Arnold Kling  

St. Petersburg Paradox

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Will Baude brings up the St. Petersburg Paradox, in which a bet with an infinite expected payoff is rejected by the typical individual. Baude points out the problem with trying to resolve the paradox by invoking diminishing marginal utility.


here's the pet peeve. While this ad hoc (but plausible) assumption solves this particular version of the St. Petersburg Paradox, it does nothing about a modified version of this game with much higher payoffs. Suppose, for example, that I increased the payoffs for this game by exponentiating 5. That is, when I was going to pay you $1 in the first version, I'll now give you $5. When I was going to pay you 2, now you can have 5^2 = $25. Instead of $4, you can have $625 instead. It's pretty easy to show that this new set of payoffs will result in a quite infinite payoff, even if you have that diminishing marginal utility of wealth. Indeed, for any proposed diminising marginal utility of wealth function that's unbounded, I can concoct some crazy version of this game with sky-high payoffs that will still have an "infinite" value.

If you are not familiar with the paradox, a good introduction is here.

A simple way to describe the paradox is this. Suppose that you were offered a bet where if you flip a coin ten times and it comes up heads every time you win $1 billion. Otherwise, you lose $10,000. Would you take the bet? On average, you stand to win by taking the bet, but most people would not take it. Even if somebody raises the payoff for winning to $10 billion or $10 trillion, most people would not take the bet. Why not?

I think that the answer is that people mentally truncate the upper value of what you might win. It is hard to believe that somebody is really going to pay you $1 billion if you flip ten heads in a row. So you act as if they were offering you something lower.

For Discussion. Is it rational or irrational to act as if somebody would stick to the terms of the bet, even if it meant that they had to pay an enormous sum of money?


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The author at Deinonychus antirrhopus in a related article titled St. Petersburg Paradox writes:
    Arnold Kling has an interesting post on the St. Petersburg Paradox (also worth checking out is this link Arnold provides [Tracked on November 5, 2003 1:46 AM]
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Bob Dobalina writes:

I don't have ten large in the bank right now. I'd beg, borrow, and steal if the chance of winning wasn't 1/1024.

I'd do it for $100, though, to win $10 million.

Am I a "typical individual?" Or do I miss the point entirely?

Boonton writes:

Perhaps the resolution comes from a mental budget that caps how much people will devote to gambling. Obviously people are happy to lose $1 for a much smaller chance of winning millions.

In theory a casino could simply offer this 'game' for a $100 bet. They would expect to raise $1,025,000 for every person who won the ten coin flips so they could profit by making the payout an even million. I think people would take it.

Sol writes:

Seems to me that Arnold is pointing at a nice (probably not original) solution to the Paradox. In the real world, it's ridiculous to talk about 1000 trillion dollar payouts -- not just hard to believe someone is going to do it, it's completely impossible. And as soon as you cap the payout, the infinite series goes away, and the expected return of the bet is something like $25. (2^50 is 1,100 trillion or so, and each term of the series is worth fifty cents.)

Bruce Bartlett writes:

If I could bet $10,000 for 10 flips infinitely, of course I would do it. One out of 100,000 times the coin is bound to come up heads 10 times in a row. But if I am given a one-time only opportunity for 10 flips, I would turn it down. Also, if I have to pay out $10,000 in cash each time I lose, I couldn't afford to do it enough times to put the odds in my favor. In other words, I think there is an unstated assumption that you can make the bet as many times as you want, which changes the parameters of the bet.

Rob writes:

I don't think people calculate the expected payoff the way one should - statistically. I think most people simply figure out if the odds are that they win or lose, and most times you will lose this game.

Sean writes:

Wouldn't a person's income or accumulated wealth factor into the decision to enter into certain bets?

I can imagine that a person with $10 million in the bank would be far more willing, in the event of a loss, to give up $10,000 than a person with with $10 in the bank.

Are poor people more likely to bet in a lottery, wheere the total potential loss is just the price of the ticket, than people with relatively large amounts of money?

Is the question of income or wealth even relevant?

Mats writes:

Interesting explanation. What is your view on the Allais paradox? That's a psychological one two, and perhaps more important from an economic/preference-studies perspective:

http://blogofpandora.blogspot.com/2003_11_01_blogofpandora_archive.html#106802749125727892

Lawrance George Lux writes:

People miss the second function here. It is not the simple odds which apply, but the number of times an Individual has to attempt the bet. The Individual is not betting the odds will work in his favor, but the odds they will work in his favor before he loses the ability to bet. lgl

As previous posters have alluded to, it's not really correct to say that "on average, you stand to win by taking the bet."

Just because the expected value is very high doesn't mean that "on average" the participant will come out ahead, due to the extremely large variance.

It will, in fact, be extremely rare for the participant to come out ahead, and it is therefore a rational decision not to take the bet given any level of risk aversion whatsoever.

Robert W Vivian writes:

I have argued in a recent article 'Solving the St Petersburg Paradox-the paradox which is not and never was' SAJEMS 6 (2) 2003 that the St Petersburg game does not lead to a paradox at all. The source of the confusion lies in the traditional solution to the game which holds that the expected value is infinite. This is a special case - where the game is played an infinite times. For any finite number of times the game is played it finite and modest, of the same order that people will offer to play the game.

Dr Shahil Shandil writes:

What most comments are implying is, in most cases, theoretically incorrect in all situations. St Petersburg's Paradox is purely based upon mathematics, which in the end simply is a reoccurrence of what has been already preceded. You see when n-1, k=1 and 2^k then the answer must be simply 2^n-2 dollars. In this case the player has won $2 (which is 2^n dollars). Taking into account the probability of winning the nth toss is (1/2)^n, then p=1/2 is in all cases.

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