The St. Petersburg Paradox: a flimsy critique of expectation theory by people who don’t know math or economics

As someone who uses math and economics for a living, I come across many opposing arguments. There are always people who wish to prove to me economics is fundamentally flawed.

My first encounter of this type came after my freshman year of study at Stanford. I was sharing some economics ideas to friends when someone interjected: “Yeah, this is interesting, but it’s based on expectation theory which is critically flawed. Haven’t you heard of the Saint Petersburg paradox?”

It’s an interesting paradox, I answered, but far from a take-down of expectation theory. Here is the paradox and two ways to resolve it.

The St. Petersburg Paradox

The paradox is about a game of chance.

Here is how the game works. A fair coin is tossed until the first heads appears, which ends the game. The payoff to you depends on the number of tosses. The payoff starts at 2 dollars and doubles on each successive toss.

That means you get 2 dollars if the first toss is a head, 4 dollars if the first toss is a tails and the second is a heads, 8 dollars if the first two tosses are tails and the third is a head, and so on. In other words, you get paid 2^k where k is the number of tosses for the first heads.

The question to you is how much should you be willing to pay to play this game? In other words, what is a fair price for this game?

The typical way to answer this question is to compute the expectation (or the “average”) of the payouts. This is done by multiplying the various payouts by their probability of occurrence and adding it up. To say it another way, the payouts are weighted by their likelihood.

The respective probabilities are easy to compute. The chance the first toss is a heads is 1/2, the chance the first toss is a tails and the second is a heads is 1/2 x 1/2, and the third toss being the first heads is 1/2 x 1/2 x 1/2, so the pattern is clear that the game ending on the k toss is (1/2)^k

So with probability 1/2 you win 2 dollars, with probability 1/4 you win 4 dollars, with probability 1/8 you win 8 dollars and thus the expectation is:

The surprising result is the expectation is infinity. This means this game–if played exactly as described–offers an infinite payout. With an astronomical payout, a rational player should logically be willing to pay an astronomical amount to play this game, like paying a million dollars, a trillion dollars, and so on until infinity.

The fair price of infinity is paradoxical because the game does not seem like it is worth much at first. Few would be willing to pay more than 10 dollars to play this game, let alone 100 dollars or 1,000 dollars.

But expectation theory seemingly says that any amount of money is justifiable. Banks should be willing to offer loans so people could play this game; venture capital firms should offer more money than they do to start-ups; individuals should be willing to mortgage their house, take a cash advance on their credit card, and take a payday loan.

What’s going on here? Why is the expectation theory fair price so different from common sense?

It turns out there are a variety of explanations.

Resolution 1: Payouts should be realistic

Imagine you are playing this game with a friend. You hit a lucky streak. The first nine tosses have been tails and you’re still going. If the tenth toss is a heads, then you get 1,204 dollars as a payout. If it’s a tails, you have a chance to win 2,408 dollars, and even more.

At this point your friend realizes he’s made a mistake. He thought he’d cash out with your 10 dollar entry fee, but he now sees he cannot afford to risk any more.

He pleads with you to stop. He’ll gladly pay you the 512 dollars you’ve earned–so long as you keep this whole bet a secret from his wife. What would you do in this situation?

Most of us would take the cash and show some mercy here. There is no joy in winning if it means crippling a friend financially. And this concocted scenario leads to one of the unrealistic assumptions of the St. Petersburg paradox.

In the hypothetical coin game, you’re supposed to believe the other side can pay out infinitely large sums of money. It doesn’t happen often, but if you get to 20 coin tosses, you fully expect to be paid 1,048,576 dollars.

This is unrealistic if you’re playing with a friend or even a really, really rich friend. It might be possible with a casino, but even a casino may have a limited bankroll.

The truth is that payouts cannot be infinite. If such a game were to exist in our reality, there must be a maximum, finite payout.

This means the expectation is not an infinite sum but rather a finite sum of several terms. Depending on the size of the bankroll, the St Petersburg gamble has a finite payout.

I will spare you the details, but here are a few examples of the expectation when using a maximum payout using a Wikipedia table for illustration):

(note: these calculations are based on payouts of 1, 2, 4, etc rather than what I used of 2, 4, 8, etc.)

As you can see, expectation theory now implies the fair price of the game is something like 25 dollars or less. With a more realistic model of the game, the expectation result matches common sense.

This should settle matters for anyone concerned with reality and practice, but there are people who don’t accept this explanation. Such philosophers think an infinite payout is possible and so the paradox still exists.

So for these people, I will offer the following alternate resolutions that don’t rely on limiting the bankroll.

Resolution 2: diminishing marginal utility

A quantity like 1,000 dollars has meaning to most people. If you were to ask a friend for such a loan, they would ask how you can pay it back, what you would use it for, and so on.

But there are times when 1,000 dollars seems to lose its value. I like to think about the show Deal or No Deal where contestants play a multi-stage lottery to win 1,000,000 dollars. At various points in the game the contestants can either keep pursuing the big prizes or they can accept smaller consolation prizes. As the prospect of a big prize increases, the contestants start to care less and less about smaller prizes like 1,000 dollars.

This is an example of the famous concept of diminishing marginal utility–the idea that at larger levels of consumption, incremental units are worth less. The concept is applicable for wealth decisions because at some point incremental earnings mean less to a person.

What this means for the St. Petersburg Paradox is that the payouts should be altered. The payouts should not be measured in dollars but rather as the utility that wealth will provide.

One way to model this is to use a logarithm function. Instead of saying the payout for the first toss is 2 dollars, we will say it is log(2) units of utility, and accordingly for the other payouts.

Using a log utility function, the St Petersburg game now has a finite payout. Here is my hand-written derivation:

This is a small payout but the actual quantity does not matter: it is just that the payout is less than infinity.

Conclusion

The St. Petersburg Paradox is not a blow to expectation theory. It is a made-up game that is interesting in its own right, was historically important to economics, and is fun to resolve.