Until very recently, I thought there was only one argument for the idea that, barring deontic concerns and the like, rationality is connected to the maximization of expected utilities, namely the long-run advantage argument based on the Law of Large Numbers. But there is another: an argument from a plausible set of axioms for rational preferability. Fix a probability space Ω. Say that a gamble is a bounded real-valued random variable on Ω. Suppose that there is a rational preferability ordering < on gambles, where we write A<B if B is preferable to A. Here are some plausible axioms for <:
- Transitivity: < is transitive
- Domination: If A(ω)≤B(ω) for all ω∈Ω, then for all C, if C<A, then C<B, and if B<C, then A<C.
- Sure Thing: If A and B are gambles that have certainty of getting payoffs a and b respectively, with a<b, then A<B.
- Additivity: If A<C and B<D, then A+B<C+D.
- Equivalence: If A and B are probabilistically equivalent (i.e., P(A∈U)=P(B∈U) for every measurable U), then for all C we have A<C if and only if B<C, and C<A if and only if C<B.
The most controversial will be, I suppose, Additivity. But there is a very simple argument for it: If you should choose C over A, and D over B, then you should choose the combination of C and D over the combination of A and B.
Add a handy technical assumption:
- Continuity: There is a collection of events Ea, for 0<a<1, such that P(Ea)=a and Ea is a subset of Eb when a<b.
Theorem: Assume (1)-(6). If E(A)<E(B) for gambles A and B, then A<B.
The proof is given in this footnote: [note 1].
Personally, I am suspicious of transitivity in general, but I am less suspicious of it in the case of real-valued bounded gambles.