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Cracking the St. Petersburg Paradox

May 8, 2018

It was a puzzle first posed by the Swiss mathematician, Nicolas Bernoulli, in a letter to Pierre Raymond de Montmort, on Sept. 9, 1713, and published in the Commentaries of the Imperial Academy of Science of St. Petersburg. Mercifully it is simple to state. Less mercifully, it is supposedly a nightmare to solve. To state the paradox, imagine tossing a coin until it lands heads-up, and suppose that the payoff grows exponentially according to the number of tosses you make. If the coin lands heads-up on the first toss, then the payoff is £2. If it lands tails on the first toss, you receive £1. If it lands heads-up on the second toss, the payoff is £4; if it takes three tosses, the payoff is £8; and so forth, ad infinitum. You can play as many rounds of the game as you wish.

Now the odds of the game ending on the first toss is ½; of it ending on the second toss is (1/2)^2 = ¼; on the third, (1/2)^3 = 1/8, etc., so your expected win from playing the game = (1/2 x £1) + (1/2 x £2) + (1/4 x £4) + (1/8 x £8) + …, i.e. £0.5 + £1 + £1 + £1 … = infinity. It follows that you should be willing to pay any finite amount for the privilege of playing this game. Yet it seems irrational to pay very much at all.

According to this reasoning, any finite stake is justified because the eventual payout increases infinitely through time, so you must end up with a profit whenever the game ends. Yet most people are only willing to pay a few pounds, or at least not much more than this. So is this yet further evidence of our intuition letting us down?

That depends on why most people are not willing to pay much. There have been very many explanations proposed over the years, some more satisfying than others, but none has been universally accepted as getting near to a convincing explanation.

The best attempt, and one which I find the most convincing, is to address the issue of infinity. It is true, of course, that you will, if you play an infinite number of rounds of the game, win an infinite amount. But what happens in the real finite world? And here is the problem. Because playing to infinity pays an infinite amount, this does not mean that the game in finite time never stops paying out money. The key question in finite time is WHEN does the game turn profitable? The answer depends on the size of the stake per round. If this stake is £2, and you repeat the game over and over again, you are likely to make a lot of money very quickly. As the stake size increases, the number of rounds it takes to turn a profit becomes increasingly longer. Take the example of a stake of £4. In this case, you only make a profit if you throw three heads in a row, which is a 1 in 8 chance. You now need to factor in the losses you made in rounds where you didn’t throw three heads in a row. This extends the number of rounds it will take to turn a profit. So the game is not profitable at any stake size unless we are willing and able to play an infinite number of rounds. It is, in theoretical terms, profitable at any stake size, however large, but it will take forever to guarantee a profit. In a world of finite rounds and time scales, however, winnings generated by the game are easily countervailed by some specified level of stake size.

So what is the optimal stake size for playing the St. Petersburg game?

This depends on how many rounds you are willing to play and how likely you wish to be to come out ahead in that timescale.

This has been modelled empirically, using a computer program to calculate the outcome at different staking levels. What does it show? Well, if you stake a pound a round, you have a better than even chance of being in profit after just three rounds. If you pay £2 a round, the even-money chance of coming out ahead takes rather more rounds – about seven. At £3 a round we are looking at more than 20 rounds, at £4 approaching 100 rounds and £5 more than 300 rounds. By the time we are staking £10 a go, more than 350,000 rounds are needed to give you more than an even chance of being ahead of the game. An approximation that generates the 50-50 point to any staking level is 4 to the power of the stake, divided by 2.9. So what’s a reasonable spend per round to play the game? That depends on the person and the exact configuration of the game. Either way, it’s not that high.

Perhaps the median (the mean of the two middle values of the series), rather than the mean offers a pretty good approximation to the way most people think about this.

Let’s say that in the game as proposed, the game is run 1000 times. In this case, 500 of the values result in tails on the first toss with a return of £1. The next 25% of values result in tails on the second toss with a return of 2. The rest of the values are not then relevant. The 500th value is 1 and the 501st value is 2. The median is the mean of £1 and £2, i.e. £1.50.

Whichever of the two ways proposed here we look at it, the solution is much closer to most people’s intuitive answer than it is to the answer implied by the classic formulation of the St. Petersburg problem.

 

Reading

Koelman, J. Statistical Physics Attacks St. Petersburg: Paradox Resolved.

http://www.science20.com/hammock_physicist/statistical_physics_attacks_st_petersburg_paradox_resolved-96549

Fine, T.A. The Saint Petersburg Paradox is a Lie.

https://medium.com/@thomasafine/the-saint-petersburg-paradox-is-a-lie-62ed49aeca0b

Hayden, B.Y. and Platt, M.L. (2009), The mean, the median, and the St. Petersburg Paradox. Judgment and Decision Making, 4 (4), June, 256-272.

http://journal.sjdm.org/9226/jdm9226.html

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