Learning Nash Equilibria in Congestion Games
This addresses convergence guarantees in multi-agent systems for researchers in game theory and machine learning, but it is incremental as it builds on existing replicator dynamics.
The paper investigates whether decentralized strategy updates by players in repeated congestion games lead to convergence to Nash equilibria, showing that sequences converge in Cesàro means but strong convergence requires specific algorithms like discounted Hedge.
We study the repeated congestion game, in which multiple populations of players share resources, and make, at each iteration, a decentralized decision on which resources to utilize. We investigate the following question: given a model of how individual players update their strategies, does the resulting dynamics of strategy profiles converge to the set of Nash equilibria of the one-shot game? We consider in particular a model in which players update their strategies using algorithms with sublinear discounted regret. We show that the resulting sequence of strategy profiles converges to the set of Nash equilibria in the sense of Cesàro means. However, strong convergence is not guaranteed in general. We show that strong convergence can be guaranteed for a class of algorithms with a vanishing upper bound on discounted regret, and which satisfy an additional condition. We call such algorithms AREP algorithms, for Approximate REPlicator, as they can be interpreted as a discrete-time approximation of the replicator equation, which models the continuous-time evolution of population strategies, and which is known to converge for the class of congestion games. In particular, we show that the discounted Hedge algorithm belongs to the AREP class, which guarantees its strong convergence.