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New results on the existence of open loop Nash equilibria in discrete time dynamic games via generalized Nash games

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  • Mathew P. Abraham

    (Indian Institute of Technology Bombay)

  • Ankur A. Kulkarni

    (Indian Institute of Technology Bombay)

Abstract

We address the problem of finding conditions which guarantee the existence of open-loop Nash equilibria in discrete time dynamic games (DTDGs). A classical approach to DTDGs involves analyzing the problem using optimal control theory. Sufficient conditions for the existence of open-loop Nash equilibria obtained from this approach are mainly limited to linear-quadratic games (Başar and Olsder in Dynamic noncooperative game theory, 2nd edn, SIAM, Philadelphia, 1999). Another approach of analysis is to substitute the dynamics and transform the game into a static game. But the substitution of state dynamics makes the objective functions of the resulting static problems extremely hard to analyze. We introduce a third approach in which the dynamics are not substituted, but retained as constraints in the optimization problem of each player, resulting thereby in a generalized Nash game. Using this, we give sufficient conditions for the existence of open-loop Nash equilibria for a class of DTDGs where the cost functions of players admit a quasi-potential function. Our results apply with nonlinear dynamics and without stage additive cost functions, and allow constraints on state and actions spaces, and in some cases, yield a generalization of similar results from linear-quadratic games.

Suggested Citation

  • Mathew P. Abraham & Ankur A. Kulkarni, 2019. "New results on the existence of open loop Nash equilibria in discrete time dynamic games via generalized Nash games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 157-172, April.
    • Handle: RePEc:spr:mathme:v:89:y:2019:i:2:d:10.1007_s00186-018-0644-2
    DOI: 10.1007/s00186-018-0644-2
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    References listed on IDEAS

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    1. Pierpaolo Battigalli & Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2015. "Self-Confirming Equilibrium and Model Uncertainty," American Economic Review, American Economic Association, vol. 105(2), pages 646-677, February.
    2. Agnieszka Wiszniewska-Matyszkiel, 2017. "Redefinition of Belief Distorted Nash Equilibria for the Environment of Dynamic Games with Probabilistic Beliefs," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 984-1007, March.
    3. Agnieszka Wiszniewska-Matyszkiel, 2016. "Belief distorted Nash equilibria: introduction of a new kind of equilibrium in dynamic games with distorted information," Annals of Operations Research, Springer, vol. 243(1), pages 147-177, August.
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    Cited by:

    1. Mathew P. Abraham & Ankur A. Kulkarni, 2020. "Price-coupling games and the generation expansion planning problem," Annals of Operations Research, Springer, vol. 295(1), pages 1-19, December.

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