Reviews: Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos

The paper revisits the convergence result of multiplicative weighted update (MWU) in a congestion game, due to Kleinberg, Piliouras, and Tardos (STOC 2009), and establishes a connection between MWU and Baum-Welch algorithm in a neat and elegant style. By showing the monotonicity of the potential function in a congestion game, the authors prove that any MWU with linear updating rule converges to the set of fixed points, which is a superset of all the Nash equilibria of the congestion game. The Baum-Eagon inequality offers a new interpretation of the dynamics of the linear updating rule in MWU and their results in congestion games are quite general, and so are the conditions on initialization and isolated Nash equilibrium in a congestion game with finite set of agents and pure strategies. The results in this paper hold for any congestion game irrespective of the topology of the strategy sets by the nature of their game, however, one should emphasize the assumption that, individual earning rates \epsilon_i are bounded above, in order for Baum-Eagon inequality to work in their context.