I’ve been thinking about what assumptions generate the St Petersburg paradox. As stated, the paradox depends on the assumption that we should maximize expected utility, an assumption that will be rejected by those who think risk aversion is rational.
But one can run the St Petersburg paradox without expected utility maximization, and in a context compatible with risk aversion. Suppose finite utilities can be represented by finite real numbers. Assume also:
Domination: If a betting portfolio B is guaranteed to produce at least as good an outcome as A no matter what, then B is at least as good as A.
Archimedeanism: For any finite utility U and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.
Transitivity: If C is better than B and B is at least as good as A, then C is better than A.
(Note: For theistic reasons, one might worry about Construction when the Vi are very negative, but we can restrict Construction to positive finite utilities if we add the assumption in Archimedeanism that V can always be taken to be positive.)
For, given these assumptions, one can generate a gambling scenario that has only finite utilities but that is better than the certainty of any finite utility. Proceed as follows. For each positive integer n, let Vn be any finite utility such that probability 1/2n of Vn is better than certainty of n units of utility (this uses Archimedeanism; the apparent use of the Axiom of Choice can be eliminated by using the other axioms, I think) and Vn ≥ Vn − 1 if n > 1. Toss a fair coin until you get heads. Let your payoff be Vn if it took n tosses to get to heads.
Fix any finite utility U. Let n be a positive integer such that U < n. Then the gambling scenario offers a probability of 1/2n of getting at least Vn, so by Domination, Transitivity and the choice of Vn, it is better than U.
And the paradoxes in this post apply in this case, too.
If we have expected utility maximization, we can take Vn = 2n and get the classic St Petersburg paradox.
Given the plausibility of Domination and Transitivity, and the paradoxes here, it looks like the thing to reject is Archimedeanism. And that rejection requires holding that there is a probability ϵ so small and finite utility U so large that no finite benefit with that probability can outweigh U.
8 comments:
I agree. We can easily see this by example: let probability ϵ be 1 in a googolplex. And let U be anything good you please. No reasonable human being would choose the bet, regardless of what V is (in fact even if V were actually infinite.)
What this really comes down to is saying that human beings care about things to a finite degree since they are themselves finite. The Archimedean postulate basically says that we can care infinitely about something, but this is false.
Alex,
I'd like to raise the issue of existence in your argument and computing power:
Your argument seems to assume that for every n, there is Vn such that probability of Vn is 1/2^n.
However, human brains - or human brains + computers, etc., or the brains of aliens, etc. - are not able to compute 1/2^n (for example) for arbitrary n. We understand a symbol "n", but we would not be able to assign probabilities in such manner in the real world (even if it's metaphysically possible, it's not within our power). For every agent with finite computing power (or any finite number of cooperating agents, each with finite computing power), and for all but finitely many values of n (there doesn't have to be a precise number for any agent, since it may change with the circumstances), 1/(2^n) is comprehension-transcendent, and there is no way for the agent to attribute the require probability to make choices on that basis.
A similar difficulty results from the assumption that for any n, there are n units of utility.
A slight weakening of the axiom: There is a positive utility U0 such that for any finite utility U < U0 and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.
I think you could make this work without numerical utility, using only (complete) preferences.
Note that Archimedianism is very different from the von Neumann – Morgenstern Archimedian axiom.
Probably, but only if you have a convex sum operation that allows one to combine countably infinitely many gambles.
Yes, these are very different Archimedean axioms. The vN-M axioms can all be satisfied even if there is a maximum utility. The axiom I give, of course, cannot.
In the real world nobody is an Archimedean; that is certainly the principle to reject. Lots of people will spend a dollar on a lottery ticket. Nobody will spend all their assets on lottery tickets, no matter what the expected payoff is, so long as the chance of receiving it is low.
Actually, my weakening of the axiom doesn't seem enough to generate the paradox. One really does seem to need the fuller and crazy Archimedean axiom. But one would need to be crazy to submit to a week of torture for a one in googolplex chance of some fixed finite good.
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