Congestion Games with Linearly Independent Paths: Convergence Time and Price of Anarchy

Abstract

Abstract. We investigate the effect of linear independence in the strategies of congestion games on the convergence time of best improvement sequences and on the pure Price of Anarchy. In particular, we consider symmetric congestion games on extension-parallel networks, an interesting class of networks with linearly independent paths, and establish two remarkable properties previously known only for parallel-link games. More precisely, we show that for arbitrary (non-negative and non-decreasing) latency functions, any best improvement sequence reaches a pure Nash equilibrium in at most as many steps as the number of players, and that for latency functions in class D, the pure Price of Anarchy is at most ρ(D). As a by-product of our analysis, we obtain that for symmetric congestion games on general networks with latency functions in class D, the Price of Stability is at most ρ(D).

Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games

“…We can therefore conclude that the (altruistic) price of anarchy will be a non-increasing function of ρ. This is a remarkable result since, to the best of our knowledge, the classical price of anarchy is unknown (the best lower bound is given by Fotakis [11]). …”

“…Lemma 4 (Fotakis [11] The following lemma gives inefficiency results for global minima of the potential function (compared to any feasible flow). Since the local minima of correspond to the Nash equilibria of the game , it follows that the global minima of are Nash equilibria.…”

“…Since the local minima of correspond to the Nash equilibria of the game , it follows that the global minima of are Nash equilibria. Furthermore, Fotakis [11] shows that every Nash equilibrium of a symmetric congestion game on an extension-parallel graph is a global minimum of the potential function . In particular, this means that the inefficiency results in Lemma 5 hold for the price of stability of symmetric network congestion games, and the price of anarchy of symmetric extension-parallel congestion games.…”

Convergence of Best-Response Dynamics in Games with Conflicting Congestion Effects

“…This observation has led to a large amount of work that identified special classes of congestion games, where bestresponse dynamics converge to a Nash equilibrium in polynomial time or even linear time. This agenda has been the focus of [7,11] in a setting with negative congestion effects, and was also studied in a setting of positive congestion effects [2]. In particular, it has been shown that it takes at most n steps (where n is the number of users) to converge to a Nash equilibrium if the network is composed of parallel links [7], and this result has been later extended to extension-parallel networks [11].…”

“…This agenda has been the focus of [7,11] in a setting with negative congestion effects, and was also studied in a setting of positive congestion effects [2]. In particular, it has been shown that it takes at most n steps (where n is the number of users) to converge to a Nash equilibrium if the network is composed of parallel links [7], and this result has been later extended to extension-parallel networks [11]. For resource selection games (i.e., where feasible strategies are composed of singletons), it has been shown in [13] that better-response dynamics converge within at most mn 2 steps for general cost functions (where m and n are the number of resources and users, respectively).…”