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

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

“…Since the local minima of Φ correspond to the Nash equilibria of the game Γ , it follows that the global minima of Φ are Nash equilibria. Furthermore, Fotakis [9] shows that every Nash equilibrium of a symmetric congestion game on an extension-parallel graph is a global minimum of the potential function Φ. In particular, this means that the ineffiency 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.…”

“…Lemma 4 (Fotakis [9]). Let Γ be a congestion game with cost functions d e , and let Φ be an exact potential for Γ .…”

Greediness and equilibrium in congestion games

“…Fotakis et al [3] have already shown that Z(G) ⊂ NE(G) for simple congestion games. Fotakis [4] showed that if a class of congestion games that satisfy two conditions: (1) the game form is that of 'extension-parallel graph', namely one can map the resources to the set of edges in a extension-parallel graph and the strategies are the set of paths leading from a certain node in the graph (designated as the source node) to another node (designated as the target node); and (2) the resource payoff functions satisfy a property referred to as the 'Common Best Reply' requirement, met in symmetric congestion game. In particular, their result implies that Z(G) ⊂ NE(G) for simple congestion games, where strategies are the singleton resource subsets of R.…”

The Efficiency of Best-Response Dynamics

“…The observation that convergence via best response dynamics can be exponentially long has led to a large amount of work aiming to identify special classes of congestion games, where BRD converges to a Nash equilibrium in polynomial time or even linear time, as shown by [2] for games with positive congestion effects and by [16] for games with negative congestion effects. This agenda has been the focus of [25] that considered symmetric network formation games with negative congestion effect played on an extension-parallel graph, and showed that the convergence is bounded by n steps. For resource selection games (i.e., where feasible strategies are composed of singletons), it has been shown in [28] that better-response dynamics converges within at most mn 2 steps for general cost functions (where m is the number of resources).…”