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St. Petersburg, Menger and slippery infinities
October 16, 2011Posted by on
In the twentieth century, St. Petersburg became Petrograd, then Leningrad and finally went back to being St. Petersburg. The St. Petersburg paradox named after this city also seems to have been running around in circles during the last three centuries. The latest round in this long standing paradox has been initiated by a mathematics professor who is coincidentally named Peters. Way back in 1934, Menger proved that a generalized version of the St. Petersburg paradox invalidates all unbounded utility functions. Prominent economists like Arrow and Samuelson have accepted this conclusion. In a paper entitled Menger 1934 revisited, Peters argues that Menger made an error that has remained undiscovered during the last 77 years.
The original version of the St. Petersburg game involved a fair coin being tossed until the first time a head appears. If this happens at the n‘th toss of the coin, the payoff of the game is 2n–1. The probability of this event is 2-n and therefore this event contributes 2n–1 2-n = 1/2 to the expected payoff of the game. Summing over all n yields 1/2 + 1/2 + … and the expected payoff from the game is therefore infinite.
Bernoulli’s 1738 paper from which the paradox obtained its name argued that nobody would pay an infinite price for the privilege of playing this game. He proposed that instead of expected monetary value, one must use expected utility. If utility of wealth is logarithmic in wealth, then the expected utility from playing the game is not only finite, but is also quite small.
Menger’s contribution was to consider a Super St. Petersburg game in which the payoff was not 2n–1 but exp(2n–1). Essentially, taking logarithms of this payoff to compute utility yields something similar to the payoff of the original St. Petersburg game, and the offending infinity reappears. Menger’s solution to this generalized paradox was to require that utility functions must be bounded. In this case, there is no monetary payoff that yields very high utilities like 2n–1 for sufficiently large n.
Peters argues that there is an error in Menger’s argument. The logarithmic function diverges at both ends — for large x, ln(x) goes to infinity, but for small x (approaching zero), ln(x) goes to minus infinity. Suppose a player pays a large price (close to his current wealth) for playing the Super St. Petersburg game. Now if heads comes quickly, the players’s wealth will be nearly zero and the utility would approach minus infinity. The crux of the Peters’ paper is the assertion: “Menger’s game produces a case of competing infinities. … the diverging expectation value of the utility change resulting from the payout is dominated by the negatively diverging utility change from the purchase of the ticket.” Therefore, the ticket price that a person would pay for being allowed to play this game is finite.
I agree with Peters that even for the Super St. Petersburg game, a person would pay only a finite ticket price if the utility function is logarithmic or is of any other type that has a subsistence threshold below which there is infinite disutility. It appears to me however that a slight reformulation reintroduces the paradox. If we do not ask what ticket price a person would pay, but what sure reward a person would forego in order to play this game, the infinite disutility of the ticket price is kept out of the picture, and the infinite utility of the payoff remains. In other words, the certainty equivalent of the Super St. Petersburg game is infinite. Peters is right that a person with logarithmic utility would not pay a trillion dollars to play the game, but Menger is right that such a person would prefer playing the Super St. Petersburg game to receiving a sure reward of a trillion dollars. Peters’ contribution is to make us recognize that these are two very different questions when there is a “competing infinity” at the other end to contend with. But Menger is right that if you really want to exorcise this paradox, you must rule out the diverging positive infinity by insisting that utility functions should be bounded.
Peters also makes a very different argument by bringing the time dimension into play. He argues that the way to deal with the paradox is to use the Kelly criterion which brings us back to logarithmic functions. Peters relates this to the distinction between time averages and ensemble averages in physics. I think this argument goes nowhere. We can collapse the time dimension completely by changing the probability mechanism from repeated coin tossing to the choice of a single random number between zero and one. The first head in the coin toss can be replaced by the first one in the binary representation of the random number from the unit interval. Choosing one random number is a single event and there is no time to average over. The coin tossing mechanism is a red herring because it is only one way to generate the required sample space.
Of course, there are other solutions to the paradox. You can throw utility functions into the trash can and embrace prospect theory. You can correct for counterparty risk (Credit Value Adjustment or CVA in modern Wall Street jargon). You can argue that such games do not and cannot exist in a market, and financial economics need not price non existent instruments.
I am quite confident that three hundred years from today, people will still be debating the St. Petersburg paradox and gaining new insights from this simple game.