Posetal Games: Efficiency, Existence, and Refinement of Equilibria in Games with Prioritized Metrics

TL;DR

This paper introduces Posetal Games, a framework where players' preferences over outcomes are modeled with partially ordered metrics, enabling analysis of equilibria with hierarchical priorities in multi-agent systems.

Contribution

It formalizes Posetal Games with preference structures, provides conditions for pure Nash Equilibria existence, and introduces refinement operations to narrow equilibrium sets.

Findings

Pure Nash Equilibria exist under certain preference conditions.

Refinement operations can systematically reduce equilibrium sets.

Application to autonomous driving demonstrates interpretability of outcomes.

Abstract

Modern applications require robots to comply with multiple, often conflicting rules and to interact with the other agents. We present Posetal Games as a class of games in which each player expresses a preference over the outcomes via a partially ordered set of metrics. This allows one to combine hierarchical priorities of each player with the interactive nature of the environment. By contextualizing standard game theoretical notions, we provide two sufficient conditions on the preference of the players to prove existence of pure Nash Equilibria in finite action sets. Moreover, we define formal operations on the preference structures and link them to a refinement of the game solutions, showing how the set of equilibria can be systematically shrunk. The presented results are showcased in a driving game where autonomous vehicles select from a finite set of trajectories. The results demonstrate the interpretability of results in terms of minimum-rank-violation for each player.