Existence and uniqueness of maximal elements for preference relations: Variational approach

TL;DR

This paper reformulates the problem of finding maximal elements in preference relations as variational inequality problems, establishing existence and uniqueness results, and proposes an algorithm with convergence guarantees.

Contribution

It introduces a variational inequality framework for preference relations, proving existence and uniqueness of maximal elements, and develops a convergent algorithm inspired by steepest descent.

Findings

Existence of maximal elements under mild assumptions

Uniqueness of maximal elements via Minty variational inequality

Convergence of the proposed algorithm to a maximal element

Abstract

In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the sequence generated by this algorithm converges to a maximal element.