The St. Petersburg paradox

I recently was reading about various paradoxes. One of them is called the St. Petersburg paradox. (Its Wikipedia page is worth reading if you're not familiar with it.)

For those too lazy or impatient to read the Wikipedia page, a brief description might be in order. Consider a gambling game in which the pot starts at $1. A (fair) coin is tossed. If it comes up heads, the player wins the pot. If it comes up tails, the pot is doubled and the coin is re-tossed; the doubling is repeated as long as the coin comes up tails. Then we ask, what price should the player pay to play to make the game fair?

It's a simple exercise to work out the expected value of the pot won by the player each game. It turns out to be infinite, which means that in traditional statistical theory, the player should be willing to play the game at any price.

The paradox, of course, is that few people would be; one writer speculated that few would be willing to pay even as much as $25. As the Wikipedia page mentions, this would actually be a rational reaction, since the statistical analysis produces an infinite expectation only if the house has infinite resources. If the house's resources are limited, that is, if a limit is put on the amount the player can win, the expected payout of the game becomes finite, and, for realistic values of that limit, remarkably small.

There is one thing that comes to my mind immediately which isn't mentioned in the Wikipedia page: the time required. The game's high expected payout comes from very rare but very high payouts. For the pot to get as high as $1000, for example, requires the coin to come up tails ten times in a row. While this doubtless does happen, it takes a lot of games for it to do so, a thousand or so; the number of games the player has to play for it to be reasonable to expect to get a large payout doubles for each additional $0.50 of expected payout. Loosely put, only one game in a million has a payout as high as a million.

But, it seems to me, there is still more that needs explaining. Relatively few people are capable of doing the mathematics necessary to work out any of the above. Fewer still are capable of doing it on the spot when the game is proposed. But I agree that almost nobody would pay even a relatively small amount to play the game; I'd be surprised to find many people willing to pay $5 and I'd expect just about nobody to pay $10, even though almost all of those people have not done, and many cannot do, the analysis to discover how sensible that reaction actually is. This leads me to wonder what, then, is behind the reaction.

The Wikipedia page mentions that Nicolas Bernoulli suggested that people will neglect unlikely events, but also remarks that people tend to overweight small probabilities. The linked-to paper by Kahneman and Tversky is interesting and probably relevant, especially the remark it contains that people are less likely to buy a fairly gambled profit than to accept a fair gamble; it implies that framing the St. Petersburg question in terms of paying a fixed amount to get an uncertain amount is likely to bias people against it. (However, I haven't come up with a good rephrasing which avoids that.)

I speculate that there are two significant pieces.

One is that most people do not really understand exponential growth. Even I, and I'd like to think I understand it intellectually, find the "doubling every toss of tails" paradigm doesn't really have the emotional grabbing power that the mathematics say it should (FWVO `should') have. Consider the story about the inventor of chess requesting one grain of rice for the first square of the board, two for the second, four for the third, eight for the fourth, etc. Few people who don't routinely work with powers of two realize just how enormous the numbers get. If, instead of grains of rice, we use something as tiny as one nanogram (that's about the mass of an average human cell), the amount for the whole board is over eighteen thousand tonnes. With actual grains of rice the figure would be larger by a factor of somewhere around twenty-five million; it would mass—if it could be assembled, of course—about half as much as all the carbon in Earth's biosphere.

The other is lack of experience, leading to, I think, Bernouilli being right in this case. Overweighting low-probability events is, for example, likely to be responsible for many people playing lotteries even though they have negative expectation—but I speculate that this works only because people are very much used to the notion that someone wins the lottery. Since the St. Petersburg gamble is not commonly played, there are no stories in circulation of people very rarely winning large amounts at it; I suspect that people completely discount the extremely unlikely outcomes in some fuzzy sense, that they effectively cut off the long improbable tail of high payoffs that is the reason the game has infinite expected value. It seems to me this is likely compounded by the other effect, by failing to fully realize just how high those rare payoffs actually are.