Санкт-Петербург парадоксы: Ақша тастау ойынындағы қайшылық

Paradox involving a game with repeated coin flipping

The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions to the paradox have been proposed, including the impossible amount of money a casino would need to continue the game indefinitely. The problem was invented by Nicolas Bernoulli, who stated it in a letter to Pierre Raymond de Montmort on September 9, 1713. However, the paradox takes its name from its analysis by Nicolas' cousin Daniel Bernoulli, one time resident of Saint Petersburg, who in 1738 published his thoughts about the problem in the Commentaries of the Imperial Academy of Science of Saint Petersburg.

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time tails appears. The first time heads appears, the game ends and the player wins whatever is the current stake. Thus the player wins 2 dollars if heads appears on the first toss, 4 dollars if tails appears on the first toss and heads on the second, 8 dollars if tails appears on the first two tosses and heads on the third, and so on. Mathematically, the player wins dollars, where is the number of consecutive tails tosses. The apparent paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value. that a square root function describing the diminishing marginal benefit of gains can resolve the problem. However, unlike Daniel Bernoulli, he did not consider the total wealth of a person, but only the gain by the lottery. This solution by Cramer and Bernoulli, however, is not completely satisfying, as the lottery can easily be changed in a way such that the paradox reappears. To this aim, we just need to change the game so that it gives even more rapidly increasing payoffs. For any unbounded utility function, one can find a lottery that allows for a variant of the St. Petersburg paradox, as was first pointed out by Menger. Recently, expected utility theory has been extended to arrive at more behavioral decision models. In some of these new theories, as in cumulative prospect theory, the St. Petersburg paradox again appears in certain cases, even when the utility function is concave, but not if it is bounded.

Buffon Keynes, in particular, insisted that the relative risk of an alternative could be sufficiently high to reject it even if its expectation were enormous.

An early resolution containing the essential mathematical arguments assuming multiplicative dynamics was put forward in 1870 by William Allen Whitworth. An explicit link to the ergodicity problem was made by Peters in 2011. These solutions are mathematically similar to using the Kelly criterion or logarithmic utility. General dynamics beyond the purely multiplicative case can correspond to non logarithmic utility functions, as was pointed out by Carr and Cherubini in 2020.

Although this paradox is three centuries old, new arguments have still been introduced in recent years.

A solution involving sampling was offered by William Feller. Intuitively Feller's answer is "to perform this game with a large number of people and calculate the expected value from the sample extraction". In this method, when the games of infinite number of times are possible, the expected value will be infinity, and in the case of finite, the expected value will be a much smaller value.

Paul Samuelson resolves the paradox by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected loss to the host. No one could be observed paying to play the game because it would never be offered. As Samuelson summarized the argument, "Paul will never be willing to give as much as Peter will demand for such a contract; and hence the indicated activity will take place at the equilibrium level of zero intensity."

Many variants of the St Petersburg game are proposed to counter proposed solutions to the game. let be the number of coin flips; if is odd, the player gains units of ; else the player loses units of utility. The expected utility from the game is then However, since the sum is not absolutely convergent, it may be rearranged to sum to any number, including positive or negative infinity. This suggests that the expected utility of the Pasadena game depends on the summation order, but standard decision theory does not provide a principled way to choose a summation order.