## Thursday, 7 August 2008

### The Saint Petersburg Paradox

The St. Petersburg game is played by flipping a fair coin until it comes up tails, and the total number of flips, n, determines the prize, which equals \$2^n. Thus if the coin comes up tails the first time, the prize is \$2^1 = \$2, and the game ends. If the coin comes up heads the first time, it is flipped again. If it comes up tails the second time, the prize is \$2^2 = \$4, and the game ends. If it comes up heads the second time, it is flipped again. And so on. There are an infinite number of possible 'consequences' (runs of heads followed by one tail) possible. The probability of a consequence of n flips (P(n)) is 1 divided by 2^n, and the 'expected payoff' of each consequence is the prize times its probability. The following table lists these figures for the consequences where n = 1 ... 10:

 n P(n) Prize Expected  payoff 1 1/2 \$2 \$1 2 1/4 \$4 \$1 3 1/8 \$8 \$1 4 1/16 \$16 \$1 5 1/32 \$32 \$1 6 1/64 \$64 \$1 7 1/128 \$128 \$1 8 1/256 \$256 \$1 9 1/512 \$512 \$1 10 1/1024 \$1024 \$1

The 'expected value' of the game is the sum of the expected payoffs of all the consequences. Since the expected payoff of each possible consequence is \$1, and there are an infinite number of them, this sum is an infinite number of dollars. A rational gambler would enter a game iff the price of entry was less than the expected value. In the St. Petersburg game, any finite price of entry is smaller than the expected value of the game. Thus, the rational gambler would play no matter how large the finite entry price was. But it seems obvious that some prices are too high for a rational agent to pay to play. Many commentators agree with Hacking's (1980) estimation that “few of us would pay even \$25 to enter such a game.” If this is correct—and if most of us are rational—then something has gone wrong with the standard decision-theory calculations of expected value above. This problem, discovered by the Swiss eighteenth-century mathematician Daniel Bernoulli is the St. Petersburg paradox. It's called that because it was first published by Bernoulli in the St. Petersburg Academy Proceedings (1738; English trans. 1954).