“St Petersburg paradox”的概念、定义、翻译、参考文献-科学参考

St Petersburg paradox

Statistics

A paradox that was the source of much correspondence among eighteenth-century mathematicians. Originally posed by Nicolaus Bernoulli to Montmort, it became well known as a result of the solution given by Daniel Bernoulli in 1738 in the journal of the St Petersburg Academy.
Bernoulli’s scenario was essentially as follows. Two players, A and B, play the following game. Player A repeatedly tosses a coin, stopping when a head is obtained. If A has to toss the coin k times, then A pays £2^k to B. Bernoulli’s question is ‘How much should B pay A in order to make the game fair?’ The answer is that B must pay to A the average amount that A pays B. Half the time (assuming a fair coin) this will be £2. Half the remaining time it will be £4, and so on. However, $St Petersburg paradox$ Since the required number of tosses has no upper limit, this sum is infinite. Thus, for this game to be fair, B must pay A an infinite amount of money, even though B is certain to receive only a finite amount of money in exchange (and less than £10 on 87.5% of occasions).

Philosophy

Paradox in the theory of probability published by Daniel Bernoulli in 1730 in the Commentarii of the St Petersburg academy. Someone offers you the following opportunity: he will toss a fair coin. If it comes up heads on the first toss he will pay you one dollar; if heads does not appear until the second throw, two dollars, and so on, doubling your winnings each time heads fails to appear on another toss. The game continues until heads is first thrown, when it stops. What is a fair amount for you to pay for this opportunity? The incredible reply is: an infinite amount. For your expectation of gain is given by the series ½+(2×¼)+(4×⅛)+…, which has an infinite sum. This little offer is apparently worth more to you than all the wealth in the world. Yet nobody in their right mind would pay much at all for it. The paradox has been taken to show the incoherence of allowing infinite utilities into decision theory. Once they are allowed, then it is worth staking any finite sum on any indefinitely small chance of an infinite payoff. If it is specified that the St Petersburg game has to stop when, for example, the payoff reaches the size of the National Debt, then the calculation of what you should pay to enter the game remains quite reasonable.

Economics

The observation that experimental subjects will only offer to pay a small fee to enter a coin-tossing game with an expected pay-off that is infinite. Consider being offered the chance to play in a game in which a fair coin is repeatedly tossed until a tail appears. If the first tail appears on the tth toss the pay-off is 2^t−1. The expected pay-off from entering the game is

$E P = \frac{1}{2} 2^{0} + \frac{1}{4} 2^{1} + \frac{1}{8} 2^{2} + … = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + … = \infty .$

The fact that most subjects are willing to pay only a small fee to enter the game can be explained by the diminishing marginal utility of money. There are stronger paradoxes that can be explained only by bounded utility or behavioural economics. See also anomalies.

单词	St Petersburg paradox
释义	St Petersburg paradox Statistics A paradox that was the source of much correspondence among eighteenth-century mathematicians. Originally posed by Nicolaus Bernoulli to Montmort, it became well known as a result of the solution given by Daniel Bernoulli in 1738 in the journal of the St Petersburg Academy. Bernoulli’s scenario was essentially as follows. Two players, A and B, play the following game. Player A repeatedly tosses a coin, stopping when a head is obtained. If A has to toss the coin k times, then A pays £2^k to B. Bernoulli’s question is ‘How much should B pay A in order to make the game fair?’ The answer is that B must pay to A the average amount that A pays B. Half the time (assuming a fair coin) this will be £2. Half the remaining time it will be £4, and so on. However, $St Petersburg paradox$ Since the required number of tosses has no upper limit, this sum is infinite. Thus, for this game to be fair, B must pay A an infinite amount of money, even though B is certain to receive only a finite amount of money in exchange (and less than £10 on 87.5% of occasions). Philosophy Paradox in the theory of probability published by Daniel Bernoulli in 1730 in the Commentarii of the St Petersburg academy. Someone offers you the following opportunity: he will toss a fair coin. If it comes up heads on the first toss he will pay you one dollar; if heads does not appear until the second throw, two dollars, and so on, doubling your winnings each time heads fails to appear on another toss. The game continues until heads is first thrown, when it stops. What is a fair amount for you to pay for this opportunity? The incredible reply is: an infinite amount. For your expectation of gain is given by the series ½+(2×¼)+(4×⅛)+…, which has an infinite sum. This little offer is apparently worth more to you than all the wealth in the world. Yet nobody in their right mind would pay much at all for it. The paradox has been taken to show the incoherence of allowing infinite utilities into decision theory. Once they are allowed, then it is worth staking any finite sum on any indefinitely small chance of an infinite payoff. If it is specified that the St Petersburg game has to stop when, for example, the payoff reaches the size of the National Debt, then the calculation of what you should pay to enter the game remains quite reasonable. Economics The observation that experimental subjects will only offer to pay a small fee to enter a coin-tossing game with an expected pay-off that is infinite. Consider being offered the chance to play in a game in which a fair coin is repeatedly tossed until a tail appears. If the first tail appears on the tth toss the pay-off is 2^t−1. The expected pay-off from entering the game is $E P = \frac{1}{2} 2^{0} + \frac{1}{4} 2^{1} + \frac{1}{8} 2^{2} + … = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + … = \infty .$ The fact that most subjects are willing to pay only a small fee to enter the game can be explained by the diminishing marginal utility of money. There are stronger paradoxes that can be explained only by bounded utility or behavioural economics. See also anomalies.
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