# Revenue Maximization with an Uncertainty-Averse Buyer

## TL;DR

This paper explores revenue-maximizing mechanisms for buyers with behavioral risk attitudes modeled by prospect theory, showing that simple posted pricing can be effective under certain conditions, but risk aversion complicates revenue approximation.

## Contribution

It introduces a model for buyer behavior beyond expected utility, characterizes optimal mechanisms under prospect theory, and demonstrates the effectiveness of posted pricing with bounded risk aversion.

## Key findings

- Posted pricing achieves a constant approximation under bounded risk aversion.
- Optimal mechanisms may need to be randomized due to buyer risk attitudes.
- Risk aversion can hinder revenue approximation in dynamic settings.

## Abstract

Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.