糖心原创

Paradox Lost: The St. Petersburg Gamble Revisited

In this paper I show that appropriate attention to the geometric distribution of the number of successes before a failure in Bernoulli’s famous St. Petersburg  gamble implies that the monetary winnings one should expect from a single play of the St. Petersburg gamble is $2, and not an arbitrarily large payoff.  Interestingly, the argument and calculation that Bernoulli made to show that, using a (base 2) logarithmic function of wealth, the gamble should be seen to be worth 2 in monetary terms, is exactly the calculation needed (but for a completely different reason) to show that the expected number of successes in playing the gamble is 1, and thus that the “expected” value of the gamble is 2.  That is, one should expect to have one success before having a failure, and thus one should expect to get the prize, $2, associated with one success.  More generally, I show that the expected payoff Bernoulli identified for the monetary gamble is applicable only in the sense that the average payoff as the number of plays of the gamble increases also increases. Technically speaking, the limit defined by the expected value does not exist.  It is not infinite.  Rather, it is simply divergent, meaning, in our context, that for any finite number one might choose, there is some number of plays of the gamble that one can expect to yield an average payoff higher than that finite number. I argue that a deeper appreciation for the physical nature of dynamic situations such as the St. Petersburg Gamble should precede conjectures about what sort of preferences might account for the behavior one observes.