## 05 November 2007

### An implication of the St. Petersburg paradox

Many of you are probably familiar with the St. Petersburg paradox. This is the following paradox. I offer to play the following game with you: I will flip a fair coin repeatedly until it comes up heads. If the first time it comes up heads is on the nth toss, I will pay you 2n dollars. How much are you willing to play this game?

Well, the probability that the coin comes up heads on the first toss is 1/2; in this case you get 2 dollars. The probability that it comes up heads for the first time on the second toss is 1/4; in this case you get 4 dollars. In general, the probability that the coin comes up heads for the first time on the nth toss is 1/2n, and in this case you get 2n dollars. So your expected winnings are

2(1/2) + 4(1/4) + 8(1/8) + 16(1/16) + ...

and each term here is 1; the series diverges. So you should be willing to pay an infinite amount of money to play this game. Yet you're not. (If you are, let me know. I'd like to have an infinite amount of money. Notice that you will only win a finite amount of money playing this game, so even after I pay you I will still have an infinite amount of money.)

Yet you're not. You may suspect that this is because you know the person you're betting against doesn't have an infinite amount of money, so your expected winnings don't come from actually summing the whole infinite series. For example, let's say Bill Gates is willing to play this game with you; let's say his net worth is 236 dollars. Then your expected winnings are

2(1/2) + 4(1/4) + 8(1/8) + ... + 235(1/235) + 236(1/236) + 236(1/237) + ...

where the first thirty-six terms are all 1; then what follows is 1/2, 1/4, 1/8, and so on. So you should be willing to spend \$37 to play this game if the Gates fortune is backing it.

Of course, you might argue -- and a lot of economists have -- that \$2n is not worth twice as much to you as \$n. The usual assumption here is that the utility of \$n is something like log2(n) "utils" (I'm not sure how they handle the problem of the units here), and that people play to maximize their expected number of utils, not their expected number of dollars. Then the "expected value" of the previous game, in utils, is
1(1/2) + 2(1/4) + 3(1/8) + ... = 2
and you should be willing to pay that amount of money which is worth 2 utils to you, namely \$4.
But I can construct a similar paradox. If the coin comes up heads on the first toss, I pay you \$2. If it comes up heads on the second toss, I pay you \$4. If it comes up heads on the third toss, I pay you \$16. If it comes up heads on the fourth toss, I pay you \$256, and so on... then you receive 1 util if the coin comes up heads on the first toss, 2 on the second toss, 4 on the third toss, 8 on the fourth toss, and so on. So you should still be willing to pay infinitely much to pay this game. And in general one can construct such a payoff sequence for any unbounded utility function.

The somewhat counterintuitive resolution that I heard for this recently is that utility functions must be bounded. So say \$1 gives me a certain amount of utility. Then in order to make it impossible to construct a St.-Petersburg-style wager, for which I would be willing to pay an infinite amount of money, there must be some K such that any amount of money gives me at most K times as much utility as \$1. I'm not sure I believe this, either... it just goes to show that sometimes expected value might not be the way to go.

Aaron said...

I'm not sure I believe this, either... it just goes to show that sometimes expected value might not be the way to go.

I agree. In my gut, I think this paradox has nothing to do with utility functions, and everything to do with some fundamental problem with our conception of expected value. In particular, I suspect the problem might stem from some sort of cultural preference for summary statistics over the full appreciation of variation... and if I were drunk, I might even speculate that such a cultural preference might be related to some sort of Platonistic belief that averages are more "real" than distributions.

If you asked me how much I would pay to play the St. Petersburg game, my unconscious reasoning might go something like this:

My probability of winning 2n dollars or more is (1/2)^n + (1/2)^(n+1) + (1/2)^(n+2) + ... = (1/2)^(n-1). Therefore, if I pay you 2n dollars, my probability of walking away with non-negative net winnings (whatever "probability" means for a single trial!) is (1/2)^(n-1). So if I want to have at least a 50% chance of not losing, I should pay at most \$4!

This kind of reasoning, which ignores the relative costs and benefits of losing and winning, is useful for the St. Petersberg paradox, but it fails miserably for games like Russian roulette, where expected-value reasoning is a better model of how I would actually behave. I suspect that the way we unconsciously reason about costs and benefits is somewhere in between, and varies depending on the situation!

John Armstrong said...

There must be some K such that any amount of money gives me at most K times as much utility as \$1. I'm not sure I believe this, either.

Are you saying you don't believe that that's the resolution, or that you don't believe utility functions work that way? I'd be willing to believe that utility functions work that way.

Despite the fact that we can name arbitrarily large numbers, we really don't think about them the same way we do small numbers. There's some point at which it just becomes "a lot", like in Watership Down, where any number over four is "too many to count".

Bill Gates' net work may be on the order of \$60B, but psychologically 60 billion is practically infinite. Honestly, I'd see no difference in my life between having \$30B and \$60B.

Isabel said...

John Armstrong wrote:
Bill Gates' net work may be on the order of \$60B, but psychologically 60 billion is practically infinite. Honestly, I'd see no difference in my life between having \$30B and \$60B.
Let's say \$30B is what it costs to buy a small country. Wouldn't you rather own two small countries than one?
Or if you don't have such aspirations, and you see yourself as a "good person", let's say that there are two Bad Things about the world that you can just make go away for \$30B each. Wouldn't you rather get rid of both of them?
I'm not saying this proves that utility functions must be unbounded; just that it leaves open a crack in the door. We can't imagine what it would be like to have billions of dollars.

John Armstrong said...

Sure, you can construct hypothetical widgets costing \$30B. But they don't really exist. Nothing really costs \$30B. Nobody can buy sovereignity, there's no practical difference between having one and two private islands, and when Bill Gates donates \$30B to charity, he's not so much donating \$30B specifically as he's donating "a lot".

Aaron said...

Sure, you can construct hypothetical widgets costing \$30B. But they don't really exist. Nothing really costs \$30B.

According to Wikipedia, the Apollo spacecraft cost about \$28 billion (in 2006 dollars) to develop. ;)

John Armstrong said...

And I would respond that Apollo cost a total of \$30B. That total was parcelled out among so many salaries or materials costs that were each a relatively small amount.

Again, you've both got this hidden assumption that people actually think more or less linearly about these sorts of things, which is really begging the question.

Let's move it away from dollars. Who hasn't heard the adage that "one death is a tragedy, a million deaths is a statistic"? It's the same effect: humans just don't think about millions. We can manipulate them on paper, but the visceral impact just isn't there like it is for a hundred, or maybe even a thousand.

Anonymous said...

If the paradox here is why I wouldn't want to spend an infinite amount of money the resolution seems simple: I don't have an infinite amount of money, and even if I had I would spend it on something more fun than watching someone flip a coin an infinite amount of times.

Actually I have trouble understanding exactly what is to be learned from this example. It is too simplified to be possible to extract any meaningful information from.

I expect there are sufficiently many people that finds betting money on coin flips stupid that the question becomes
"If you had to bet money on this stupid game and try to get as much as you can, how much would you bet?"

And that's honestly a test of intuitive application of game theory and not a test of their economic reasoning.

/Rettaw

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