The mathematics of expecting disappointment

Why might someone value a risky opportunity below its average payoff? One explanation is that falling short of expectations matters more than exceeding them by the same amount. But this raises a circular question: if disappointment depends on the opportunity’s value, how is that value determined?

In the theoretical preprint Disappointment Aversion and Expectiles, Fabio Bellini, Fabio Maccheroni, Tiantian Mao, Ruodu Wang and Qinyu Wu develop new foundations and representations for Faruk Gul’s disappointment-aversion model. Their central connection is to an established statistical quantity called an expectile. They also prove that a stylized learning process can converge to the same valuation. These are mathematical results, not new experimental evidence about human choices or brains.

A benchmark generated by the choice itself

The model first translates outcomes into utility: a numerical representation of how desirable they are. Disappointment occurs when realized utility falls below the risky choice’s own advance valuation. Outcomes above that benchmark produce elation.

A parameter called beta controls the asymmetry. Positive beta means disappointment aversion; zero recovers ordinary expected utility; negative beta, restricted to values above minus one, represents elation seeking.

The authors show that the valuation is the unique level at which the expected size of upward utility deviations equals the expected size of downward deviations multiplied by one plus beta. A disappointment-averse decision maker therefore requires extra expected elation to balance expected disappointment.

This balance defines an expectile of utility. It differs from a percentile, which identifies a position in a distribution: an expectile depends on the magnitudes of deviations on both sides of its benchmark. With beta equal to zero, it becomes the mean utility.

Under a finite-second-moment assumption, the same valuation minimizes average squared deviations, with below-benchmark deviations receiving a weight of one plus beta. The statistical connection supplies an optimization formula for a value originally defined through a self-consistency equation.

Beyond monetary gambles

The paper extends the model from monetary lotteries to general outcomes. Choices are represented as acts: plans specifying which outcome occurs in each possible state of the world. A probability distribution over those states supplies the uncertainty.

This matters because two choices can disappoint in exactly the same circumstances. Looking only at their separate outcome distributions would lose that relationship.

The authors introduce a behavioral condition called disappointment hedging. Suppose two acts are equally attractive, and a third disappoints in precisely the same states as the first. Combining that third act with the first offers no protection against shared disappointing states. The condition says that combining it with the second should be at least as attractive. These combinations average utility state by state, rather than randomly selecting one whole act.

Together with assumptions about consistent utility measurement and preference for better outcome distributions, disappointment hedging characterizes the expectiled-utility model. This is an equivalence within the specified mathematical framework, not a demonstration that everyone follows those assumptions.

Caution as probability reweighting

For nonnegative beta, the authors also express the valuation as the lowest expected utility across a particular collection of reweighted probability distributions.

The minimizing distribution increases the odds of disappointing states by a factor of one plus beta, while preserving conditional probabilities within that event and its complement. This gives the model a cautious interpretation: evaluate a choice as though disappointing circumstances had become more salient.

The reweighting is an equivalent decision rule, not a claim that real-world probabilities change. Increasing beta lowers the model’s valuation; at zero, the reweighting disappears.

Learning the same value

A further theorem considers an estimate repeatedly updated after independent draws from an unchanged, bounded reward distribution. Positive surprises move the estimate upward; negative surprises move it downward with relative strength one plus beta.

With positive learning steps whose sum diverges but whose squared sum remains finite, the estimate converges almost surely to the reward’s expectile. In other words, the learning mechanism reaches the same balance as the decision model.

The authors relate this rule to earlier neuroscience research on asymmetric reward-prediction errors. Their theorem does not establish that biological learning universally implements it. Bounded, independent rewards and the prescribed learning schedule are substantive restrictions.

The contribution is a connection between descriptions of choice, statistical estimation and learning dynamics. It offers clearer formulas and testable behavioral implications, while leaving empirical validation of those implications to further work.