Abstract This paper studies a discrete-time dynamic stochastic game featuring two players: a Leader (Player 1) and a Follower (Player 2), distinguished by its hybrid nature. Specifically, the system’s dynamics adhere to a Markovian-like property, further divided into two sub-dynamics: one regular and the other impulsive, both exerting influence on state evolution. Moreover, players are equipped with two action types: regular and impulsive. The activation of each sub-dynamics depends on the chosen action type. The follower seeks a strategy, conceptualized as a sequence of actions, aimed at maximizing a specified infinite-horizon discounted payoff criterion, where the discount factor is contingent upon the history of state-action pairs. Conversely, the leader aims to minimize such a payoff through its strategy, giving rise to the concept of a minimax equilibrium. Our analysis hinges upon the dynamic programming method, which enables us to (i) characterize the optimal value function as a solution to certain dynamic programming equations and (ii) establish the existence of an optimal minimax equilibrium. We conclude by presenting the application of our findings through a pollution management scenario involving multiple contingency levels often encountered by foreign industries or companies (followers) operating within a particular country (leader).
Correa-Rozo et al. (Thu,) studied this question.