Friday, February 18, 2022

How not to value wagers

Given the Axiom of Choice, there is a rotationally invariant finitely additive probability measure defined for all subsets of a circle. We can use such a finitely probability measure to define an expected value Ef or integral of a bounded function f on the circle, and we might want to have a decision theory based on this expected value. Given a wager that pays f(z) at a uniformly randomly chosen location z on the circle, we are indifferent to buying the wager at price Ef, we must accept the wager at lower prices, and we must reject it at higher prices.

This procedure, however, leads to the following interesting thing: There will be bounded wagers that pay more than y no matter what, but where one is indifferent with respect to buying the wager at price y. To see this, let x be an irrational number, and as in my previous post, let u be a bounded function on the circle such that u(ρz) > u(z) for all z where ρ is rotation by x degrees. Then let f(z) = u(ρz) − u(z). Because of the additivity of integrals with respect to finitely additive measures and rotational invariance, we have Ef = ∫f(ρz)dP(z) − ∫f(z)dP(z) = 0. But f(z) > 0 for all z. So the decision theory tells us to be indifferent to the game where you get payoff f(z) at z when the game is offered for free, even though no matter what the outcome of the game, you will received a strictly positive amount.

More generally, given the Axiom of Choice, there is no finitely-additive rotationally-invariant expected value assignment for bounded utilities that respects the principle that any gamble that is sure to pay more than y ought to be accepted at price y.

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