Abstract
In this paper 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 (positively or negatively) influenced over time by the competing firms through costly investments. It is shown that under a wide range of market introduction costs a now or never equilibrium co-exists with a continuum of delay equilibria, with each of them inducing a different time of product introduction.
Highlights
The main agenda of this paper is 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
Our analysis of a simple model of dynamic competition under potential new product introduction illustrates the properties of all three types of equilibria identified in Section 3 and shows that delay equilibria, maximum delay equilibra and or never equilibria, might co-exist for some parameter constellations
Summary
The main agenda of this paper is 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.
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