Control of parametric games

This work studies a class of multi-player games in which the players' decisions can be influenced by a superplayer. We define a game with n players and parameterized utilities u (., a) where the superplayer controls the value of a. The regular players follow Markovian repeated play dynamics that encompass a wide class of learning dynamics including strict best response. The objective of the superplayer is to control a dynamically to achieve a desired outcome in the game-play, which in this work we define as the realization of target joint strategies. We introduce the class of parametric games and reformulate the superplayer control problem as a Markov decision process (MDP). Reachability criteria are developed, allowing the superplayer to determine which game-play may occur with positive probability. With a reachable goal joint strategy, a cost-optimal policy can be computed using standard tools in dynamic programming. A sample MDP reward function is presented such that a reachable target joint strategy is guaranteed to be played almost surely. Finally, an application in a cyber-security context is provided to illustrate the use of the proposed methodology and its effectiveness.