Multi-Leader Congestion Games with an Adversary

TL;DR

This paper analyzes a multi-leader congestion game with an adversary attacking the most loaded resource, establishing the existence of a tight approximate equilibrium and providing algorithms to compute such equilibria efficiently.

Contribution

It introduces the first polynomial-time method to find a tight approximate equilibrium in multi-leader congestion games with adversarial attacks.

Findings

Existence of a tight 1.1974-approximate equilibrium for all instances.

A polynomial-time algorithm computes a 1.1974-approximate equilibrium.

The approximation factor 1.1974 is proven to be optimal.

Abstract

We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads, causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria may fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a $K$-approximate equilibrium can always be guaranteed, where $K \approx 1.1974$ is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a $K$-approximate equilibrium. The factor $K$ is tight, meaning that there is an instance that does not admit an $\alpha$-approximate equilibrium for any $\alpha<K$. Thus $\alpha=K$ is the smallest possible value of $\alpha$ such that the existence of an $\alpha$-approximate equilibrium can be guaranteed for any instance of the considered game. Secondly, we focus on approximate equilibria of a given fixed instance. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible $\alpha$ among all $\alpha$-approximate equilibria of the given instance.