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Is there a solution to the St. Petersburg paradox?

June 19, 2015

It was a puzzle first posed by the Swiss mathematician, Nicolas Bernoulli, in a letter to Pierre Raymond de Montmort, on Sept. 9, 1713, and published in the Commentaries of the Imperial Academy of Science of St. Petersburg. Mercifully it is simple to state. Less mercifully, it is a nightmare to solve. To state the paradox, imagine tossing a coin until it lands heads-up, and suppose that the payoff grows exponentially according to the number of tosses you make. If the coin lands heads-up on the first toss, then the payoff is £2; if it lands heads-up on the second toss, the payoff is £4; if it takes three tosses, the payoff is £8; and so forth, ad infinitum. Now the odds of the game ending on the first toss is ½; of it ending on the second toss is (1/2)^2 = ¼; on the third, (1/2)^3 = 1/8, etc., so your expected win from playing the game = (1/2 x £2) + (1/4 x £4 + 1/8 x £8) + …, i.e. £1 + £1 + £1 … = infinity. It follows that you should be willing to pay any finite amount for the privilege of playing this game. Yet it seems irrational to pay very much at all. So what is the solution? There have been very many attempts at a solution over the years, some more satisfying than others, but none totally so. For the best attempt to date, I think we should go back to 1923, and the classic Moritz explanation, offered by R.E. Moritz, writing in the American Mathematical Monthly. “The mathematical expectation of one chance out of a thousand to secure a billion dollars is a million dollars, but this does not mean that anyone in his senses would pay a million dollars for a single chance of winning a billion dollars”. Of the more recent attempts, I like best that offered by Benjamin Hayden and Michael Platt, in 2009. “Subjects … evaluate [the gamble] … as if they were taking the median rather than the mean of the payoff distribution … [so] this classic paradox has a straightforward explanation rooted in the use of a statistical heuristic.” This lies between one and two pounds, in the earlier specification. Surprisingly, though, there is still no real consensus on the solution to this puzzler of more than three hundred years vintage. If and when we do finally solve it, we will have made a giant step toward establishing a more complete and precise understanding of the meaning of rationality and the working of the human economic mind. Care to try?

Further Reading and Links

Moritz, R. E. (1923). Some curious fallacies in the study of probability. The American Mathematical Monthly, 30, 58–65. Hayden B. and Platt, M. (2009). The mean, the median, and the St. Petersburg Paradox. Judgment and Decision Making, 4, no. 4, June, 256-272. Link:

From → Economics, Puzzles

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