Minimax identity with robust utility functional for a non-concave utility

TL;DR

This paper investigates the minimax identity for non-concave, upper-semicontinuous utility functions under mild growth conditions, extending classical results by relaxing the concavity assumption and analyzing the effects of concavification.

Contribution

It introduces a novel analysis of the minimax identity for non-concave utilities using concave envelopes, providing new inequalities and equalities in this broader setting.

Findings

Established minimax equalities for non-concave utility functions.

Derived bounds relating utility of original and concavified functions.

Extended results to models with upper bounds on endowment.

Abstract

We study the minimax identity for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope of the utility function we obtain equalities and inequalities between the robust utility functionals of an initial utility function and its concavification. Furthermore, we prove similar equalities and inequalities in the case of implementing an upper bound on the final endowment of the initial model.