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

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).

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.

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 *E*_{p},
which satisfies these plausible axioms:

Linearity:

*E*_{p}(*a**f*+*b**g*) =*a**E*_{p}*f*+*b**E*_{p}*g*for*a*and*b*in*V*Probability-match:

*E*_{p}1_{A}=*p*(*A*) for any event*A*, where 1_{A}is 1 on*A*and 0 elsewhereDominance: if

*f*≤*g*everywhere, then*E*_{p}*f*≤*E*_{p}*g*, and if*f*<*g*everywhere, then*E*_{p}*f*<*E*_{p}*g*.

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 *V*_{2}-valued *V*_{1}-linear functionals on
*V*_{1}-vector spaces
*V*_{1} ⊆ *V*_{2}
are real or hyperreal field, except that our extending functional may
need to take values in a field of hyperreals even larger than *V*_{2}. 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 *V*_{2} that
contains a supremum for every subset of *V*, and embed *V*_{2} in *V*_{3} which has a supremum
for every subset of *V*_{2}. Let *Ω* be our probability space.

Let *X* be the space of
bounded *V*_{2}-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 *V*_{2}
(which has values in *V*_{3}) is then a seminorm
dominating *ϕ*. Extend *ϕ* to a *V*-linear function *ϕ* on *X* dominated by *V*_{2}. Note that if *f* > 0 everywhere for *f* with values in *V*, then *f* > *α* > 0 everywhere
for some *α* ∈ *V*_{2}, and
hence *ϕ*(−*f*) ≤ − *α* by
seminorm domination, hence 0 < *α* ≤ *ϕ*(*f*).
Letting *E*_{p}
be *ϕ* restricted to the *V*-valued functions, our construction
is complete.

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

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