The hyperreals and the von Neumann - Morgenstern representation theorem

This is all largely well-known, but I wanted to write it down explicitly. The von Neumann–Morgenstern utility theorem says that if we have a total preorder (complete transitive relation) ≾ on outcomes in a mixture space (i.e., a space such that given members a and b and any t ∈ [0,1], there is a member (1−t)a + tb satisfying some obvious axioms) and satisfying:

• Independence: For any outcomes a, b and c and any t ∈ (0, 1], we have a ≾ b iff ta + (1−t)c ≾ tb + (1−t)c, and

• Continuity: If a ≾ b ≾ c then there is a t ∈ [0,1] such that b ≈ (1−t)a + tc (where x ≈ y iff x ≾ y and y ≾ x)

the preorder can be represented by a real-valued utility function U that is a mixture space homomorphism (i.e., U((1−t)a+tb) = (1−t)U(a) + tU(b)) and such that U(a) ≤ U(b) if and only a ≾ b.

Clearly continuity is a necessary condition for this to hold. But what if we are interested in hyperreal-valued utility functions and drop continuity?

Quick summary:

• Without continuity, we have a hyperreal-valued representation, and

• We can extend our preferences to recover continuity with respect to the hyperreal field.

More precisely, Hausner in 1971 showed that in a finite dimensional case (essentially the mixture space being generated by the mixing operation from a finite number of outcomes we can call “sure outcomes”) with independence but without continuity we can represent the total preorder by a finite-dimensional lexicographically-ordered vector-valued utility. In other words, the utilities are vectors (u₀,...,u_n − 1) of real numbers where earlier entries trump later ones in comparison. Now, given an infinitesimal ϵ, any such vector can be represented as u₀ + u₁ϵ + ... + u_n − 1ϵ^n − 1. So in the finite dimensional case, we can have a hyperreal-valued utility representation.

What if we drop the finite-dimensionality requirement? Easy. Take an ultrafilter on the space of finitely generated mixture subspaces of our mixture space ordered by inclusion, and take an ultraproduct of the hyperreal-valued representations on each of these, and the result will be a hyperreal-valued utility representing our preorder on the full space.

(All this stuff may have been explicitly proved by Richter, but I don’t have easy access to his paper.)

Now, on to the claim that we can sort of recover continuity. More precisely, if we allow for probabilistic mixtures of our outcomes with weights in the hyperreal field F that U takes values in, then we can embed our mixing space M in an F-mixing space M_F (which satisfies the axioms of a mixing space with respect to members of the larger field F), and extend our preference ordering ≾ to M_F such that we have:

• F-continuity: If a ≾ b ≾ c then there is a t ∈ F with 0 ≤ t ≤ 1 such that b ≈ (1−t)a + tc (where x ≈ y iff x ≾ y and y ≾ x).

In other words, if we allow for sufficiently fine-grained probabilistic mixtures, with hyperreal probabilities, we get back the intuitive content of continuity.

To see this, embed M as a convex subset of a real vector space V using an embedding theorem of Stone from the middle of the last century. Without loss of generality, suppose 0 ∈ M and U(0) = 0. Extend U to the cone C_M = {ta : t ∈ [0, ∞), a ∈ M} generated by M by letting U(ta) = tU(a). Note that this is well-defined since U(0) = 0 and if ta = ub with 0 ≤ t < u, then b = (1−s) ⋅ 0 + s ⋅ a, where s = t/u, and so U(b) = sU(a). It is easy to see that the extension will be additive. Next extend U to the linear subspace V_M generated by C_M (and hence by M) by letting U(a−b) = U(a) − U(b) for a and b in C_M. This is well-defined because if a − b = c − d, then a + d = b + c and so U(a) + U(d) = U(b) + U(c) and hence U(a) − U(b) = U(c) − U(d). Moreover, U is now a linear functional on V_M. If B is a basis of V_M, then let V_M^F be an F-vector space with basis B, and extend U to an F-linear functional from V_M^F to F by letting U(t₁a₁+...+t_na_n) = t₁U(a₁) + ... + t_nU(a_n), where the a_i are in B and the t_i are in F. Now let M^F be the F-convex subset of V_M^F generated by M. This will be an F-mixing space (i.e., it will satisfy the axioms of a mixing space with the field F in place of the reals). Let a ≾ b iff U(a) ≤ U(b) for a and b in M^F. Then if a ≾ b ≾ c, we have U(a) ≤ U(b) ≤ U(c). Let t between 0 and 1 in F be such that (1−t)U(a) + tU(c) = U(b). By F-linearity of U, we will then have U((1−t)a+tc) = U(b).