Markov perfect equilibria in differential games with regime switching

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14 January 2014

We propose a new methodology exploring Markov perfect equilibrium strategies in differential games with regime switching. Specically, 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 between alternative and consecutive regimes. 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 her impact on the other player’s actions and welfare, vice versa. In order to determine the optimal timing between regime changes, the notion of erroneous timing is introduced and necessary conditions for a particular timing to be erroneous are derived. We then apply this original material to an exhaustible resource extraction game. Sucient conditions for the existence of an interior solution are compared to those characterizing an erroneous timing. The impact of feedback strategies for adoption time on the equilibrium depends on 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 with no direct switching cost, the interplay between the two ensures that the first-mover adopts the new technology in nite time. Interestingly, this result differs from what is obtained in a non-game theoretic framework, i.e. immediate adoption.