The St Petersburg Paradox

  The St. Petersburg paradox is presented in Daniel Bernoulli, Exposition of a New Theory on the Measurement of Risk (1738) found in Econometrica, Jan 1954 (see §17 on p. 31). The problem was not first formulated by him and is similar to the wheat and chessboard problem. A coin is repeatedly tossed until a tails first appears. If this occurs on the first toss the player gets 1 coin, on the second he get 2 coins, on the 3rd: 4 coins, etc., with the number of coins increasing exponentially. What is the expected number of coins that he will get for playing this game? Surprisingly the number is infinite.

Instead of allowing the number of tosses go on indefinitely suppose that it is agreed before the start that the player stops on the nth toss. The expected value for the game is then,

Additionally, one can offer a bonus, b, for getting n heads in a row.  If each player has to make a wager, w, to play then the net gain for the kth trial is 2^k-1 - w and for n heads in a row it is b - w. The expected value then becomes,

For a fair game the expected value is zero and the expression for the wager is simplified if the bonus is 2ⁿ so,

If one sets the limit at 8 tosses of the coin one does not win anything until one gets four tosses in a row. The maximum that one can win is 251 coins for 8 heads in a row.

The paradox is that the expected winnings is infinite for an unlimited number of tosses. Experience will show that the usual outcome for playing is quite low. When the limit on the number of tosses is not too large, the wager is affordable but with larger wagers one can expect to lose more often. The game is one of survival.

Suppose we consider a game of losses instead of winnings. Experience might show that something is profitable based on a limited amount of experience but there are cases where the risks can snowball.