During our last sesion on Decision theory, we were discusing on St. Peterburg paradox.
We, at least partially, agree that there is a paradox even if there is no infinite utilities. I will briefly defend that this position does not resist a simple mathematical analysis.
On the asumption that there are no infinite utilities the St. Peterburg game is perfectly acceptable:
I would bet 2utilities for getting 2utilities if the coin lands heads and 4utilities if the second time that I flip the coin it lands heads again. The game seems to be completelly fair. And so are the following games where:
The fist column represents the maximum price of the game. This would be 2utilities if the coin is flipped only once, 4 if it is flipped at most 2 times, and so on. In general 2 to the power of n where n is the number of times that the coin can as much be flipped.
The second represents the probability of each case.
The third column represents the expected utility (how many utilities should I pay to play the game).
For the example assumme that I have a lineal utility function regarding money between 0 and 1million euro (hard to believe but assumme that that is the case) and that the utility of 100M € equals the utility of 1M for me. The function saturates at 1M. (if you are not convince, for 30€ you can earn up to 1billion €, and I think that that is enought to saturate definitely the utility function of all of us).
The fouth column shows the money I would win or lose depending on the result of the game.
If you are having doubts on whether to play the game or not is because the utility of money is not linal for you and therefore: U(1M€) is not equal to 50000*U(20€).
In this case you would pay less money to play the game, but this is completely compatible with decision theory. Think of something wich utility is lineal in this range and you accpet the game (psichological reasons to avoid betting are out of the question) as you clearly see when the game is propossed to win just 4€.
The paradox is expressed in terms of utilities so have to find something which utility is lineal between 0 and 1M.
The real problem arises just in case we consider infinite utilities (no matter whether they are lineal or not). Imagine that more money has always a higher utility, so the utility function of money is a monotonically strictly increasing function in any interval. Then there is a problem, because at the limit the price is infinite...
The expected utility of any lottery involving an infinite price cost infinite no matter what the probability is. This two lotteries has the same cost (infinite):
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You should prefer the second lottery to all that you have and that is obviously unacceptable. The solution: there are not infinite utilities.
The problem with lower probabilities is just that we are not able to find any utility that satisfies that lottery and therefore it is difficult to find an interpretation of paying 250 utilities to play this lottery.
But that says absolutely nothing against the decision theory.
The St. Petersburg game is only a problematic if we consider infinite utilities.