Multiplicative Weights Update in Zero-Sum Games

Abstract

We study the classic setting where two agents compete against each other in a zero-sum game by applying the Multiplicative Weights Update (MWU) algorithm. In a twist of the standard approach of [Freund and Schapire 1999], we focus on the K-L divergence from the equilibrium but instead of providing an upper bound about the rate of increase we provide a nonnegative lower bound for games with interior equilibria. This implies movement away from equilibria and towards the boundary. In the case of zero-sum games without interior equilibria convergence to the boundary (and in fact to the minimal product of subsimplexes that contains all Nash equilibria) follows via an orthogonal argument. In that subspace divergence from the set of NE applies for all nonequilibrium initial conditions via the first argument. We argue the robustness of this non-equilibrating behavior by considering the following generalizations: Step size: Agents may be using different and even decreasing step sizes. Dynamics: Agents may be using Follow-the-Regularized-Leader algorithms and possibly apply different regularizers (e.g. MWU versus Gradient Descent). We also consider a linearized version of MWU. More than two agents: Multiple agents can interact via arbitrary networks of zero-sum polymatrix games and their affine variants. Our results come in stark contrast with the standard interpretation of the behavior of MWU (and more generally regret minimizing dynamics) in zero-sum games, which is typically referred to as converging to equilibrium. If equilibria are indeed predictive even for the benchmark class of zero-sum games, agents in practice must deviate robustly from the axiomatic perspective of optimization driven dynamics as captured by MWU and variants and apply carefully tailored equilibrium-seeking behavioral dynamics.