Bounding the Uncertainty of Graphical Games: The Complexity of Simple Requirements, Pareto and Strong Nash Equilibria

TL;DR

This paper explores the computational complexity of bounding uncertainty in graphical games, revealing that adding simple requirements can make equilibrium existence and computation intractable, especially under stronger conditions.

Contribution

It provides new complexity results showing that even simple additional constraints can lead to intractability in computing Nash equilibria in graphical games.

Findings

Existence of Nash equilibria becomes uncertain with simple additional requirements.

Computing equilibria is intractable when simple constraints are added.

Stronger equilibrium conditions lead to higher complexity, reaching the second level of the polynomial hierarchy.

Abstract

We investigate the complexity of bounding the uncertainty of graphical games, and we provide new insight into the intrinsic difficulty of computing Nash equilibria. In particular, we show that, if one adds very simple and natural additional requirements to a graphical game, the existence of Nash equilibria is no longer guaranteed, and computing an equilibrium is an intractable problem. Moreover, if stronger equilibrium conditions are required for the game, we get hardness results for the second level of the polynomial hierarchy. Our results offer a clear picture of the complexity of mixed Nash equilibria in graphical games, and answer some open research questions posed by Conitzer and Sandholm (2003).