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St. Petersburg paradox
Jul. 12th, 2006 01:07 pmImagine a game in which you:
* win $2 with probability 1/2
* win $4 with probability 1/4
* win $8 with probability 1/8
* win $16 with probability 1/16
* win $32 with probability 1/32
and so on, ad infinitum.
Assume also that your payoff will be determined and paid up instantly.
How much would you pay to play this game?
The expected value of the game is
St. Petersburg paradox
And I think this behavior is perfectly rational.
Your answer to the question above is a measure of your risk-aversity. But we should also investigate the effects of a lower-payoff vs a higher payoff game.
A game paying twice as much as the above game:
* win $4 with probability 1/2
* win $8 with probability 1/4
* ...
is obviously a better game to play (almost twice as good), and yet, the expected value of the game is also infinity.
My solution would be to make a transformation on the probabilities that makes the sum converge. The idea is that we have a "horizon", and very remote probabilities should count for less. But a 1/8 probability should count almost the same as a 1/2 probability. Any concrete solutions?
One problem is that you can always make a game with payoffs that make this sum diverge: just make the payoffs proportional to 1/p. This is not too problematic however: IMHO, the real problem is unbounded utility. Founding utility on human happiness is a good way out (I think).
* win $2 with probability 1/2
* win $4 with probability 1/4
* win $8 with probability 1/8
* win $16 with probability 1/16
* win $32 with probability 1/32
and so on, ad infinitum.
Assume also that your payoff will be determined and paid up instantly.
How much would you pay to play this game?
The expected value of the game is
1/2 * 2 + 1/4 * 4 + 1/8 * 8 + ... =
1 + 1 + 1 + ... = infinitySt. Petersburg paradox
A naive decision theory using only this expected value would therefore suggest that any fee, no matter how high, would be worth paying for this opportunity. In practice, no reasonable person would pay more than a few dollars to enter. This seemingly paradoxical difference led to the name St. Petersburg paradox. [huh? why this name? -GL]
And I think this behavior is perfectly rational.
Your answer to the question above is a measure of your risk-aversity. But we should also investigate the effects of a lower-payoff vs a higher payoff game.
A game paying twice as much as the above game:
* win $4 with probability 1/2
* win $8 with probability 1/4
* ...
is obviously a better game to play (almost twice as good), and yet, the expected value of the game is also infinity.
My solution would be to make a transformation on the probabilities that makes the sum converge. The idea is that we have a "horizon", and very remote probabilities should count for less. But a 1/8 probability should count almost the same as a 1/2 probability. Any concrete solutions?
One problem is that you can always make a game with payoffs that make this sum diverge: just make the payoffs proportional to 1/p. This is not too problematic however: IMHO, the real problem is unbounded utility. Founding utility on human happiness is a good way out (I think).
