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Selection of a Markov Perfect Nash Equilibrium in a Class of Differential Games

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Abstract

This paper revisits the problem of how to select an equilibrium in a differential game in the case of multiplicity of Nash equilibria. Most of the previous applied dynamic games literature has considered pre-play negotiations between players, implicitly or explicitly, with the aim of reaching an agreement on the selection of the pair of strategies. The main objective of this paper is to determine what would be the equilibrium to be played without pre-play communications. We study the linear and nonlinear Markov perfect Nash equilibria for a class of well-known models in the literature if pre-play communications are eliminated. We analyze both symmetric and nonsymmetric strategies. We show that the nonlinear strategies are not always the optimal strategies implemented when pre-play communications are removed. We conclude that in the presence of multiple equilibria and without pre-play communications the equilibria actually implemented are symmetric piecewise linear Markov perfect Nash equilibria at least for a range of initial values of the state variable.

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Notes

  1. Other examples of papers that explicitly consider pre-play negotiations between players are the following (the list is not exhaustive): Piga [22], Ihori and Itaya [19], Wirl and Feichtinger [36], Benchekroun and Long [2], and Kossioris et al. [17].

  2. The problem of the sustainability of the coordinated outcomes over time is already solved when the efficient solution is in itself an equilibrium, as in Chiarella et al. [4], Rincón-Zapatero et al. [24], and Martín-Herrán and Rincón-Zapatero [21].

  3. Dockner and Sorger [9] follow a different approach and construct an infinite number of Nash equilibria in a dynamic game of joint exploitation of a productive asset. They show that the efficient stationary stock can approximately be supported as Nash equilibria in stationary Markovian strategies when the discount rate approaches zero.

  4. One exception is Zagonari [37] where no agreement on the selection of the pair of strategies is required because unilateral initiatives are considered. The paper analyzes a model where two group of countries differ in their preferences for consumption goods as well as in their attention to environmental issues. Another exception is Tasneem et al. [30] where the empirical relevance of the nonlinear equilibria in a two-player common property resource game is examined. Their results show that nonlinear equilibria cannot be ruled out as irrelevant on behavioral grounds.

  5. Most of the works in this literature study different problems formulated as linear-quadratic differential games. However, there are some exceptions like Kossioris et al. [17] that analyze the shallow lake pollution control differential game presenting nonlinear dynamics.

  6. Dockner and Wagener [10] consider strategies with a local support.

  7. See Dockner and Wagener [10] for a (in this case) mathematically equivalent alternative approach using a shadow price auxiliary system of ordinary differential equations.

  8. We focus on globally defined strategies to avoid the discussion about whether or not our results depend on the local or global character of the strategies.

  9. Rowat [26] analyzes the general case without the hypothesis of boundedness of the strategies. The results in that paper show that “a fortiori” the optimal strategies satisfy \(u_j(t)=\phi _j(x(t))\le A\) for all \(t\ge 0\).

  10. Dockner and Long [7] consider locally defined nonlinear strategies, whereas in this paper we only consider globally defined nonlinear strategies.

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Authors and Affiliations

  1. IMUVA and Dpto. Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid, Campus Miguel Delibes, Paseo Belén 7, 47011, Valladolid, Spain

    Javier de Frutos

  2. IMUVA and Dpto. Economía Aplicada, Facultad de Ciencias Económicas y Empresariales, Universidad de Valladolid, Avda. Valle Esgueva, 6, 47011, Valladolid, Spain

    Guiomar Martín-Herrán

Authors
  1. Javier de Frutos
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  2. Guiomar Martín-Herrán
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Correspondence to Guiomar Martín-Herrán.

Additional information

We are grateful to the editor and two anonymous reviewers for valuable comments and suggestions on an earlier draft of this paper. This research is partially supported by MINECO under projects MTM2013-42538-P, MTM2016-78995-P (AEI) (Javier de Frutos) and ECO2014-52343-P and ECO2017-82227-P (AEI) (Guiomar Martín-Herrán), and by Junta de Castilla y León VA024P17 co-financed by FEDER funds (EU).

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de Frutos, J., Martín-Herrán, G. Selection of a Markov Perfect Nash Equilibrium in a Class of Differential Games. Dyn Games Appl 8, 620–636 (2018). https://doi.org/10.1007/s13235-018-0257-7

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