Risk Neutrality Incompleteness and Degenerate Indifference

Corollary 3.
A binary relation on a X is a Preorder that satisfies Incompleteness, Independence, Strict
Monotonicity and Restricted Hemicontinuity if and only if there exists a compact, convex set P ⊆
∆(S) with nonempty interior such that, for all x, y ∈ X
x y ⇐⇒ px ≥ py for all p ∈ P (5)
x = y ⇐⇒ px = py for all p ∈ P (6)
Moreover,
(x y ⇔ x ≥ y) ⇐⇒ P = ∆(S) (7)
It is an implication of Proposition A.1 in Ghirardato, Maccheroni, and Marinacci (2004) that
the stated axioms are sufficient for the existence of a set P ⊆ ∆(S) that represents as in (5).
It is easy to verify that Strict Monotonicity implies a nonempty interior for P . In light of this
fact, the indifference-degeneracy implication (6) again follows from Proposition 1. Given (6), it is
easily seen that (7) is true. This equivalence shows that, in the present setting, Bewley preferences
coincide with the usual partial ordering if and only if P contains all possible priors, hence that the
model becomes uninformative in this special case.
Importantly, Corollary 3 can also be stated as a preference-completion theorem. To this end,
we write explicitly as an axiom the requirement that there exist two distinct acts between which
the decision maker is indifferent.
Nontrivial Indifference
There exist distinct x, y ∈ X such that x ∼ y.
Corollary 4.
If is a Preorder on X that satisfies Independence, Strict Monotonicity, Nontrivial Indifference
and Restricted Hemicontinuity, then is complete.
This result parallels the well-known and general preference-completion theorem that is due to
Schmeidler (1971). The latter provides natural topological conditions on the preference domain
(i.e. connectedness) as well on the relation and its asymmetric part ≻ (i.e. hemicontinuity)
that are sufficient for to be complete. Although the domain in Corollary 4 is a special case in
the class of domains that Schmeidler’s theorem allows for, we note that, unlike the latter result,
almost every condition in the statement of Corollary 4 is a preference postulate with a behavioral
content.
Appendix
Proof of Proposition 1.
We will first show that there is a set {q 1 , . . . , q n } of n linearly independent elements in rint(P ).
Let p ∈ rint(P ). For ɛ > 0, let B ɛ (p) := {v ∈ R n ++ : d(v, p) < ɛ}, where d denotes Euclidean
distance. For all i ≤ n, define q i ∈ R n ++ by
q i := p + ɛ 2 ei .
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