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Comparing Open-Loop with Markov Equilibria in a Class of Differential Games


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  • Ngo Van Long


  • Koji Shimomura
  • Harutaka Takahashi


We consider a class of differential games with transition equations that are homogeneous of degree one. For any game G with a discount rate r, consider a Markov perfect equilibrium (MPE) with strategies that are linear in the state variables. We show that the time paths of the control variables of this equilibrium constitute an open-loop equilibrium of a corresponding game that differs from G only in that its rate of discount is equal to r plus a suitably chosen constant. In the context of a resource depletion game, this implies that the open-loop solution is more conservationist. In an alternative formulation, we consider a game G with a rate of stock depreciation and a Markov perfect equilibrium with strategies that are linear in the state variable. Then the resulting equilibrium time paths of the control variables can be used to form an open-loop equilibrium of a corresponding game with a rate of depreciation that is equal to plus a suitably chosen constant. The interpretation for the case of a resource game is similar. We also prove a converse result: under certain assumptions, an open loop equilibrium of a differential game G can be used to construct a MPE of a modified game . This result is useful because it means a MPE can be found by first solving for an open loop equilibrium. On considère une classe de jeux dynamiques où l'évolution du stock est une fonction homogène du premier degré. Étant donné un jeu G avec le taux d'actualisation r , on considère un équilibre Markov-parfait dont les stratégies sont linéaires par rapport aux variables d'état. On montre que les sentiers des variables de contrôle de cet équilibre constituent un équilibre en boucle ouverte d'un jeu qui correspond à , et qui a un taux d'actualisation ( . Dans le contexte d'un jeu d'exploitation de ressources naturelles, celà implique que l'équilibre en boucle ouverte tends à conserver les ressources. Alternativement, on considère un jeu G où le taux de dépréciation du stock est et où les stratégies de l'équilibre Markov-parfait sont linéaires. On montre que les sentiers des variables de contrôle donnent un équilibre en boucle ouverte d'un jeu dans laquelle le taux de dépréciation du stock est . On obtient aussi un résultal dans l'autre direction: sous certaines hypothèses, on peut utiliser un équilibre en boucle ouverte d'un jeu G pour construire un équilibre Markov-parfait d'un jeu modifié .

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 97s-22.

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Date of creation: 01 Apr 1997
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Handle: RePEc:cir:cirwor:97s-22

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Keywords: Differential games; Markov perfect equilibria; open-loop equilibria; Jeux différentiels; équilibre Markov-parfait; équilibre en boucle ouverte;

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  1. Van Long, Ngo & Shimomura, Koji, 1998. "Some results on the Markov equilibria of a class of homogeneous differential games," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 557-566, January.
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Cited by:
  1. Ngo Long, 2011. "Dynamic Games in the Economics of Natural Resources: A Survey," Dynamic Games and Applications, Springer, vol. 1(1), pages 115-148, March.
  2. Talat S. Genc & Georges Zaccour, 2010. "Investment Dynamics: Good News Principle," Working Papers 1006, University of Guelph, Department of Economics and Finance.
  3. Murray C. Kemp & Ngo Van Long, 2007. "Development Aid in the Presence of Corruption: Differential Games among Donors," CIRANO Working Papers 2007s-23, CIRANO.
  4. Sébastien Rouillon, 2014. "Do Social Status Seeking Behaviors Worsen the Tragedy of the Commons?," Dynamic Games and Applications, Springer, vol. 4(1), pages 73-94, March.
  5. Charles Figuières, 2009. "Markov interactions in a class of dynamic games," Theory and Decision, Springer, vol. 66(1), pages 39-68, January.


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