Expectiles Reframe Disappointment Aversion in Savage Choice

Gul’s 1991 model of disappointment aversion is usually read as a non-expected-utility theory over lotteries. This paper moves the theory into a Savage setting, where acts map states to general outcomes and subjective probability is part of the representation. The technical move is to show that the core object behind disappointment aversion is an expectile: the same asymmetric-loss functional introduced by Newey and Powell for econometric estimation.

The payoff is mostly conceptual rather than empirical. The authors do not report experiments or simulations. Instead, they give representation theorems, elicitation formulas, and a learning argument showing how asymmetric reward-prediction-error updates can converge to an expectiled value.

Core Contribution

The central claim is that disappointment aversion can be pinned down by standard Savage-style structure plus one behavioral asymmetry. In Theorem 4, a binary relation over acts is represented by expectiled utility if and only if it is probabilistically sophisticated, invariant biseparable, and satisfies disappointment hedging. In the disappointment-averse case there is a continuous nonconstant utility index $u$ and a parameter $\beta \geq 0$ such that

The same theorem covers the dual elation-seeking case with $-1 < \beta \leq 0$. This makes the sign and magnitude of $\beta$ do real decision-theoretic work: it measures the asymmetry between losses relative to the endogenous reference point and gains above it.

Technical Approach

The paper’s main mechanism is to translate Gul’s fixed-point representation into expectile form. In the proofs, the authors use the equivalence between Gul’s equation and the expectile first-order condition from asymmetric least squares. The proof of Proposition 1 shows that a solution $v$ of Gul’s equation is also a solution of the expectile condition by rewriting the positive and negative deviations of $u(X)$ around $v$. Theorem 2 then invokes the Newey-Powell characterization of expectiles as asymmetric least-squares minimizers.

The Savage extension relies on axioms that separate belief, taste, and disappointment. The appendix spells out the background structure for invariant biseparable preferences: weak order, dominance, essentiality, event-monotonicity, event-continuity, event-substitution, and a preference-average construction. Against that baseline, disappointment hedging says that mixing can be preferred when it avoids accumulating disappointment in the same states. Its dual, elation speculation, reverses the asymmetry.

A useful part of the paper is the elicitation result. Theorem 5 shows that $\beta$ can be recovered from a single indifference involving a sure midpoint and a binary act. If $F$ is the matching event, then

The same section notes that $P(F)>1/2$ corresponds to disappointment aversion, $P(F)=1/2$ to expected utility, and $P(F)<1/2$ to elation seeking. Once $\beta$ is known, the utility value of an intermediate outcome can be elicited from another binary-act indifference through the paper’s equation (23). That is a cleaner identification route than first recovering an entire expected-utility core.

Results and Analysis

Because this is a theory paper, the evidence is formal. Theorem 6 gives a behavioral foundation for expectiled reward over bounded random rewards: monotonicity, continuity, probabilistic sophistication, and disappointment aversion are equivalent to representation by $\mathbb{E}_{\beta}[U]$. The result says that aversion to accumulated disappointments is not just compatible with expectiles; under the paper’s assumptions it characterizes them.

The neurocomputational section adds a second foundation. The authors model temporal-difference learning with different gains for positive and negative reward prediction errors:

With the normalization $w_+=1$ and $w_-=1+\beta$, the long-run learned value is the expectiled reward. This is not an experiment, but it links the representation to dopamine-based reinforcement-learning findings where positive and negative prediction errors are updated asymmetrically.

The most convincing part is the unification: Gul’s disappointment aversion, Savage acts, expectile risk measures, asymmetric least squares, and asymmetric TD learning all point to the same parameterized functional. The main caveat is scope. The claims are as strong as the axioms and convergence arguments; the paper does not test whether real decision makers’ choices or neural data satisfy the representation.