You Can't Always Get What You Want: Games of Ordered Preference

TL;DR

This paper introduces a recursive method and relaxations for finding generalized Nash equilibria in noncooperative games with players' preferences arranged in a hierarchy, applicable to real-world scenarios like traffic management.

Contribution

It develops a recursive approach for optimality conditions and proposes relaxations that can be solved as mixed complementarity problems, advancing the analysis of hierarchical preference games.

Findings

The approach reliably converges to equilibrium solutions.

It accurately captures players' ordered preferences.

The relaxations improve computational tractability.

Abstract

We study noncooperative games, in which each player's objective is composed of a sequence of ordered- and potentially conflicting-preferences. Problems of this type naturally model a wide variety of scenarios: for example, drivers at a busy intersection must balance the desire to make forward progress with the risk of collision. Mathematically, these problems possess a nested structure, and to behave properly players must prioritize their most important preference, and only consider less important preferences to the extent that they do not compromise performance on more important ones. We consider multi-agent, noncooperative variants of these problems, and seek generalized Nash equilibria in which each player's decision reflects both its hierarchy of preferences and other players' actions. We make two key contributions. First, we develop a recursive approach for deriving the first-order optimality conditions of each player's nested problem. Second, we propose a sequence of increasingly tight relaxations, each of which can be transcribed as a mixed complementarity problem and solved via existing methods. Experimental results demonstrate that our approach reliably converges to equilibrium solutions that strictly reflect players' individual ordered preferences.