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Some results on the Markov equilibria of a class of homogeneous differential games

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  • Van Long, Ngo
  • Shimomura, Koji

Abstract

We consider the class of differential games with transition dynamics and constraints that are homogeneous of degree one. We show that if the integrand of the objective function is homogeneous of degree , then best replies to linear homogeneous Markov strategies are linear homogeneous, and the value function is homogeneous of degree . A parallel result holds when one applies logarithmic transformation to the integrand. Examples are provided. Nous étudions la classe de jeux différentiels dont l'équation de transition et les contraintes sont caractérisées par l'homogénéité du premier degré. Nous prouvons que si la fonction d'objectif possède l'homogénéité du degré , alors la meilleure réponse aux stratégies markoviennes qui possèdent l'homogénéité du premier degré doit avoir la même propriété, et la fonction de valeur est caractérisée par l'homogénéité du degré . On obtient un résultat similaire dans le cas d'une transformation logarithmique de la fonction d'objectif. L'article contient trois exemples.
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  • 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.
    • Handle: RePEc:eee:jeborg:v:33:y:1998:i:3-4:p:557-566
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    References listed on IDEAS

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    1. Fischer, Ronald D. & Mirman, Leonard J., 1992. "Strategic dynamic interaction : Fish wars," Journal of Economic Dynamics and Control, Elsevier, vol. 16(2), pages 267-287, April.
    2. Léonard,Daniel & Long,Ngo van, 1992. "Optimal Control Theory and Static Optimization in Economics," Cambridge Books, Cambridge University Press, number 9780521331586.
    3. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, December.
    4. Fischer, Ronald D. & Mirman, Leonard J., 1996. "The Compleat Fish Wars: Biological and Dynamic Interactions," Journal of Environmental Economics and Management, Elsevier, vol. 30(1), pages 34-42, January.
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    Cited by:

    1. Gerhard Sorger, 2005. "A dynamic common property resource problem with amenity value and extraction costs," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(1), pages 3-19, March.
    2. Guillaume Bataille, 2024. "An Explicit Solution to Harvesting Behaviors in a Predator-Prey System," AMSE Working Papers 2405, Aix-Marseille School of Economics, France.
    3. Ngo Van Long & Koji Shimomura & Harutaka Takahashi, 1999. "Comparing Open-loop With Markov Equilibria in a Class of Differential Games," The Japanese Economic Review, Japanese Economic Association, vol. 50(4), pages 457-469, December.
    4. 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.
    5. Murray C. Kemp & Ngo Van Long, 2007. "Development Aid in the Presence of Corruption: Differential Games among Donors," CIRANO Working Papers 2007s-23, CIRANO.
    6. Eggert, Wolfgang & Itaya, Jun-ichi & Mino, Kazuo, 2011. "A dynamic model of conflict and appropriation," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1), pages 167-182.
    7. 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.
    8. List, John A. & Mason, Charles F., 2001. "Optimal Institutional Arrangements for Transboundary Pollutants in a Second-Best World: Evidence from a Differential Game with Asymmetric Players," Journal of Environmental Economics and Management, Elsevier, vol. 42(3), pages 277-296, November.
    9. Murray C. Kemp & Ngo Van Long, 2009. "Foreign Aid in the Presence of Corruption: Differential Games among Donors," Review of International Economics, Wiley Blackwell, vol. 17(SI), pages 230-243, May.

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    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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