The St. Petersburg Paradox

Imagine this: you're in a casino playing a very simple game in which a fair coin is tossed. If it lands on heads, the stake is doubled, so if you bet £2 you would have £4 in the prize pot if the coin lands on heads. If it lands on tails, the game is over. Since this is a fair coin, there is an equal chance of it landing on heads or tails, and thus a 0.5 (50%) probability of landing on heads each time the coin is flipped. The probability of two consecutive heads is 0.5 multiplied by 0.5, which gives us a probability of 0.25 (25%). The game repeats until the coin lands on tails, in which case the contestant walks away with their accumulated winnings. The key question is: how much would you be willing to pay to play this game?

Let's take a look at the expected value of the payout from this game, which is calculated by summing the products of the probabilities and the winnings at each stage. Supposing an initial stake of £2, our calculation would be as follows:

(0.5*2) + (0.25*4) + (0.125*8) + (0.0625*16) + ...

= 1 + 1 + 1 + 1 + ... = ∞ (infinity)

In other words, the expected payout from this game would be infinite! Leaving aside concerns of bankrupting the casino, the conclusion to be drawn from the mathematics here is surely that you should bet any money you can get your hands on in this game: your life savings, the last few pounds in your bank account, the money you've been saving towards that first house... all of it! Yet this advice might make you feel uncomfortable and indeed the paradox is that on the one hand, there is the mathematical truth of this infinite expected payout and, on the other hand, most of us would be highly reluctant to stake much more than £20 to play this game.

A possible explanation for this is the diminishing marginal utility of money, which is the observation that the satisfaction gained by increasing one's wealth from, say, £700 to £800 is lower than the satisfaction gained by increasing one's wealth from £100 to £200. Alternative explanations point out the unrealistic nature of the game- perhaps our scepticism is the result of learning through experience that payouts such as these are not to be trusted. There is also more complex debate about the usefulness of expected value in our decision-making in scenarios such as these.

Regardless of which explanation seems most convincing, this paradox constitutes an interesting example of the overlap between mathematics and economics. If you have found this interesting and have perhaps been left with lots of burning questions, why not consider tuition with me? I provide stimulating and challenging sessions that inspire our students. I want to teach my students to go out and solve real-world problems and, crucially, to learn to think for themselves, rather than simply regurgitating formulae from a textbook. If this sounds good, look no further for your tutor!