Pure Nash Equilibria in Resource Graph Games

TL;DR

This paper characterizes the cost functions that guarantee the existence of pure Nash equilibria in resource graph games, providing conditions for both weighted and unweighted players and exploring special cases like matroid strategies.

Contribution

It offers complete characterizations of cost functions ensuring pure Nash equilibria in resource graph games, including new results for weighted players and matroid strategy sets.

Findings

Pure Nash equilibria exist under symmetric linear cost functions for unweighted players.

Existence of equilibria for weighted players requires linear or exponential cost functions with symmetry.

Hardness results are provided for deciding equilibrium existence in network routing and matroid games.

Abstract

This paper studies the existence of pure Nash equilibria in resource graph games, which are a general class of strategic games used to succinctly represent the players' private costs. There is a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function that depends on the load vector of the resources in a specified neighborhood. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. 1. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players' strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. 2. For games with weighted players, pure Nash equilibria are guaranteed to exist for any choice of the players' strategy space if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. 3. For the special case that the players' strategy sets are matroids, we show that pure Nash equilibria exist under a local monotonicity property, even when cost functions are player-specific. We point out an application of this result to bilevel load balancing games, which are motivated by the study of network infrastructures that are resilient against external attackers and internal congestion effects. 4. Finally, we derive hardness results for deciding whether a given strategy is a pure Nash equilibrium for network routing games and matroid games, respectively.