# General Solution to St. Petersburg Paradox

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• #21646
Participant

Dr Murphy,

In your 4th lecture, when addressing the St. Petersburg Paradox, you state that for any given utility function, the payout structure of the casino could be adjusted in such a way as to get infinite expected utility.

I don’t think that’s true. Simply use a bounded utility function and the result will obviously be finite, no matter the nature of the game. In practical terms, it also makes a lot more sense to use a bounded function since there is a limited amount of things you can do with even an arbitrarily large sum of money.

#21647
bob.murphy.ancap
Participant

Hi Jon,

OK yes, if I said “for any given utility function” then that is obviously not true. For example, if you got 0 utility regardless of how much wealth you had, then you can’t get an infinite amount of expected utility.

What I was trying to get at (and apparently fumbled the point) is that it doesn’t truly solve the original paradox to introduce risk aversion.

#21648
Participant

I thought Bernoulli’s approach was about diminishing marginal utility, not risk aversion. And while Bernoulli’s choice of a logarithmic utility function does not generalise to all possible games of chance, his approach does generalise if we choose a different function. Nor do we have to choose something silly like a constant function. Consider the following simple utility function (where X is the amount of money you have and negative values of X are not allowed):

X/(X+1000)

This is a strictly monotonously increasing function (satisfying the assumption that extra money is always at least a little bit useful) with an upper bound of 1, so no matter the payout structure, the expected utility can’t possibly exceed 1. This is also a more realistic utility function than the one offered by Bernoulli since it seems intuitively obvious that money alone can’t make you arbitrarily happy or satisfied.

#21649
bob.murphy.ancap
Participant

The standard way to model risk aversion mathematically is to have DMU from additional amounts of money (or wealth). For example, if your utility from wealth is U = SQRT(W), then you would be risk averse. Note that you would get more utility from a certain \$1000 in wealth, rather than a 50% chance of \$500 and a 50% chance of \$1500, if we assume you act to maximize the expectation of utility.

If you want to see more on this, the Wikipedia article is a good place to start:
https://en.wikipedia.org/wiki/Expected_utility_hypothesis#Risk_aversion

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