Malicious Bayesian Congestion Games

TL;DR

This paper introduces malicious Bayesian congestion games where players can be malicious or rational, explores the existence and complexity of pure Bayesian Nash equilibria, and analyzes how malicious types affect overall system performance.

Contribution

It formally defines malicious Bayesian congestion games, proves the NP-completeness of equilibrium existence, and analyzes the impact of malicious players on social cost through the Price of Malice.

Findings

Pure Bayesian Nash equilibria may not exist in general.

Deciding equilibrium existence is NP-complete even with linear latency functions.

Malicious types can sometimes reduce social cost, bounded by a tight factor.

Abstract

In this paper, we introduce malicious Bayesian congestion games as an extension to congestion games where players might act in a malicious way. In such a game each player has two types. Either the player is a rational player seeking to minimize her own delay, or - with a certain probability - the player is malicious in which case her only goal is to disturb the other players as much as possible. We show that such games do in general not possess a Bayesian Nash equilibrium in pure strategies (i.e. a pure Bayesian Nash equilibrium). Moreover, given a game, we show that it is NP-complete to decide whether it admits a pure Bayesian Nash equilibrium. This result even holds when resource latency functions are linear, each player is malicious with the same probability, and all strategy sets consist of singleton sets. For a slightly more restricted class of malicious Bayesian congestion games, we provide easy checkable properties that are necessary and sufficient for the existence of a pure Bayesian Nash equilibrium. In the second part of the paper we study the impact of the malicious types on the overall performance of the system (i.e. the social cost). To measure this impact, we use the Price of Malice. We provide (tight) bounds on the Price of Malice for an interesting class of malicious Bayesian congestion games. Moreover, we show that for certain congestion games the advent of malicious types can also be beneficial to the system in the sense that the social cost of the worst case equilibrium decreases. We provide a tight bound on the maximum factor by which this happens.