Malicious Bayesian Congestion Games

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.

💡 Research Summary

The paper introduces a novel extension of congestion games called Malicious Bayesian Congestion Games. In the classical model every player is a rational agent who selects a strategy to minimize her own latency. The authors enrich this setting by allowing each player to be of one of two possible types: a rational type (R) that behaves as in the standard model, and a malicious type (M) whose sole objective is to maximize the total latency experienced by the other players. The type of each player is drawn independently from a known probability distribution; a player knows her own type but only the distribution of the others. This yields a Bayesian game in which strategies may be contingent on the realized type.

Model and Definitions

• Players: a finite set N = {1,…,n}.
• Resources: a set E, each equipped with a latency (delay) function ℓ_e : ℝ_+ → ℝ_+. The paper mainly studies linear latencies ℓ_e(x)=a_e·x+b_e, but the definitions hold for any non‑decreasing functions.
• Strategies: each player i has a strategy set S_i ⊆ 2^E (often restricted to singleton sets in the hardness proofs).
• Types: a random variable θ_i ∈ {R,M} with Pr