St Petersburg paradox

[Steven G]

I am at a loss as to why everyone is talking about utility in the St Petersburg paradox (If you are unfamiliar with the game the rules are below). I understand that the expected value is 1+1+1+...=infinity and that can't be right. Why people bring in utility is strange to me. I was hoping that someone could give me a mathematical reason why the expected value is not correct.

The standard version of the St. Petersburg paradox is derived from the St. Petersburg game, which is played as follows: A fair coin is flipped until it comes up heads the first time. At that point the player wins$2ⁿ,where n is the number of times the coin was flipped. How much should one be willing to pay for playing this game?

 

[Prof]

The expected value is infinity. It results from the accumulation of infinitely many extremely high-paying but highly improbable outcomes. No human can play the game forever so in practice, the amount we are willing to pay is likely to be related to how many times (finite) we would actually play the game. Any finite version of the game would have a finite expected value.

 

[Steven G]

The expected value is infinity. It results from the accumulation of infinitely many extremely high-paying but highly improbable outcomes. No human can play the game forever so in practice, the amount we are willing to pay is likely to be related to how many times (finite) we would actually play the game. Any finite version of the game would have a finite expected value.

I do not buy what you say completely, as you can play enough so that the expected value is $50. I just do not think that it is advisable to pay $50.
Am I understanding the game correctly? If you agree that a fair price is $50, then you pay $50 each time you play? That is absolutely crazy to pay that much to play. Your point that you can play only for a finite amount of time is an excellent point-thanks!

 

[Prof]

I think you would pay $50 to be able to have up to 50 flips (and you would hope for 50 tails in a row and get a payoff of $2^50). But it is quite likely (15/16 or 94%) that you would only get a payoff of 2, 4, 8, or $16 which makes it seem crazy, as you say. That's why people discuss this problem - because in practice, it would be hard to believe in the math when you are going to lose money in about 97% (31/32) of the games. in this example. Once you get to 6 tails in a row ($64 or more payoff), you are making a profit. There is a 1/32 chance of getting 6 or more tails in a row. Twenty or more tails in a row and you would get $1 million dollars or more. Seems something like playing the lottery as people lose most of the time and are hoping for the jackpot.

 

[Steven G]

The more I think about it you are correct. If you have enough $50 bills it is to your advantage to play the game.
Thanks for you time. I really appreciate it.

 

[JeffM]

Utility is a basic concept in economics.

The basic idea is that the subjective value (utility) of anything is never linear everywhere. For a heterosexual man, a wife can be a very good thing, but having ten may be so much of a good thing that many men would rather have fewer. (The idea of having ten wives strikes me as horrific: I am almost insane from having three cats, one dog, and one baby in the house, and none of them has the power of speech to express what seem to be very limited appetites. Wives on the other hand ...)

So if you think in terms of expected value of utility, the Petersburg Paradox disappears because the paradox assumes a linear utility function on money.

Pay me $100,000, and I’ll sell you a 1% probability of a $10 million after-tax reward. Unless you view a 1% chance of 10 million as having greater utility than a certainty of 100,000, you will not take the bet. Utility is a mathematical way to describe human behavior.

If you are aware of Savage, Ramsey, de Finetti, et. al. and the theory of “rational decision,” utility simply supplies a vocabulary to render the paradox a trivial mistake. Of course, if you think the theory of rational choice is nonsense, utility does nothing to address the paradox.

In a lot of economic literature, the utility function is ASSUMED to be twice differentiable with a positive first derivative and a negative second derivative. That assumption is known to be empirically false (households buy lottery tickets as well as fire insurance), but it probably is true empirically for most people with respect to high proportions of their wealth. Lottery tickets are not sold in units of thousands of dollars.