Abstract
We study Markov perfect equilibria (MPE) of two-player multi-mode differential games with controlled state dynamics, where one player controls the transition between modes. Different types of MPE are characterized distinguishing between delay equilibria, inducing for some initial conditions mode switches after a positive finite delay, and now or never equilibria, under which, depending on the initial condition, a mode switch occurs immediately or never. These results are applied to analyze the MPE of a game capturing the dynamic interaction between two incumbent firms among which one has to decide when to extend its product range by introducing a new product. The market appeal of the new product can be influenced over time by both firms through costly investments. Under a wide range of market introduction costs a now or never equilibrium coexists with a continuum of delay equilibria, each of them inducing a different time of product introduction.
Highlights
The main agenda of this paper are to improve our understanding of strategic effects arising in dynamic economic and managerial settings characterized by potential structural breaks, which induce jumps in the payoff functions of the economic actors, in the law governing the dynamics of relevant state variables, or both
The innovation of this paper relative to the literature discussed above is that we characterize a Nash equilibrium of multi-mode differential games where both the controls of the players steering the state variable and the decision, whether a mode switch occurs at given point in time, are determined by Markovian strategies and depend solely on the current value of the state and the mode
We study Markov perfect equilibria in two-player multi-mode differential games where both players can influence the dynamics of the state variable and one player controls the timing of the mode switches
Summary
The main agenda of this paper are to improve our understanding of strategic effects arising in dynamic economic and managerial settings characterized by potential structural breaks, which induce jumps in the payoff functions of the economic actors, in the law governing the dynamics of relevant state variables, or both. We are interested in characterizing Markov perfect equilibria (MPE) in multi-mode differential games with a finite number of continuously evolving states, the dynamics of which are controlled by the actions of all players, and a set of modes, where the time of the transition between the modes is determined by one of the players; for easier exposition we assume this is player 1 Problems of this kind arise, for example, in dynamic competition models, like capital accumulation games or dynamic models of reputation formation, where one of the competitors through the introduction of new products or technologies to the market can change the demand structure. The innovation of this paper relative to the literature discussed above is that we characterize a (simultaneous move) Nash equilibrium of multi-mode differential games where both the controls of the players steering the state variable and the decision, whether a mode switch occurs at given point in time, are determined by Markovian strategies and depend solely on the current value of the state and the mode.
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