Uniform Mixed Equilibria in Network Congestion Games with Link Failures

Motivated by possible applications in fault-tolerant routing, we introduce the notion of uniform mixed equilibria in network congestion games with adversarial link failures, where players need to route traffic from a source to a destination node. Given an integer rho >= 1, a rho-uniform mixed strategy is a mixed strategy in which a player plays exactly rho edge disjoint paths with uniform probabilities, so that a rho-uniform mixed equilibrium is a tuple of rho-uniform mixed strategies, one for each player, in which no player can lower her cost by deviating to another rho-uniform mixed strategy. For games with weighted players and affine latency functions, we show existence of rho-uniform mixed equilibria and provide a tight characterization of their price of anarchy. For games with unweighted players, instead, we extend the existential guarantee to any class of latency functions and, restricted to games with affine latencies, we derive a tight characterization of both the prices of anarchy and stability

Altruism in Atomic Congestion Games

This paper studies the effects of introducing altruistic agents into atomic congestion games. Altruistic behavior is modeled by a trade-off between selfish and social objectives. In particular, we assume agents optimize a linear combination of personal delay of a strategy and the resulting increase in social cost. Our model can be embedded in the framework of congestion games with player-specific latency functions. Stable states are the Nash equilibria of these games, and we examine their existence and the convergence of sequential best-response dynamics. Previous work shows that for symmetric singleton games with convex delays Nash equilibria are guaranteed to exist. For concave delay functions we observe that there are games without Nash equilibria and provide a polynomial time algorithm to decide existence for symmetric singleton games with arbitrary delay functions. Our algorithm can be extended to compute best and worst Nash equilibria if they exist. For more general congestion games existence becomes NP-hard to decide, even for symmetric network games with quadratic delay functions. Perhaps surprisingly, if all delay functions are linear, then there is always a Nash equilibrium in any congestion game with altruists and any better-response dynamics converges. In addition to these results for uncoordinated dynamics, we consider a scenario in which a central altruistic institution can motivate agents to act altruistically. We provide constructive and hardness results for finding the minimum number of altruists to stabilize an optimal congestion profile and more general mechanisms to incentivize agents to adopt favorable behavior.Comment: 13 pages, 1 figure, includes some minor adjustment

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.Comment: 18 pages, submitted to WAOA'0

Robust Price of Anarchy for Atomic Games with Altruistic Players

We study the inefficiency of equilibria for various classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i's perceived cost is a convex combination of 1-\beta_i times his direct cost and \beta_i times the social cost. Tuning the parameters \beta_i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of linear congestion games, fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concepts. Thereto, we suitably adapt the smoothness notion introduced by Roughgarden and show that it captures the essential properties to determine the robust price of anarchy of these games. Our bounds reveal that for congestion games and cost-sharing games the worst-case robust price of anarchy increases with increasing altruism, while for valid utility games it remains constant and is not affected by altruism. We also show that the increase in price of anarchy is not a universal phenomenon: for symmetric singleton linear congestion games we derive a bound on the price of anarchy for pure Nash equilibria that decreases as the level of altruism increases. Since the bound is also strictly lower than the robust price of anarchy, it exhibits a natural example in which Nash equilibria are more efficient than more permissive notions of equilibrium