🖐 Petersburg Paradox

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Petersburg paradox is not a paradox because no formal contradiction is derived. The standard version of the St. If he had, he would have discovered that the expected value of the first series 1, 2, 4, 8, 16, etc. Petersburg game is not finite, but the actual outcome will always be finite. We can leave it open exactly what the prizes consists of. A fair coin is flipped until it comes up heads. Bernoulli, 15 November However, he never performed any calculations. In the St. For a discussion of the distinction between actual and potential infinities, see Linnebo and Shapiro In discussions of the St. However, to claim that a rational agent should pay millions, or even billions, for playing this game seems absurd. Bernoulli to Montmort, 9 September Montmort responded that these problems. Montmort to N. If one wishes to suppose that the moral value of goods was as the square root of the mathematical quantities … my moral expectation will be. No actual infinities are required for constructing the paradox, only potential ones. Even so, if the coin keeps landing tails every time it is flipped, the agent wins 0 units of utility. Petersburg game the monetary values of the outcomes and their probabilities are easy to determine. Petersburg game is finite. We will discuss this distinction in Section 2.

The St. Decision theorists advise us to apply the principle of st petersburg paradox casino expected value. As we will see in the next couple of sections, many scholars argue that the value of the St. Many discussions of the St. The paradox consists in this that the calculation gives for the equivalent that A must give to B an infinite sum, which would seem absurd.

Petersburg game to be about 2. Petersburg game, even though it is almost certain that the player will win a very modest amount. Now this sum, if I reason as a sensible man, st petersburg paradox casino not more for me, does not make more pleasure for me, does not engage me more to accept the game, than if it would be only 10 or 20 million coins.

Petersburg paradox is derived from the St. James M. Petersburg game is infinite read: not finite. The paradox can be restored by increasing the values of the outcomes up to the point at which murphy casino agent is fully compensated for her decreasing marginal utility of money see Menger [].

How much money the bank has in the vault when the player plays the game is irrelevant. In a strict logical sense, the St. Although for the most part these problems are not difficult, you will find however something most curious. It would thus be a mistake to dismiss the paradox by arguing that no actual prizes can have infinite utility.

According to this principle, the value of an uncertain prospect is the sum total obtained by multiplying the value of each possible outcome with its probability and then adding up all the terms see the entry on normative theories of rational choice: expected utility.

This is important because, as noted in section 2, the amount the player actually wins will always be finite.

The player thus knows that paying more than what one actually wins cannot be the best means to the end of maximizing utility. Similar objections were raised in the eighteenth century by Buffon and Fontaine see Dutka What is wrong with evaluating a highly idealized game we have little reason to believe we will ever get to play? Petersburg game have infinite value. This is absurd given that we are confining our attention to bettors who value wagers only as means to the end of increasing their fortune. So 2 would still hold true. Joyce However, this seems to presuppose that actual infinities do exist. No matter how many times the coin is flipped, the player will always win some finite amount of utility. Petersburg game is. To this we have to add the aggregated value of the first m possible outcomes, which is obviously finite. Therefore, the expected monetary value of the St. Daniel Bernoulli proposed a very similar idea in his famous article mentioned at the beginning of this section. It need not be money. This is the first clear statement of what contemporary decision theorists and economists refer to as decreasing marginal utility: The additional utility of more money is never zero, but the richer you are, the less you gain by increasing your wealth further. The St Petersburg game can be regarded as the limit of a sequence of truncated St Petersburg games, with successively higher finite truncation points—for example, the game is called off if heads is not reached by the tenth toss; by the eleventh toss; by the twelveth toss;…. All that matters is that the bank can make a credible promise to the player that the correct amount will be made available within a reasonable period of time after the flipping has been completed. It is also worth keeping in mind that the St. If so, we could perhaps interpret Joyce as reminding us that no matter what finite amount the player actually wins, the expect utility will always be higher, meaning that it would have been rational to pay even more. They offer the following argument for accepting 1 :. Petersburg game may not be as unrealistic as Jeffrey claims. However, the mathematical argument presented by Nicolaus himself was also a bit sketchy and would not impress contemporary mathematicians. Some authors claim that the St. The expected utility of the St. Petersburg paradox have focused on 1. Petersburg game should be dismissed because it rests on assumptions that can never be fulfilled. Petersburg paradox discussed in the modern literature can thus be formulated as follows:. Bernoulli to Montmort, 20 February In order to render the case more simple I will suppose that A throw in the air a piece of money, B undertakes to give him a coin, if the side of Heads falls on the first toss, 2, if it is only the second, 4, if it is the 3rd toss, 8, if it is the 4th toss, etc. So it seems that we, at the very least, have a counterexample to the principle of maximizing expected value. If rationality forces us to liquidate all our assets for a single opportunity to play the St. The good news is that his conclusion was correct:. Petersburg game, then it seems unappealing to be rational. However, modern decision theorists agree that this solution is too narrow. Petersburg paradox was introduced by Nicolaus Bernoulli in It continues to be a reliable source for new puzzles and insights in decision theory. Some authors have discussed exactly what is problematic with the claim that the expected utility of the modified St. How much should one be willing to pay for playing this game? For a discussion of this possibility, see Williamson Some events that have probability 0 do actually occur, and in uncountable probability spaces it is impossible that all outcomes have a probability greater than 0. Daniel Bernoulli [ 33]. The version of the St. In the final version of the text, Daniel openly acknowledged this:. The least controversial claim is perhaps 2. However, he pointed out that his solution works even if he the value of money is strictly increasing but the relative increase gets smaller and smaller 21 May :. Petersburg game, which is played as follows: A fair coin is flipped until it comes up heads the first time. Any nonzero probability times infinity equals infinity, so any option in which you get to play the St. The fact that the bank does not have an indefinite amount of money or other assets available before the coin is flipped should not be a problem. It is worth stressing that none of the prizes in the St. Joyce notes that. Thus we have a principled reason for accepting that it is worth paying any finite amount to play the St Petersburg game. Some would say that the sum approaches infinity, not that it is infinite. Petersburg paradox. Bernoulli Nicolaus asked de Montmort to imagine an example in which an ordinary dice is rolled until a 6 comes up:. Petersburg game with a nonzero probability has infinite expected utility. Petersburg game is, for one reason or another, finite. Bernoulli, 21 May That which renders the mathematical expectation infinite, is the prodigious sum that I am able to receive, if the side of Heads falls only very late, the th or th toss. It is, of course, logically possible that the coin keeps landing tails every time it is flipped, even though an infinite sequence of tails has probability 0.