Congestion games viewed from M-convexity

Congestion games have extensively been studied till recently. An interesting special class of congestion games on networks is the class of congestion games on extension-parallel networks considered by Holzman and Law-yone (2003). It is shown by Fotakis (2010) that for every congestion game on an extension-parallel network, any best-response sequence reaches a pure Nash equilibrium of the game in n steps, where n is the number of players. We show that the fast convergence of best-response sequences results from Mconvexity (introduced byMurota (1996)) of the potential function associated with the game. It is revealed that the best-response dynamic process corresponds to a greedy algorithm for minimizing M-convex functions. We also give a characterization of M-convex functions in terms of greedy algorithms.