[jon_gunnarsson]

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.

[bob.murphy.ancap]

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.

[jon_gunnarsson]

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.

[bob.murphy.ancap]

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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