What is your favorite paradox? You mean you don’t have one? I thought everybody had a favorite paradox…
In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a tail first appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc. In short, you win 2k−1 dollars if the coin is tossed k times until the first tail appears.
What would be a fair price to pay for entering the game? To answer this we need to consider what would be the average payout: With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. The expected value is thus
This sum diverges to infinity, and so the expected win for the player of this game, at least in its idealized form, in which the casino has unlimited resources, is an infinite amount of money. This means that the player should almost surely come out ahead in the long run, no matter how much he pays to enter; while a large payoff comes along very rarely, when it eventually does it will typically be far more than the amount of money that he has already paid to play. According to the usual treatment of deciding when it is advantageous and therefore rational to play, one should therefore play the game at any price if offered the opportunity.
Pretty straightforward. The paradox is that, even though the expected win of this “game” is infinite, nobody in their right mind would give more than a few dollars to play.
There are various solutions given for this game, though one stands out in my mind: Nobody has an infinite amount of money!
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 backers and their bankroll W (with the assumption that if you win more than the bankroll you will be paid what the bank has):
Backer 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 $22.79 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 $10^12 (one million times one million dollars). A “googolaire” is a hypothetical person worth a googol dollars ($10^100).
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.
If you have your own “favorite” paradox, share them with me by posting a comment below.