Learning to Control Unknown Strongly Monotone Games

Abstract

Consider a game where the players' utility functions include a reward function and a linear term for each dimension, with coefficients that are controlled by the manager. We assume that the game is strongly monotone, so gradient play converges to a unique Nash equilibrium (NE). The NE is typically globally inefficient. The global performance at NE can be improved by imposing linear constraints on the NE. We therefore want the manager to pick the controlled coefficients that impose the desired constraint on the NE. However, this requires knowing the players' reward functions and action sets. Obtaining this game information is infeasible in a large-scale network and violates user privacy. To overcome this, we propose a simple algorithm that learns to shift the NE to meet the linear constraints by adjusting the controlled coefficients online. Our algorithm only requires the linear constraints violation as feedback and does not need to know the reward functions or the action sets. We prove that our algorithm converges with probability 1 to the set of NE that satisfy target linear constraints. We then prove an L2 convergence rate of near-$O(t^{-1/4})$.