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. The prize doubles at every step while the chance of reaching that step halves. Each possible stopping point therefore contributes half a ducat to the expected payment, and there are infinitely many stopping points. The expectation is infinite. The difficulty begins when this correct arithmetic is presented to a person with a finite purse and called the price he should pay.
This is the St. Petersburg game in the form Daniel Bernoulli examined in 1738. His answer changed the thing being averaged. Money was to be valued by its utility to the person who would possess it; an increment could matter less when added to a larger fortune. In the logarithmic specification he developed, the indefinitely increasing prizes no longer produced an infinite expectation of utility. The logarithm grows slowly enough for the diminishing probabilities to control the sum. The example separates three quantities that an incautious account can run together: an expected payment, an expected utility and the certain amount of money judged equivalent to the gamble.
The monetary equivalent requires a return journey from the utility scale. Imagine someone with a hundred pounds who is given a ticket paying either nothing or a hundred, each with equal probability. His final wealth will be one hundred or two hundred. With logarithmic utility, the certain wealth that matches the average of the two logarithms is their geometric mean: about one hundred and forty-one pounds and forty-two pence. Relative to his original wealth, the ticket is worth a certain addition of about forty-one pounds and forty-two pence. Its expected cash prize is fifty. The gap has a reason within the example: moving from the mean fortune of one hundred and fifty to two hundred adds less utility than falling from that mean to one hundred removes.
Now ask how much the same person would pay to acquire the ticket. The price comes out of his hundred pounds before either outcome is known. At a price of forty-one pounds and forty-two pence, the two possible final fortunes would be lower than in the gift calculation, and their utility average would fall below that of keeping his original hundred. The largest acceptable price is instead about thirty-eight pounds and twenty pence. Bernoulli expressly distinguished possession of a prospect from paying for it. A utility expectation is not a price waiting to be read off the page. The transaction changes the circumstances under which the expectation must be calculated.
Bernoulli supplied a particular valuation of wealth. Von Neumann and Morgenstern asked a more general question: what relations among choices would permit consequences to be assigned numbers whose probability-weighted average represents the ranking of lotteries? The answer does not begin by measuring pleasure. It begins with alternatives, their probabilities and the person’s ordering of them. Choices must have the appropriate consistency and continuity; the combination of chances must obey the mixture conditions. When these requirements hold, expected utility becomes a representation of those choices. The theorem supplies the arithmetic from the relations, rather than declaring that the arithmetic must describe every person offered a bet.
A small construction shows what the additional information buys. Take three consequences, called worst, middle and best. Assign zero to the worst and one to the best. Suppose the person is indifferent between the middle consequence for certain and a forty-per-cent chance of the best, with the worst otherwise. On the expected-utility scale, the middle consequence receives forty hundredths. The number is pinned down by a comparison among chances. Merely knowing that the middle consequence lies between the other two could not supply it. The probability mixture has become an operation against which the numerical intervals can be tested.
The unit and origin remain free. Multiply all three utilities by two and add seven: the values become seven, seven and eight-tenths, and nine. The same mixture still averages to the same middle value. Every lottery comparison survives because probabilities sum to one and the positive multiplier preserves the direction of each difference. But square the original values before averaging and the middle receives sixteen hundredths, while the forty-per-cent chance of the best still averages to forty hundredths. The original indifference has vanished. An arbitrary increasing relabelling preserves the ranking of the three certain consequences; it need not preserve the expected-utility representation of the lotteries.
This is cardinality with a specified job. The relative intervals must respect the mixture comparisons, but the construction has not established that the person experiences twice as much pleasure at one outcome as at another. Nor can one person’s interval be laid against another’s merely because both have been normalized between zero and one. Each person’s best and worst may be different; even identical consequences need not have identical significance. Von Neumann and Morgenstern kept the single-person construction distinct from interpersonal comparison. Risk restores more structure than a ranking of sure outcomes, while leaving the social sum in the preceding plate still to be justified.
The gamble gives the scale an operation. It does not give the number a sensation.
Aversion to risk is a further property, rather than a synonym for expected utility. For money or wealth, a concave utility function values the average fortune at least as highly as the average utility of the fortunes on either side. Linear utility gives a different answer, and convex utility another. The representation accommodates these possibilities when the requisite choice relations hold. Even within a concave specification, the price of a particular bet depends on the starting wealth and the rest of the consequences. A willingness to pay cannot be interpreted without knowing what paying leaves behind.
The range of the gamble matters as well. A theorem stated for finite lotteries does not, by itself, settle every lottery with unbounded prizes. The St. Petersburg calculation needs the convergence argument that its particular probabilities and logarithmic valuation permit. Change the prizes enough and logarithmic utility can again have an infinite expectation. The lesson is more exact than the reassurance that diminishing utility dissolves infinity. One must specify the outcomes, verify that the expected utility exists in the required sense, and only then use it to compare the alternatives. A finite answer earned for one game cannot be carried into another by changing its name.
The willingness to bet therefore reveals a relationship among alternatives under stated circumstances. A sufficiently rich and orderly pattern can support a numerical representation; one isolated bet leaves many possible preferences and, when chances are uncertain, many possible beliefs. Whether the person actually keeps the required relations must be asked of the choices themselves. Whether he ought to keep them requires a reason for endorsing the restrictions. The Axiom will return to both questions. Before that, The Gulf follows the difficulty that even a successful construction leaves open: how a scale belonging to one life could become a measure between lives.