The game has stopped, but the money has not ceased to belong to someone. Two players have put their stakes upon the table; neither has yet won the contest; an interruption prevents them from finishing. Returning each man's contribution would erase what the play has already accomplished. Giving everything to the player who leads would treat a possible victory as an accomplished one. Between these answers lies a claim upon events that will never occur. To divide the stakes is to price that claim: to ask what the unfinished future was worth at the moment the door opened.
In the correspondence of Pascal and Fermat, the future becomes divisible because its alternatives can be counted. Pascal considers two equally matched players, each staking thirty-two pistoles, in a contest won by the first to take three rounds. One has won two rounds and the other one. If the leader wins the next round, he takes all sixty-four pistoles. If he loses it, the players stand level and each can fairly claim thirty-two. The leader therefore has thirty-two assured, together with an equal chance of the remaining thirty-two. His share is forty-eight; the other's is sixteen. The score of two to one does not make the division two to one. What matters is the contest still to be played, with its rules and its chances, rather than the proportion of rounds already won.
The operation is simple enough to hide the change it makes. A future payment is multiplied by the chance of receiving it; the products are added; the resulting number stands in for the prospect. In Pascal's example the leader's chance of eventually winning is three-quarters, so his expected receipt is three-quarters of sixty-four. A stake in an unfinished game has become a present quantity of money without anyone pretending to know its outcome. Probability does not remove the uncertainty. It gives the alternatives weights, allowing the uncertainty to enter an account. The apparatus can settle the division because the players have a common rule for translating their remaining chances into shares of the common pot.
A fair price acquires a precise, limited meaning. Imagine a ticket that pays one hundred pounds if a fair coin falls heads and nothing if it falls tails. Its expected payment is fifty pounds. At a price of fifty, the buyer's expected net gain is zero; at forty, it is ten. Calling the fifty-pound price actuarially fair identifies this equality between the price and the probability-weighted payment. It says nothing yet about whether the purchaser should enter, whether the promoter has disclosed the rules, or whether fifty pounds is a trivial expense or the month's food. Arithmetic can describe the terms of an exchange before the exchange becomes a sensible thing for a particular person to make.
The plate's question presses too hard when it asks why no one will pay the expected value. Someone may. A buyer who values the relevant changes in wealth linearly is indifferent between a fifty-pound certainty and the ticket just described, provided the promise is credible and no other costs intervene. Someone with an offsetting exposure may welcome the ticket because it pays precisely when another receipt fails. Even the same person can value the same printed ticket differently after taking on another obligation. The expected payment belongs to the prospect as specified. The price that makes its holder indifferent belongs to the prospect in that holder's circumstances. Confusing those two quantities makes an explanation of choice look like a correction to multiplication.
Repeated play explains why the expected payment nevertheless has such authority. If independent tickets have the same bounded payment rule, the average payment across many tickets becomes increasingly concentrated around their common expectation. A seller conducting many such transactions has a reason to attend to that average. Yet the theorem concerns an average, under stated conditions. It does not promise that the next ticket will pay fifty, that total deviations from the expected sum become smaller, or that a purchaser with limited resources can survive every sequence needed to reach the long run. The person who loses the rent on the first toss does not receive an extension from the law of large numbers. Repetition can change the exposure; the bare possibility of repeating a wager does not establish the means to do so.
The average is a property of the prospect. Paying for it is an event in a life.
The gap becomes impossible to overlook in the game associated with St. Petersburg. Toss a fair coin until the first head. In the version discussed by Daniel Bernoulli, a head on the first toss pays one ducat, on the second two, on the third four, and so on, doubling with every additional toss. The probabilities halve as the payments double. Each possible stopping point contributes half a ducat to the expectation: one times a half, two times a quarter, four times an eighth. There are infinitely many stopping points. The sum has no finite bound. No mistaken probability, hidden commission or unfavorable rule is needed to produce the trouble. The arithmetic recommends an unlimited price for a prospect that ordinarily ends with a small payment.
The extraordinary expectation is sustained by payments almost never observed. Half the games end at one ducat; three-quarters end at no more than two. But the tail cannot simply be discarded because it is remote. The expectation includes it by definition, and ever larger possible payments compensate for their ever smaller probabilities. A finite simulation that fails to encounter those payments has not disproved the sum. Nor does a very large payment in one trial establish that the theoretical promise could actually have been honored. The game separates three questions that ordinary speech often joins: what the mathematical contract promises, what a finite run happens to deliver, and what resources stand behind the promise.
Putting a ceiling on the payment changes the first of those questions. In a constructed version, let the coin continue until heads but cap the payment at 1,024 ducats. The first ten stopping points contribute half a ducat each. All later stopping points together have probability one in 1,024 and pay 1,024, adding one more ducat. The expectation is therefore six ducats. This is a cap on payment, not a rule that pays nothing if the game lasts too long. A finite undertaking can be priced as a finite undertaking; its expectation need not resemble the unbounded game's. Yet a six-ducat expectation still does not force a person to pay six. A credible ceiling repairs the description of the promise. It does not settle what bearing the remaining variation is worth.
Bernoulli's further move was to change what is averaged. A ducat added to a small fortune and a ducat added to an immense one can have different significance to their recipients. Suppose that the utility of positive final wealth is its logarithm. The expected utility of the St. Petersburg prospect can then be finite even though its expected payment is infinite: the utility attached to the growing prizes increases much more slowly than the prizes themselves. The answer uses a hypothesis about the bearer of the gamble. It is not a proof that money has stopped adding up, or that every conceivable gamble becomes harmless once logarithms are introduced. The monetary expectation continues to be what it was. Another quantity now enters the decision.
Even that decision needs an accurately drawn transaction. Owning a prospect and deciding what certain sum would replace it is different from owning cash and deciding how much to spend on acquiring the prospect. In the first case the gamble is already part of one's resources. In the second its price must be deducted before the uncertain payment arrives. Bernoulli expressly distinguishes the two. A price large enough to exhaust the buyer's means cannot be treated as if it left his starting position intact. The distinction is especially important near ruin, where subtracting the admission price may matter more than adding a spectacular possible prize.
The fair division of the interrupted game survives these distinctions. Pascal's forty-eight pistoles remains the probability-weighted share under the agreed rules. What changes is the claim one may make for that number. It supplies an account of the remaining chance, not a complete theory of every bargain that could be struck over it. A player needing cash at once may accept less. Another may already hold a claim that pays when this one loses. Costs, credibility and preferences can enter without making the calculation of the underlying expectation incorrect. To calculate a prospect precisely is to make these additional questions visible, since their effects can no longer be hidden inside an unexplained impression of good or bad fortune.
The odds give risk a language in which disagreements can be located. One person disputes the probabilities; another doubts the promised payment; a third accepts both and declines the price. Only the last disagreement belongs wholly to valuation. The calculus has accomplished something durable when it keeps the three apart. It tells us how much the prospect pays on average, where that average exists, and what assumptions the statement carries. To learn why a person would surrender part of that average for a certainty, we must follow the payment away from the gaming table and into the wealth of the person who receives it.