Variational Reformulation of Generalized Nash Equilibrium Problems with Non-ordered Preferences

TL;DR

This paper introduces a variational reformulation for generalized Nash equilibrium problems with non-ordered preferences, enabling analysis without numerical preference representation and extending to incomplete and non-transitive preferences.

Contribution

It characterizes equilibrium problems using variational inequalities without preference representation and establishes conditions linking quasi-variational solutions to Nash equilibria.

Findings

Reformulation applies to non-ordered preferences.

Conditions for equilibrium existence with incomplete preferences.

Application to Arrow-Debreu economy under uncertainty.

Abstract

The preferences of players in non-cooperative games represent their choice in the set of available options, which meet the completeness property if players are able to compare any pair of available options. In the existing literature, the variational reformulation of generalized Nash games relies on the numerical representation of players' preferences into objective functions, which is only possible if preferences are transitive and complete. In this work, we first characterize the jointly convex generalized Nash equilibrium problems in terms of a variational inequality without requiring any numerical representation of preferences. Furthermore, we provide the suitable conditions under which any solution of a quasi-variational inequality becomes an equilibrium for the generalized Nash game with non-ordered (incomplete and non-transitive) non-convex inter-dependent preferences. We check the solvability of these games when the strategy maps of players are (possibly) unbounded. As an application, we derive the occurrence of competitive equilibrium for Arrow-Debreu economy under uncertainty.