Finite St. Petersburg lotteries
The classical St. Petersburg lottery assumes that the casino has infinite resources. This assumption is often criticized as unrealistic, particularly in connection with the paradox, which involves the reactions of ordinary people to the lottery. Of course, the resources of an actual casino (or any other potential backer of the lottery) are finite. More importantly, the expected value of the lottery only grows logarithmically with the resources of the casino. As a result, the expected value of the lottery, even when played against a casino with the largest resources realistically conceivable, is quite modest. If the total resources (or total maximum jackpot) of the casino are W dollars, then L = 1 + floor(log2(W)) is the maximum number of times the casino can play before it no longer covers the next bet. The expected value E of the lottery then becomes:
The following table shows the expected value E of the game with various potential bankers and their bankroll W (with the assumption that if you win more than the bankroll you will be paid what the bank has):
Banker Bankroll Expected value of lottery
Friendly game $100 $4.28
Millionaire $1,000,000 $10.95
Billionaire $1,000,000,000 $15.93
Bill Gates (2008) $58,000,000,000 $18.84
U.S. GDP (2007) $13.8 trillion $22.78
World GDP (2007) $54.3 trillion $23.77
Googolaire $10100 $166.50
Notes: The estimated net worth of Bill Gates is from Forbes. The GDP data are as estimated for 2007 by the International Monetary Fund, where one trillion dollars equals $1012 (one million times one million dollars). A “googolaire” is a hypothetical person worth a googol dollars ($10100).
A rational person might not find the lottery worth even the modest amounts in the above table, suggesting that the naive decision model of the expected return causes essentially the same problems as for the infinite lottery. Even so, the possible discrepancy between theory and reality is far less dramatic.