  The interplay between exploration and exploitation in competitive multi-agent
learning is still far from being well understood. Motivated by this, we study
smooth Q-learning, a prototypical learning model that explicitly captures the
balance between game rewards and exploration costs. We show that Q-learning
always converges to the unique quantal-response equilibrium (QRE), the standard
solution concept for games under bounded rationality, in weighted zero-sum
polymatrix games with heterogeneous learning agents using positive exploration
rates. Complementing recent results about convergence in weighted potential
games, we show that fast convergence of Q-learning in competitive settings is
obtained regardless of the number of agents and without any need for parameter
fine-tuning. As showcased by our experiments in network zero-sum games, these
theoretical results provide the necessary guarantees for an algorithmic
approach to the currently open problem of equilibrium selection in competitive
multi-agent settings.
