Plate v in print

The Wager

expected utility; cardinality returns, but only as a shadow cast by choice among risks.

A choice among chances can give utility a scale without giving pleasure a measure.

— the argument of this plate, after Bernoulli and von Neumann–Morgenstern

A coin is tossed until the first head appears. A head on the first toss pays one ducat; a head on the second pays two; a head on the third pays four, and the prize continues to double with each further toss. The chance of each prize halves as its size doubles. Every possible stopping point therefore contributes half a ducat to the expected payment, and there are infinitely many stopping points. The expected payment is infinite. This is the St. Petersburg game in the form Daniel Bernoulli examined in 1738. Its difficulty is not a mistake in adding the prizes. It is the passage from that correct sum to a claim about what someone ought to sacrifice for admission. A person with a finite fortune must decide what surrendering a definite part of it means, while the expectation is sustained by prizes of an indefinitely increasing size. The arithmetic of the payments has left the value of taking the chance unsettled.1

1. Daniel Bernoulli, “Exposition of a New Theory on the Measurement of Risk” (1738), trans. Louise Sommer, Econometrica 22 (1954), §§17–19, pp. 31–33, with Karl Menger's notes 9–10. For a first head on toss n, probability is 2−n and the prize is 2n−1. Their product is ½. This is an idealized promise with no finite prize ceiling, not a claim about an institution capable of honoring it.

Bernoulli proposed applying the probabilities to the utility of wealth. In his logarithmic specification, an additional ducat contributes less when it is added to a larger fortune. Consider a person whose final wealth would be fifty or a hundred and fifty with equal probability. The expected wealth is a hundred. The average of the logarithms corresponds to certain wealth of about eighty-six and six-tenths: the square root of fifty times a hundred and fifty. On that specification, the difference of about thirteen and four-tenths is the premium the person would surrender to replace the gamble with its mean for certain. The utility average and the monetary equivalent are different quantities, connected by the chosen utility function. Buying a gamble requires a further step because its price reduces wealth in every outcome. Bernoulli explicitly distinguished the value of already possessing the St. Petersburg prospect from the amount someone would pay to acquire it. A finite expected utility does not, by itself, name a ticket price. The price must be put inside the wealth calculation and compared with keeping the money.2

2. Bernoulli, §§3–13, pp. 24–29, and §19, pp. 32–33. With u(w) = log w, the stated certainty equivalent is exp[½ log 50 + ½ log 150] = √7500 ≈ 86.603; the risk premium is 100 − √7500 ≈ 13.397. For initial wealth a, admission price c and the original game, indifference instead requires Σn≥12−n log(a − c + 2n−1) = log a, with positive wealth in every outcome. This is a preference specification, not a measured universal law.

The later achievement of von Neumann and Morgenstern was to ask what structure in preferences could justify such an arithmetic without first measuring pleasure. Consider choices among lotteries with stated probabilities over a finite set of consequences. The chooser must be able to order the prospects consistently; an intermediate prospect must be comparable with suitable mixtures of better and worse ones; and combining chances must respect their probabilities. In the expected-utility representation, a ranking also survives mixing each of two alternatives with the same third alternative in the same proportion. That last restriction is consequential: the common part of a gamble cannot change which of the differing parts is preferred. Under the appropriate ordering, continuity and mixture conditions, numbers can be assigned to the consequences so that each lottery is ranked by its probability-weighted average. The theorem makes an average legitimate for the represented preferences. It neither requires everyone to value money linearly nor discovers a quantity of feeling by inspecting the numbers printed on a prize.3

3. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (first published 1944), ch. I, §3, especially §§3.3–3.7, pp. 17–29 in the linked chapter. Their axioms organize ordering and the combination of alternatives, including reduction of successive mixtures. The common-mixture restriction stated here is the independence property of the resulting expected-utility representation. The linked excerpt does not identify its printing.
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A small construction shows what kind of scale has returned. Suppose three final outcomes are a receipt of nothing, forty, or a hundred. Normalize the best outcome's utility to one and the worst outcome's to zero. If the chooser is indifferent between forty for certain and a one-quarter chance of a hundred, with nothing otherwise, forty receives a utility of one-quarter. The number comes from the probability at which those prospects are judged equivalent. It need not equal forty per cent of the utility of a hundred, and it says nothing about whether another chooser would accept the same exchange. Once the mixture conditions hold, the resulting numbers have more structure than the ranks of certain outcomes alone. Replacing every utility by twice itself plus seven preserves all lottery comparisons: averaging adds the same seven to every prospect and doubles each difference. The unit and origin remain free, but the relative intervals are constrained by choices among chances.4

4. The 0/40/100 construction is illustrative. Its normalization gives u(0) = 0, u(40) = ¼ and u(100) = 1 if the stated indifference holds. Von Neumann and Morgenstern, §§3.3–3.6, pp. 17–27, explain probability mixtures and uniqueness up to a positive affine transformation. For v = a + bu, b > 0, expected v equals a + b times expected u because the probabilities sum to one.

An arbitrary increasing relabeling no longer preserves that arithmetic. Suppose three consequences have utility zero, one and two. A half chance of the best and a half chance of the worst then ties the middle consequence for certain. Squaring the three numbers keeps their order as individual consequences: zero remains below one, and one below four. But an average formed from the squared consequence values now assigns two to the gamble and one to the certain middle. It breaks the original indifference. One can still square the expected-utility score of each whole lottery and preserve the ranking of lotteries; what one cannot generally do is move that nonlinear transformation inside the average. This is the extra content of the scale. Risk has supplied comparisons among mixtures that a mere ordering of sure consequences did not supply. Calling the representation cardinal identifies that restricted freedom of transformation. It does not establish that a unit on the scale is a unit of sensation.

The representation also keeps several questions apart that a single word, utility, can conceal. For monetary outcomes, a concave utility function makes a fair spread in wealth unattractive and gives the certainty equivalent illustrated above. Expected-utility theory itself permits other curvatures; aversion to risk is an additional property of the preferences represented. A comparison between persons requires still more. Each person's scale can be assigned its own positive unit and origin, so one person's gain of ten index units cannot simply be placed against another's loss of five. Nor does consistency among gambles decide whether the represented choices advance the chooser's health, freedom or happiness. Those may be consequences he values, or criteria by which someone evaluates the choices, but they do not follow from the mixture theorem. The wager offers a disciplined account of how one person's uncertainty is ordered. The account becomes a welfare judgment only when the further grounds for that judgment are supplied.5

5. Von Neumann and Morgenstern, §3.7, pp. 28–29, expressly distinguish their single-person construction from comparisons between persons and discuss what the mixture axioms exclude about a separate utility of gambling. Bernoulli supplies one concave wealth specification; concavity is not a requirement of the general representation. The distinction between ordering choices and judging their value connects this plate to The Comparison and The Capability.

Cardinality has therefore returned by a route different from the original search for a substance of pleasure. The probabilities are visible features of the offered gamble, and the indifferences give the scale its discipline. Whether a particular person obeys that discipline is an empirical question; whether every departure should be corrected is a further question about the principles of choice. A theorem can be exact while its assumptions fail to describe a decision. That possibility matters most where the assumptions seem least burdensome: when a common consequence is removed from both options, or when a known chance is set beside an uncertain one. The next plate, The Paradox, takes the apparent self-evidence of the rules into those comparisons. The wager has given want a scale again. The choices that sustain it must now be asked whether they will keep it.