The Flow of Information , Harper Brothers, New York, 1958, pp. 109–111.

Estes, W. K., A random walk model for choice behavior, in Mathematical Methods in the Social Sciences (ed. K. J. Arrow, et al .), Stanford University Press, Stanford, 1960.

Oak Ridge National Laboratory Report 4290, Nuclear Energy Centers; Industrial and Agro-Industrial Complexes , November, 1968.

O’Connor, Donald J., Oxygen balance of an estuary, J. Sanitary Enging Div ., American Society of Civil Engineers, Vol. 86, No. SA3, May 1960.

Oliver, R. M., D. S. P. Hopkins and R. Armstrong, An Academic Productivity and Planning Model for a university Campus , University of California at Berkeley, February, 1970.

Richardson, L. F., Arms and Insecurity , Borwood, Pittsburgh, 1960.

Shapiro, Edward, Macro Economic Analysis , Harcourt, Brace & World, New York, 1966.

Thomann, Robert V., Mathematical model for dissolved oxygen, J. Sanitary Enging Div ., American Society of Civil Engineers, Vol. 89, No. SA5, October, 1963.

Bibliography

Bales, R. F., Interaction Process Analysis , Addison-Wesley, Reading, Massachusetts, 1951.

Bartholomew, D. J., Stochastic Models for Social Processes , Wiley, New York, 1967.

Bartos, O. J., A model of negotiation and some experimental evidence, in Mathematical Explorations in Behavorial Science , Illinois, 1965.

Bohigian, Hay E., The Foundations and Mathematical Models of Operations Research with Extensions to the Criminal Justice System , Gazette Printers, 1972.

Chaplin, J. P., and T. S. Krawiec, Systems and Theories of Psychology , Holt, Rinehart & Winston, New York, 1960.

Coleman, J. S., Reward structures and the allocation of effort, in Readings in Mathematical Social Science (ed. P. F. Lazarfeld and N. W. Henry), MIT Press, Cambridge, Massachusetts, 1966.

Horvath, William J., Stochastic models of behavior, in Management Science , Vol. 12, August 1966, pp. B513–518.

Horvath, W. J., and C. C. Foster, Stochastic models of war alliances, in J. Conflict Resolution . Vol. 7, 1963, pp. 110–116.

Isaacs, R., Differential Games , Wiley, New York, 1965.

Kupperman, Robert H., and Harvey A. Smith, Strategies of mutual deterrence, Shiver , Vol. 176, April 7, 1972.

Lazarfeld, P. F., and N. W. Henry (eds.), Readings in Mathematical Social Science , MIT Press, Cambridge, Massachusetts, 1968.

Luce, R. D. and Raiffa, H., Games and Decisions , Wiley, New York, 1957.

Nash, J., Non-cooperative games, Ann. Math . Vol. 54, No 2, 1951.

Rashevsky, N., Mathematical Biology of Social Behavior , University of Chicago Press, 1959.

Simon, H. A., Models of Man , Wiley, New York, 1957.

Simon, H. A., Some strategic considerations in the construction of social science models, in Mathematical Thinking in the Social Sciences (ed. P. F. Lazarfeld), Glenco Free Press, 1954.

Sorokin, P., Social and Cultural Dynamics , American Bank Co., Vols. I, II, III, 1937, Vol. IV, 1941.

Suppes, P., and R. C. Atkinson, Choice behavior and monetary payoff, in Mathematical Methods in Small Group Processes (ed. J. H. Criswell et al .), Stanford University Press, Stanford, 1962.

Suppes, P. and R. C. Atkinson, Markov Learning Models for Multiperson Interactions , Stanford University Press, Stanford, 1960.

Weiss, Herbert K., Stochastic models of the duration and magnitude to a “deadly quarrel”, Operations Res ., January 1963.