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The St. Petersburg paradox or St. Petersburg lottery〔''Conceptual foundations of risk theory''. By Michael D. Weiss, United States. Dept. of Agriculture. Economic Research Service. (p 36 )〕 is a paradox related to probability and decision theory in economics. It is based on a particular (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. The St. 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. Several resolutions are possible. The paradox takes its name from its resolution 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'' . However, the problem was invented by Daniel's brother Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713 .〔Eves, Howard. ''An Introduction To The History of Mathematics'' (Sixth ed.). Brooks/Cole - Thomson Learning, 1990, p. 427.〕 == The paradox == A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, 16 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2''k'' dollars, where ''k'' equals number of tosses (k must be a whole number and greater than zero). What would be a fair price to pay the casino for entering the game? To answer this, one needs to consider what would be the average payout: with probability 1/2, the player wins 2 dollars; with probability 1/4 the player wins 4 dollars; with probability 1/8 the player wins 8 dollars, and so on. The expected value is thus : :: :: Assuming the game can continue as long as the coin toss results in heads and in particular that the casino has unlimited resources, this sum grows without bound and so the expected win for repeated play is an infinite amount of money. Considering nothing but the expected value of the net change in one's monetary wealth, one should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of the game, many people expressed disbelief in the result. Martin quotes Ian Hacking as saying "few of us would pay even $25 to enter such a game" and says most commentators would agree.〔.〕 The paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「St. Petersburg paradox」の詳細全文を読む スポンサード リンク
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