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. However, unlike Daniel Bernoulli, he did not consider the total wealth of a person, but only the gain by the lottery. He demonstrated in a letter to Nicolas Bernoulli that a square root function describing the diminishing marginal benefit of gains can resolve the problem. The mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it. For example, with natural log utility, a millionaire ($1,000,000) should be willing to pay up to $20.88, a person with $1,000 should pay up to $10.95, a person with $2 should borrow $1.35 and pay up to $3.35.īefore Daniel Bernoulli published, in 1728, a mathematician from Geneva, Gabriel Cramer, had already found parts of this idea (also motivated by the St. This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay (specifically, any c that gives a positive change in expected utility). Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. The first time tails appears, the game ends and the player wins whatever is in the pot. The initial stake begins at 2 dollars and is doubled every time heads appears. 3.4 Rejection of mathematical expectationĪ casino offers a game of chance for a single player in which a fair coin is tossed at each stage.However, the problem was invented by Daniel's cousin, Nicolas Bernoulli, who first stated it in a letter to Pierre Raymond de Montmort on Septem( de Montmort 1713). The paradox takes its name from its analysis by Daniel Bernoulli, one-time resident of the eponymous Russian city, who published his arguments in the Commentaries of the Imperial Academy of Science of Saint Petersburg ( Bernoulli 1738).
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Several resolutions to the paradox have been proposed. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take.
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It is based on a theoretical lottery game that leads to a random variable with infinite expected value (i.e., infinite expected payoff) but nevertheless seems to be worth only a very small amount to the participants. Petersburg lottery is a paradox related to probability and decision theory in economics. Petersburg paradox is typically framed in terms of gambles on the outcome of fair coin tosses.