On the optimality of likelihood ratio test for prospect theory based binary hypothesis testing

TL;DR

This paper investigates the conditions under which the likelihood ratio test remains optimal in binary hypothesis testing when decision-makers exhibit behavioral biases modeled by prospect theory, revealing scenarios where randomized rules are necessary.

Contribution

It demonstrates that unlike Bayesian testing, LRT may not always be optimal under prospect theory, and provides conditions and representations for optimal decision rules involving randomization.

Findings

LRT may or may not be optimal in prospect theory-based testing.

Optimal rules can involve randomization of up to two LRTs.

Nonrandomized LRTs are not always optimal in this context.

Abstract

In this letter, the optimality of the likelihood ratio test (LRT) is investigated for binary hypothesis testing problems in the presence of a behavioral decision-maker. By utilizing prospect theory, a behavioral decision-maker is modeled to cognitively distort probabilities and costs based on some weight and value functions, respectively. It is proved that the LRT may or may not be an optimal decision rule for prospect theory based binary hypothesis testing and conditions are derived to specify different scenarios. In addition, it is shown that when the LRT is an optimal decision rule, it corresponds to a randomized decision rule in some cases; i.e., nonrandomized LRTs may not be optimal. This is unlike Bayesian binary hypothesis testing in which the optimal decision rule can always be expressed in the form of a nonrandomized LRT. Finally, it is proved that the optimal decision rule for prospect theory based binary hypothesis testing can always be represented by a decision rule that randomizes at most two LRTs. Two examples are presented to corroborate the theoretical results.