  We study the game modification problem, where a benevolent game designer or a
malevolent adversary modifies the reward function of a zero-sum Markov game so
that a target deterministic or stochastic policy profile becomes the unique
Markov perfect Nash equilibrium and has a value within a target range, in a way
that minimizes the modification cost. We characterize the set of policy
profiles that can be installed as the unique equilibrium of some game, and
establish sufficient and necessary conditions for successful installation. We
propose an efficient algorithm, which solves a convex optimization problem with
linear constraints and then performs random perturbation, to obtain a
modification plan with a near-optimal cost.
