Tuesday, November 22, 2022

Hyperreal expected value

I think I have a hyperreal solution, not entirely satisfactory, to three problems.

  1. The problem of how to value the St Petersburg paradox. The particular version that interests me is one from Russell and Isaacs which says that any finite value is too small, but any infinite value violates strict dominance (since, no matter what, the payoff will be less than infinity).

  2. How to value gambles on a countably infinite fair lottery where the gamble is positive and asymptotically approaches zero at infinity. The problem is that any positive non-infinitesimal value is too big and any infinitesimal value violates strict dominance.

  3. How to evaluate expected utilities of gambles whose values are hyperreal, where the probabilities may be real or hyperreal, which I raise in Section 4.2 of my paper on accuracy in infinite domains.

The apparent solution works as follows. For any gamble with values in some real or hyperreal field V and any finitely-additive probability p with values in V, we generate a hyperreal expected value Ep, which satisfies these plausible axioms:

  1. Linearity: Ep(af+bg) = aEpf + bEpg for a and b in V

  2. Probability-match: Ep1A = p(A) for any event A, where 1A is 1 on A and 0 elsewhere

  3. Dominance: if f ≤ g everywhere, then Epf ≤ Epg, and if f < g everywhere, then Epf < Epg.

How does this get around the arguments I link to in (1) and (2) that seem to say that this can’t be done? The trick is this: the expected value has values in a hyperreal field W which will be larger than V, while (4)–(6) only hold for gambles with values in V. The idea is that we distinguish between what one might call primary values, which are particular goods in the world, and what one might call distribution values, which specify how much a random distribution of primary values is worth. We do not allow the distribution values themselves to be the values of a gamble. This has some downsides, but at least we can have (4)–(6) on all gambles.

How is this trick done?

I think like this. First it looks like the Hahn-Banach dominated extension theorem holds for V2-valued V1-linear functionals on V1-vector spaces V1 ⊆ V2 are real or hyperreal field, except that our extending functional may need to take values in a field of hyperreals even larger than V2. The crucial thing to note is that any subset of a real or hyperreal field has a supremum in a larger hyperreal field. Then where the proof of the Hahn-Banach theorem uses infima and suprema, you move to a larger hyperreal field to get them.

Now, embed V in a hyperreal field V2 that contains a supremum for every subset of V, and embed V2 in V3 which has a supremum for every subset of V2. Let Ω be our probability space.

Let X be the space of bounded V2-valued functions on Ω and let M ⊆ X be the subspace of simple functions (with respect to the algebra of sets that Ω is defined on). For f ∈ M, let ϕ(f) be the integral of f with respect to p, defined in the obvious way. The supremum on V2 (which has values in V3) is then a seminorm dominating ϕ. Extend ϕ to a V-linear function ϕ on X dominated by V2. Note that if f > 0 everywhere for f with values in V, then f > α > 0 everywhere for some α ∈ V2, and hence ϕ(−f) ≤  − α by seminorm domination, hence 0 < α ≤ ϕ(f). Letting Ep be ϕ restricted to the V-valued functions, our construction is complete.

I should check all the details at some point, but not today.

No comments: