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Total Reward Semi-Markov Mean-Field Games with Complementarity Properties

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  • Piotr Więcek

    (Wrocław University of Science and Technology)

Abstract

We study a class of dynamic games with a continuum of atomless players where each player controls a semi-Markov process of individual states, while the global state of the game is the aggregation of individual states of all the players. The model differs from standard models of dynamic games with continuum of players known as mean field or anonymous games in that the moments when the decisions are made are discrete, but different for each of the players. As a result, the individual states of each player follow a continuous time Markov chain, but the global state follows an ordinary differential equation. Games of this type were introduced by Gomes et al. (Appl Math Optim 68:99–143, 2013) and received some attention in the literature in last few years. In our paper we introduce a novel model of this type where players maximize their cumulative payoffs over their lifetime. We show that the payoffs of the players using any stationary strategy of a certain class in a game with continuum of players are close to those obtained in n-person counterparts of this game for n large enough. This implies that equilibrium strategies in the anonymous model can well approximate equilibria in related games with large finite number of players. In the rest of the paper we concentrate on a subclass of games where the payoff and transition probability functions exhibit some strategic complementarities between players. In that case we prove that the game possesses a stationary equilibrium. Moreover, largest and smallest equilibrium strategies are nondecreasing in the states. It also turns out that these equilibria can be well approximated using a distributed iterative procedure.

Suggested Citation

  • Piotr Więcek, 2017. "Total Reward Semi-Markov Mean-Field Games with Complementarity Properties," Dynamic Games and Applications, Springer, vol. 7(3), pages 507-529, September.
  • Handle: RePEc:spr:dyngam:v:7:y:2017:i:3:d:10.1007_s13235-016-0194-2
    DOI: 10.1007/s13235-016-0194-2
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    References listed on IDEAS

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    1. Bergin, J & Bernhardt, D, 1995. "Anonymous Sequential Games: Existence and Characterization of Equilibria," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(3), pages 461-489, May.
    2. Jovanovic, Boyan & Rosenthal, Robert W., 1988. "Anonymous sequential games," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 77-87, February.
    3. Bergin, James & Bernhardt, Dan, 1992. "Anonymous sequential games with aggregate uncertainty," Journal of Mathematical Economics, Elsevier, vol. 21(6), pages 543-562.
    4. Balbus, Łukasz & Reffett, Kevin & Woźny, Łukasz, 2014. "A constructive study of Markov equilibria in stochastic games with strategic complementarities," Journal of Economic Theory, Elsevier, vol. 150(C), pages 815-840.
    5. Chakrabarti, Subir K., 2003. "Pure strategy Markov equilibrium in stochastic games with a continuum of players," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 693-724, September.
    6. Diogo Gomes & João Saúde, 2014. "Mean Field Games Models—A Brief Survey," Dynamic Games and Applications, Springer, vol. 4(2), pages 110-154, June.
    7. repec:spr:compst:v:66:y:2007:i:3:p:513-530 is not listed on IDEAS
    8. Rabah Amir, 2002. "Complementarity and Diagonal Dominance in Discounted Stochastic Games," Annals of Operations Research, Springer, vol. 114(1), pages 39-56, August.
    9. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    10. Kunreuther, Howard & Heal, Geoffrey, 2003. "Interdependent Security," Journal of Risk and Uncertainty, Springer, vol. 26(2-3), pages 231-249, March-May.
    11. Xavier Vives, 2009. "Strategic complementarity in multi-stage games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(1), pages 151-171, July.
    12. Horst, Ulrich, 2005. "Stationary equilibria in discounted stochastic games with weakly interacting players," Games and Economic Behavior, Elsevier, vol. 51(1), pages 83-108, April.
    13. Curtat, Laurent O., 1996. "Markov Equilibria of Stochastic Games with Complementarities," Games and Economic Behavior, Elsevier, vol. 17(2), pages 177-199, December.
    14. Andrzej Nowak, 2007. "On stochastic games in economics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 513-530, December.
    15. Sachin Adlakha & Ramesh Johari, 2013. "Mean Field Equilibrium in Dynamic Games with Strategic Complementarities," Operations Research, INFORMS, vol. 61(4), pages 971-989, August.
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