One of the first people to take up the theory of diminishing marginal utility in a setting that truly deserves the qualification “mathematical” was the Cambridge philosopher and mathematician Frank Ramsey (figure 6.1). The contributions of this extraordinary man during his all-too-short life—he died before the age of twenty-seven—would have sufficed to fill the careers of several scholars, not only in his chosen subjects but also in the field of economics. Ramsey was born in 1903, the eldest son of a minor academic, an author of textbooks on mathematics and physics who later became president of Magdalene College at Cambridge. The mother, a graduate of Oxford, was an ardent feminist and a pillar in Frank’s life. The family had two more daughters and a son. The latter, Michael Ramsey, would become Archbishop of Canterbury—the only sibling who remained a Christian. One sister would become a medical doctor, and the other a lecturer in economics.
FIGURE 6.1: Frank Plumpton Ramsey.
Source: Art by Patrick L. Gallegos (2017), courtesy of Wikimedia Commons
Frank showed his mettle from an early age. Entering Trinity College at Cambridge with a full scholarship, he graduated at age twenty as Senior Wrangler, the university’s top-ranked student of mathematics. But he had made a name for himself even before then, with an exceptional feat. Largely self-taught in the German language, the eighteen-year old undergraduate translated Ludwig Wittgenstein’s notoriously difficult Tractatus logico-philosophicus, one of the most important philosophical treatises of the early twentieth century, from German into English. (Wittgenstein, the youngest son of a fabulously wealthy Austrian family, who had come to Cambridge to study under the philosopher Bertrand Russell in 1911, wrote the first version of Tractatus in the trenches of the battlefields of World War I, during which he distinguished himself as a courageous officer, and later in Italian captivity. In the fall of 1923, Ramsey traveled to Austria for a fortnight to discuss the Tractatus with Wittgenstein himself, who was then teaching schoolchildren in the village of Puchberg.)
A year later, Ramsey fell in love with Margaret Pyke, a feminist and birth control activist. Unfortunately—for Ramsey at least—Margaret was married. She was the wife of a journalist and inventor who today would be described as a “nerd” or “gearhead” because of his unconventional and often unimplementable inventions. But this did not dishearten Ramsey. Intellectually mature, way beyond his years, he remained boyish at heart. Once, after a walk with her along a lake, when they were resting (she reading a book, he pretending to do so), Ramsey posed the decidedly nonphilosophical question, “Margaret, will you f*** with me?” Apparently, she was in no mood to consent, replying, “Do you think once would make any difference?”1 The indifference of her response, even though quite appropriate, must have struck him as unwontedly coldhearted.
His unreciprocated love for a married woman ten years his senior drove him to the newest fad among the intellectuals of the early twentieth century—psychoanalysis. He traveled to Austria again, this time to undergo analysis in Vienna with one of Sigmund Freud’s first disciples, the psychologist Theodor Reik. One can only surmise that his desire for an unattainable, significantly older woman must have spoken volumes about Frank’s relationship to his mother, whom he adored and who would die four years later in a car accident.
Cured of his doomed love, Ramsey returned to Cambridge. Thanks to the efforts of the eminent economist John Maynard Keynes, he was elected a Fellow of King’s College in 1924 and was made a lecturer of mathematics two years later. At the same time, he got involved with a student of psychology, Lettice Baker, who would later become a distinguished portrait photographer. To abide by the mores of the times, or at least to hide their transgressions, they were forced to keep their frolicking secret. Afraid of being found out by either his mother or by the Fellows at King’s College, they had to sneak off to Lettice’s room at Trinity College for their trysts.
Ramsey’s fears were well founded. Keynes had informed him that even his previous liaison with Margaret could constitute an impediment to his election as Fellow.2 Eventually, Ramsey’s mother found out about him and Lettice, and this hastened the young couple’s decision to get married. But theirs was not to be a traditional union. They agreed that they would be free to pursue other conquests, and they both took advantage of their liberty. Frank understood this to mean that he had to report to Lettice his innermost feelings…toward other women. Two years after the wedding, he fell in love with one of Lettice’s friends, Elizabeth Denby, a social reformer who, like Margaret, was nearly ten years older than he was.
Ramsey was rather better at dishing it out than at taking it. Once, while he was spending a Christmas holiday in France with Elizabeth, Lettice took up with an Irish writer and, as was their custom, informed Frank of the liaison. He was furious. There he was, doing nothing more than spending a little quality time with his mistress, all the time looking forward to resuming a tranquil life with his wife, and there she was, destroying it all. Considering himself the wounded party, expecting compassion, he lamented plaintively, “I had decided to give up Elizabeth, and you know how important she is to me.” Somewhat belatedly, he arrived at the conclusion that “I can’t stand the strain of this sort of polygamy and I want to go back to monogamy.” His wish was soon fulfilled, in fact, because the affair between Lettice and the Irishman came to an end after a few weeks. When she informed him how miserable she felt, he did not bother to hide his glee and relief. She responded with sarcasm: “Well, it’s something that my being unhappy has helped you…Let me cheer you by saying I’m still very gloomy and depressed.”3
Wittgenstein was lured to Cambridge by Keynes and Ramsey in 1929. In spite of his fame, the Austrian philosopher still had no academic degree, so he needed to submit a thesis before being allowed to teach. Ramsey, now a twenty-six-year-old with only a bachelor of arts degree, became the nominal PhD thesis supervisor of his forty-year-old friend. This was after their relationship had undergone a severe test: During one of their conversations, Wittgenstein had expressed the opinion that Freud, though very clever, was morally deficient. After that, the two refused to speak to each other for quite a while. The thesis that Wittgenstein eventually submitted was none other than the Tractatus, a first version of which had previously been considered as insufficient to award him a bachelor’s degree because it had no preface and no footnotes. One of the examiners at the successful thesis defense was Bertrand Russell.
The following year, Ramsey fell severely ill. He had developed jaundice that was initially thought to have been caused by a blockage of the gallbladder. An unsuccessful operation at a hospital in London revealed that he had long suffered from a liver and kidney disease without being aware of it. He died a few days after the operation, leaving behind Lettice and two daughters. One can only speculate how philosophy and mathematics would have advanced, quite apart from economics, had he lived longer.
John Maynard Keynes, Ramsey’s mentor at King’s College, was to become the most influential economist of the twentieth century. He had distinguished himself as a civil servant and as the representative of Britain at the Versailles peace conference after World War I. The twenty-three-year old Ramsey came to his attention in 1926 with the paper Truth and Probability, in which he took issue with Keynes’s own Treatise on Probability, published five years earlier. By extending the familiar deductive logic of definite conclusions to an inductive logic of partial conclusions, Ramsey defines probability theory not in the way that physicists or statisticians do, as a frequency or as the proportion of favorable outcomes to all outcomes, but in the spirit of the seventeenth-century German polymath Gottfried Wilhelm Leibniz, as a branch of logic—namely, “the logic of partial belief and inconclusive argument.”4
Following that paper, and with Keynes’s encouragement, Ramsey set his formidable mind to economics. For a philosopher of his talents, it was no more than a simple diversion. He wrote and published only two papers in this field: A Contribution to the Theory of Taxation in March 1927, and A Mathematical Theory of Saving in December 1928. Keynes, who had them published in his Economic Journal, was fulsome in his approval. In an obituary for Ramsey, he wrote, “The latter of these [referring to A Mathematical Theory of Saving] is, I think, one of the most remarkable contributions to mathematical economics ever made.”4 Although the paper was certainly of the highest caliber, this praise is somewhat exaggerated. If anything, the paper was one of the very first in the field of economics that really deserved the moniker mathematical. Hence, “the most remarkable contribution ever made” sounds suspiciously like a tautology. In his general assessment, however, Keynes was right, even if it took half a century, until the 1970s, for the paper’s groundbreaking importance to sink in.
Nonetheless, it is the paper Truth and Probability that is of interest to us here. Unfortunately, Ramsey’s writing style was somewhat convoluted at times, and a bit pompous. For example, when referring to Keynes’s interpretation of probability, he expressed his disagreement in the following manner: “I hope that what I have already said is enough to show that [Keynes’ theory] is not so completely satisfactory as to render futile any attempt to treat the subject from a rather different point of view.”5
If probability is the degree of belief (i.e., the degree of confidence one has in a certain outcome), how can it be measured? “It is not enough to measure probability,” Ramsey wrote. “In order to apportion correctly our belief to the probability, we must be able to measure our belief.”6 Because belief and confidence stem from a person’s subjective impressions, a psychological approach must be taken, and the degree of belief must be measured psychologically. But how? One can agree on a few benchmark figures: Full belief in an outcome is denoted by 1.0, full belief in the opposite by zero, and equal beliefs in the proposition and its contradiction by ½. But what is meant by a belief of 2/3? That the belief that an outcome will occur is twice as strong as the belief that it will not?
One can certainly not make do with an ordinal scale, which simply indicates which belief is greater. Such a scale, like the one devised to express the hardness of materials in the early nineteenth century by the German mineralogist Friedrich Mohs, suffices to indicate which material can scratch another. As the name says, an ordinal scale provides an ordering—in this case, listing materials in terms of increasing hardness. Mohs simply assigned a higher number to a harder material, arbitrarily giving a value of 1.0 to the soft mineral talc and a value of 10.0 to diamonds, the hardest material then known.
But his ordinal scale had an important shortcoming. While it allowed the positioning of materials on the scale of increasing hardness, one cannot say that the hardness of quartz (7) plus the hardness of calcite (3) equals the hardness of diamonds, or that gypsum (3) is three times as hard as talc. What Ramsey sought was a way to measure belief in a cardinal manner, like the measurement of length or weight. On a cardinal scale, a greater number not only indicates longer length or heavier weight, but also specifies how much longer and heavier the object is. The difference between ordinal scales and cardinal scales is that the latter can be added and subtracted.
And this is what Ramsey was after. He sought a measurement process that would allow addition and subtraction. The first alternative he considered as a proxy for his ultimate goal, the measurement of probability, was to assess the intensity of feeling. The more intensely one feels that a certain outcome will occur, the higher its probability. But even before he got to checking for addability, Ramsey dismissed the option. One cannot trust a person to assign numbers to the intensities of her or his feelings. After all, even “the beliefs which we hold most strongly are often accompanied by practically no feeling at all; no one feels strongly about things he takes for granted.”7
Ramsey concluded that in order to measure the strength of one’s belief, one should measure the extent to which one is prepared to act on it. “Our judgment about the strength of our belief is really how we should act in hypothetical circumstances.” To assess that strength, Ramsey took a page out of Daniel Bernoulli’s playbook. “The old-established way of measuring a person’s belief is to propose a bet and see what are the lowest odds which he will accept.” There are also shortcomings with that approach, of which Ramsey mentions two. First, he notes that the marginal utility of money diminishes. (He mentions this only very casually, since by then this phenomenon was “universally agreed.”)8 Hence, measurements could be distorted if the potential gains or losses are so large that they would put the gambler into a different class of wealth. Also, the amount that one is willing to wager may lead to the erroneous conclusion that a person’s belief in a certain outcome is stronger when he is rich than when he is poor.
The second shortcoming he cites is that people may have a special eagerness or reluctance to place bets. Surprisingly, Ramsey is a bit too nonchalant here. His statement merits—indeed requires—more than just an offhand aside because, as we already saw in chapter 1, diminishing marginal utility of money, cited by Ramsey just a few lines before, entails risk aversion. Hence, in theory, a reluctance to bet is tautologous to diminishing marginal utility, while an eagerness to bet should be ruled out as a contradiction. Ramsey does propose an explanation in terms of human psychology (namely, the joy or the dislike of excitement that accompanies gambling). In chapter 8, we will occupy ourselves at length with the dichotomy of risk aversion and the simultaneous eagerness to bet.
To avoid such weaknesses, Ramsey came up with an alternative proposal. He illustrated it with an elaborate example: “I am at a cross-roads and do not know the way; but I rather think one of the two ways is right. I propose therefore to go that way but keep my eyes open for someone to ask; if now I see someone half a mile away over the fields, whether I turn aside to ask him will depend on the relative inconvenience of going out of my way to cross the fields, or of continuing on the wrong road if it is the wrong road. But it will also depend on how confident I am that I am right; and clearly the more confident I am of this, the less distance I should be willing to go from the road to check my opinion. I propose therefore to use the distance I would be prepared to go to ask, as a measure of the confidence of my opinion.”9 Hence, the measure of one’s belief (i.e., of the probability that one is right) is the effort that one would be willing to extend in order to make sure. In Ramsey’s example, the degree of belief (the probability that the chosen way is the correct one) is measured in fractions of miles.
According to Ramsey, the laws of probability are an extension of formal logic to a theory of partial belief. As such, a person’s degrees of belief, proxy to the measurement of probability, must—at least in theory—be logically consistent. After all, if a measuring system is inconsistent, it is useless. Obviously, this should hold true as well for the measurement of the strengths of beliefs. For example, if my belief that “It will be sunny tomorrow” is stronger than “It will be cloudy tomorrow,” and my belief that “It will be cloudy tomorrow” is stronger than “It will be rainy tomorrow,” then my belief that “It will be sunny tomorrow” must be stronger than “It will be rainy tomorrow.”10 Ramsey specifies several definitions and axioms that must hold in order for the measurements to be consistent, and then he derives from them four laws of probable belief that he considers fundamental.
The first law is that the belief in an occurrence and in its opposite must add up to 1 (i.e., to 100 percent).11 For example, my belief that “It will rain tomorrow” plus my belief that “It will not rain tomorrow” exhausts all the possibilities, and hence the expression adds up to 1. The second law can be illustrated as follows: My belief that “The Knicks will win tomorrow’s game if it rains” plus my belief that “The Knicks will lose if it rains” must also equal 100 percent. The third law is a bit more involved: My belief that “I will live to be 120 years and die a billionaire” equals my belief that “I will live to 120” multiplied by my belief that “I will die a billionaire, given that I have lived to be 120.” The fourth, and final, law can be illustrated as follows: My belief that “I will live until 120 and die a billionaire” plus my belief that “I will live to an age of 120 years but without a billion dollars” is equal to my belief that “I will live until 120”…with or without a billion dollars.12
Anybody who does not conform to the laws because his beliefs are not coherent is vulnerable to a spoof by sneaky bookmakers. For example, if the fourth law is violated, a shrewd bookmaker could take advantage of a naïve gambler. To illustrate, let us say that the gambler is willing to bet on the winner of a tournament among several soccer clubs. If the Grasshopper Club wins, he will receive fifty dollars. He can also place a bet on the Butterflies Club and receive fifty dollars if that team wins. Assume that the gambler’s beliefs are such that he is willing to pay five dollars for each of the two simple bets. There is also a third bet, the compound bet, which pays fifty dollars if either the Grasshoppers or the Butterflies win. The gambler evaluates his belief for the compound bet at twelve dollars. Now the bookmaker could sell him the compound bet for twelve dollars, and buy the two simple bets for five dollars apiece (a total of ten dollars). She immediately pockets the two dollars that remain. If the Grasshoppers win, the gambler pays fifty dollars to the bookie because he lost one of the simple bets, but gets fifty dollars from the bookie because he won the compound bet. If the Butterflies win, he will pay fifty dollars because he lost the other simple bet, but also gets fifty dollars because he won the compound bet. If any other team wins, no money will exchange hands. In all cases, the gains and losses (if any) balance out. But the savvy bookie gets to keep her two-dollar profit, while the naïve gambler, having violated the fourth law, is sure to lose.13
The laws are so fundamental, says Ramsey, that “if anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd.”14 Well…it may be absurd, but unfortunately it is also very common. Indeed, the manner in which a question is framed, or a choice presented, often determines how people answer, decide, and act. The laws that should, in principle, govern degrees of belief are violated much more frequently than Ramsey thought. Such violations will be discussed in part III of this book.
Ramsey’s paper led to the understanding that human beings not only exhibit a subjective view of money, as epitomized by its diminishing marginal utility, but also assess probability in a subjective manner, as expressed by a person’s intensity of belief. And this person’s intensity of belief is expressed, according to Ramsey, by that person’s willingness to act based on that belief.
However, confusion about the nature of probability continued to reign, with an abundance of interpretations. It took several more years for the commotion to settle down. In the early 1930s, the Russian mathematician Andrei Kolmogorov developed a theory of probability that was not subjective, like Ramsey’s, but mathematically rigorous and objective. In the Foundations of the Theory of Probability, published in 1933, Kolmogorov stipulated three axioms for the measure of probability: nonnegativity (probabilities are always equal to or greater than zero), normalization (something is bound to happen), and finite additivity (the probability that one of several nonoverlapping events occurs is found by summing the individual probabilities).15
Kolmogorov’s theory was dry, bare of emotion. But humans do not conform to that ideal—if it even is an ideal. Frank Ramsey recognized that psychology must come to bear on the study of probability. But even he stipulated that humans, with all their frailty, must be rational and consistent in their personal beliefs. The fact that they are not will be the subject matter of part III of this book.
