A Comment on: “Expected Uncertain Utility”

Abstract

In an innovative paper, replete with many important results and insights, Gul and Pesendorfer (2014) (hereafter, GP) proposed a novel model for choice under uncertainty. They considered a setting of purely subjective uncertainty in which the objects of choice are acts that, for each state of nature , deliver a monetary prize x from a set of final prizes , with . We denote the set of acts by , and the decision-maker's preference relation defined over by a weak order ≿.

In GP's model, the decision-maker (hereafter, DM) has a prior μ defined over , a σ-algebra of what they referred to as ideal events. GP interpreted any ideal event E (in ) as one for which the DM can precisely quantify that event's uncertainty by assigning it the probability . An event is deemed ideal by the DM if both it and its complement together satisfy a version of Savage (1954)'s sure thing principle.

Unfortunately, GP's characterization fails on two accounts, as their axioms neither ensure

In this note, we show that strengthening one of GP's axioms, along with a slight modification of their continuity axiom, provides a characterization of EUU maximization. But first, we present in Section 2 an example of an EUU functional involving a state-dependent interval utility and show that the preferences generated by this example, despite satisfying all of GP's axioms, cannot be represented by an EUU function of the form in (2).

Let the state space be endowed with the Lebesgue measure μ. Let denote the set of measurable events with respect to μ. Following GP, is the (interval) envelope of an act f, with (respectively, ) denoting the lower (respectively, upper) envelope.

We show that ≿ satisfies GP's Axioms A1–A6 which we list here for the convenience of the reader. To state them, we employ the following notation: for any pair of acts f and g and any event , fCg denotes the act that agrees with f on C and with g on the complement of C. We also require the following definitions.

An event E is ideal if implies for all acts f, g, h, and . An event A is null if for all acts f, g, and h. An event D is diffuse if for every non-null ideal event E. Let (respectively, , ) be the set of all ideal (respectively, null, diffuse) events. Let denote the set of ideal simple acts.¹

As in GP, we say an event E is left (respectively, right) ideal if implies (respectively, implies ). Let and be the collection of left and right ideal sets, respectively. GP's Lemma B0 establishes .

In line with GP's use of notation, events E, , , et cetera, denote ideal events while events D, , denote diffuse events. The following are GP's six Axioms (A1–A6).

To verify ≿ satisfies the above six axioms, we utilize the fact that an event is deemed ideal by ≿ if and only if it is measurable (i.e., an element of ).

Returning to the axioms, we see each is verified as follows:

Since the preference relation ≿ generated by (3) satisfies GP's Axioms 1–6, it follows from GP's Theorem 1 that it should admit an EUU representation with prior μ.²

We retain four of GP's axioms and propose strengthening Axiom A3 and modifying Axiom A6(i) while leaving the original Axiom A6(ii) unchanged. The strengthening of Axiom A3 ensures the constancy of conditional certainty equivalents of diffuse “bets” which rules out the (counter-)example from the previous section. The modification of Axiom A6(i) enables us to establish that the set of ideal events is indeed a σ-algebra.

GP's Axiom A6(i) implies a weaker version of Arrow's monotone continuity that applies to ideal acts and ideal events. Our new A6*(i) is the monotone continuity axiom applied to all acts and ideal events.

It is straightforward to show that A6*(i) implies the countable additivity of ideal events and simplifies the proof of Theorem 1. The next property ensures the conditional certainty equivalence between diffuse acts. Its role is similar to that of P3 in Savage's axiomatization of subjective expected utility.

For simplicity, A7 can be combined with A3 into the following:³