Inverse Matrix Games with Unique Nash Equilibrium

In an inverse game problem, one needs to infer the cost function of the players in a game such that a desired joint strategy is a Nash equilibrium. We study the inverse game problem for a class of multiplayer matrix games, where the cost perceived by each player is corrupted by random noise. We provide sufficient conditions for the players' quantal response equilibrium -- a generalization of the Nash equilibrium to games with perception noise -- to be unique. We develop efficient optimization algorithms for inferring the cost matrix based on semidefinite programs and bilevel optimization. We demonstrate the application of these methods using examples where we design the cost matrices that rationalize collision avoidance in path-planning and fairness in resource allocation.