The St. Petersburg Paradox

The St. Petersburg Paradox is a classic problem in the general area of gambling, probability theory, risk aversion and the marginal utility of money. It was first posed by Nicolaus Bernoulli in 1713 and the solution was given in terms of marginal utility in 1738 by Daniel Bernoulli, his cousin. In most games we rationally expect a fair price to play the game is the expectation value, perhaps slightly less if you are risk averse. In general you should be happy to pay five or a bit less dollars for a game where, if a head comes up you win $10 and get $0 if its a tail. In many other games people are even happy to pay more than this, eg. Roulette.

Ok so what about the following game. How much would you pay to play it?

I have a pot of $2. I flip a coin. If it comes up tails then you win the pot, if it comes up heads then I double the pot and we play another round, and so forth. Each time we get a head the amount in the pot doubles and we play again. So if the first tail is in the nth round you win $2^n. So far so good, but what happens when we calculate the expectation? What do we get?

The chance of getting tail in round “n” is (1/2)^n. The pay off is 2^n, so the expected value per round is $1. However there are potentially infinite numbers of rounds. The expectation therefore is also infinite.

Clearly no one in their right mind would pay that amount of money even if they could. If you were to pay only $1000, you would only make money 0.05% of the time, even if in a small number proportion of those cases you are fabulously wealthy.

The solution proposed by Daniel Bernoulli was to have a concept of utility of money, where the value of money to a person was essentially logarithmic. If you got twices as much, it was only one extra unit of utility to you. This lets you get a finite answer to the problem, and a value that is probably more realistic to what someone would actually pay to play.

More pragmatically you could point out that our intuitive feel for the price has more to do with the fact that any such game is in reality finite. There is only so much money in the world, and given that you would reach something of the order of the total monetary value of the earth in around 40 something rounds the game should be capped there. Capping the game at that level gives you an expectation of $40 ($1 per round). Perhaps not such a paradox after all.

You could also compare this to the pay outs on actual lottery games which while different probably have the most similar pay off profiles of various gambling games, and see how much people pay to play them, and contrast this with a say 20 round St. Petersburg game (maximum pay out around $1,000,000) which should have a fair price of $20.

Looking at the nsw lotteries annual report for 2004, we find that the chance of a first prize in Lotto is (correctly picking 6 from 45) is around 1 in 8 million for a single game. First division prize is $1,000,000 for the Monday game, but only 21 million dollar payouts were made because of prize splitting around half the draws. The cost per game is $0.30 plus agents commission (3 to 5c/game, cheaper in bulk).

Considering the Monday night first prize which is presumably $52 million total over a year, and is won by around 80 odd people in this time the expected pay out is something of the order of $650,000. At odds of 1 in 8 million, the fair value is $0.08, but of course, like the St Petersburg game, there is a fair bit of value in the intermediate prizes, Lotto, like St Petersburg, paying out around half of its total prizes in smaller prizes. This makes the fair cost go up to around $0.16 per game, about half the amount actually paid. This rough calculation broadly agrees with the overall pool figures of $141.2 mil of prizes for $231.4 mil in sales, giving a fair price of $0.18 per game.

This has a fairly drastic impact on the idea of declining marginal utility, at least for the people playing lotteries. These same people would presumably pay $32 for a 20 round St. Petersburg game. Rather than marginal utility decreasing with wealth, it seems they have increasing marginal utility of money. This seems strange, but perhaps is not totally ridiculous.

For many people getting together capital to start a business or buy a house may be prohibitive at their current incomes. Lotteries for these people are perceived as at least a chance of getting together a pool of money they could never have dreamed of saving for a small cost per week.

For most people though I think the willingness to pay an unfair price is explained by the flutter of excitement as the numbers come down, which for them is well worth the couple of dollars they may have paid away each week.

For me its hard to believe that many people wouldn’t play any real St. Petersburg game at the fair price. They will play lotto for even less. The paradox only arises because ultimately, in reality, the pool is capped. People rightly value an infintesimal chance of an infinite pay off as zero rather than 1, because it is unpayable.

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