The St. Petersburg Paradox

Minutes

President Collins called the 2144th meeting to order at 8:15 p.m. on March 22, 2002. The Recording Secretary read the minutes of the 2143rd meeting and they were approved with one correction. The speaker for the evening was Klaus P. Heiss, Consulting Mathematician. The title of his presentation was “The St. Petersburg Paradox”. Another proposed scientific name for man, alluding to a characteristic behavior, is Homo ludens, man who plays. Many aspects of human behavior can be related to how the mind goes about evaluating strategies, risks, rewards and losses for games and for survival. Within 265 msec of experiencing a gambling loss, there is a response in the centers of the brain responsible for aggression. This aggressive response in many gamblers leads to a higher probability of making a larger wager in the next bet. Studying how humans play games may provide a basis for a deeper understanding of rational behavior. After the Seventeenth Century, probability theory developed largely through the impetus of gambling, and its potential for providing an analysis of reasonable behavior when participating in games. In the Twentieth Century, the mathematical theory of games began with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. The theory of games has recently been brought back into pop-cultural prominence through the movie A Beautiful Mind, based on the life of John Nash winner of the 1994 Nobel Prize for Economics for his analysis of equilibrium states in certain types of games, generalizing the von Neumann-Morgenstern minimax theorem for zero sum games to non-zero sum games, which constitute the vast majority of social conflicts and economic interactions. In 1738, the mathematician Daniel Bernoulli published a paper on a gambling problem in which mathematical analysis suggests a behavior that is not followed by most, otherwise rational, gamblers. The game goes as follows: for an agreed fixed bet, a fair coin is tossed until it comes up tails. An initial tails loses the bet, and a heads wins a prize, say $1, which is doubled for each consecutive toss of heads. St. Petersburg Paradox Game consecutive heads 1 2 3 4 …n winnings $1 $2 $4 $8 $1×2n-1 probability 1/2 1/4 1/8 1/16 2-n expected payoff $0.50 $0.50 $0.50 $0.50 $0.50 The “expected value” of a game for a bettor is the sum of the expected payoffs for all possible outcomes. Since the expected payoff of this game is constant, and there are an infinite number of outcomes, the expected value of the game is infinite. By this analysis, a rational gambler should be willing to wager any amount to play the game, assuming the “house” could make an infinite payoff. The paradox is that few reasonable people would enter this game for more than some relatively modest bet. A number of explanations have been advanced for this behavior. One explanation for the paradoxical behavior, proposed by Karl Menger with Wald, Gödel and Morgenstern (“the Vienna Circle”) in 1934, is based on “the marginal utility of money”. As the potential winnings increase without limit, the player apparently acts as though there were not a linear increase in its value, but diminishing incremental value, a flatter curve of expectation. At the same time, as the potential loss of the fixed bet increases without limit, the player acts as though there were not a linear decrease in value, but a steeply increasing incremental loss. Human players act as though they overestimate the probability of a large loss, or of the value of its cost to them, and underestimate the probability of large gains, or of the value of the reward. In some respects this game is the reverse of a lottery in which the perceived cost of improving the probability of winning (buying more tickets) increases rapidly, while the perceived gain in winning levels off. It has also been pointed out that this curve of marginal utility is not the same as the utility curve of “losing other people's money”, as for insurance where a small premium is used to secure a valuable gain. Other analyses have been based on “the utility of risk”, the risk perceived by the gambler is a function of the total wealth at risk, corresponding to the negative part of the marginal utility curve. This may explain the irrational sense of loss experienced by some bettors when someone else wins a lottery. Another important explanation may be the “non-Archimedean property” in certain decisions, an outright discontinuity, that may explain otherwise apparently irrational behaviors. For a lottery-type game, the additive contribution derived from investment in an alternative strategy, such as a regular salary, does not have the expected gain of a lottery win, so the cost of lottery tickets seems a good investment. Or, in strategic matters, no amount of tugboats ever will add up to one nuclear submarine, and so on. Similar utilitarian or discontinuous behaviors explain insurance and such things as the traditional Northwest Indian “potlatch” involving the destruction or squandering of wealth. As an example of “intuition” over “rationality”, would Queen Isabella have given Columbus better ships if she had known what the expected gain of the exploration really was? She gave him the cheapest possible ships, potentially endangering the probability of a successful return, because it was all Isabella was willing to bet. Exploration is a game where we often rise above “rational” thinking despite all the scientific “evidence” to the contrary. Mr. Heiss kindly answered a number of questions from the floor. President Collins thanked Mr. Heiss for the society, and welcomed him to its membership. The President made the announcements about the next meeting, parking, and refreshments, and adjourned the 2143rd meeting to the social hour at 9:34 p.m. Attendance: 43 Temperature: -0.6°C Weather: partly cloudy Links: http://plato.stanford.edu/entries/paradox-stpetersburg/ Respectfully submitted, John S. Garavelli Recording Secretary