Markov perfect equilibria in differential games with regime switching strategies

We propose a new methodology exploring Markov perfect equilibrium strategies in differential games with regime switching. Specifically, we develop a general game with two players having two kinds of strategies. Players choose an action that influences the evolution of a state variable, and decide on the switching time from one regime to another. Compared to the optimal control problem with regime switching, necessary optimality conditions are modified for the first-mover. When choosing her optimal switching strategy, this player considers the impact of her choice on the other player's actions and payoffs. In order to determine the equilibrium timing of regime changes, the notion of wrong timing is introduced and necessary conditions for a particular timing to be wrong are derived. We then apply this new methodology to an exhaustible resource extraction game. Sufficient conditions for the existence of an interior solution are compared to those characterizing a wrong timing. The impact of feedback strategies for the equilibrium adoption time depends on the balance between two conflicting effects: the first mover incurs an indirect cost due to the future switching of her rival (incentive to delay the switch). But she is able to affect the other player's switching decision (incentive to switch more rapidly). In a particular case without direct switching cost, the interplay between the two ensures that the first-mover adopts the new technology in finite time. Interestingly, this result differs from what is obtained in a non-game theoretic framework, i.e. immediate adoption.