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Nash equilibria of games with increasing best replies


  • CALCIANO, Filippo L.

    () (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium and Department of Economics, University of Rome 3, Italy)


The intuitive idea of two activities being complements, for example tea and lemon, is that increasing the level of one makes somehow desirable to increase the level of the other (Samuelson, 1974). Hence complementarity, in its very nature, is a sensitivity property of the set of solutions to an optimization problem. In the context of games, complementarity should then be captured by properties of the joint best reply. We introduce notions of increasingness for the joint best reply which capture properly this intuitive idea of complementarity among players’ strategies. We show, by generalizing the fixpoint theorems of Veinott (1992) and Zhou (1994), that the Nash sets of our games are nonempty complete lattices. Hence we extend the class of games with strategic complementarities

Suggested Citation

  • CALCIANO, Filippo L., 2009. "Nash equilibria of games with increasing best replies," CORE Discussion Papers 2009082, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2009082

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    References listed on IDEAS

    1. Jaskold Gabszewicz, J. & Thisse, J. -F., 1979. "Price competition, quality and income disparities," Journal of Economic Theory, Elsevier, vol. 20(3), pages 340-359, June.
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    3. Irmen, Andreas & Thisse, Jacques-Francois, 1998. "Competition in Multi-characteristics Spaces: Hotelling Was Almost Right," Journal of Economic Theory, Elsevier, vol. 78(1), pages 76-102, January.
    4. Boccard, N. & Wauthy, X.Y., 2010. "Equilibrium vertical differentiation in a Bertrand model with capacity precommitment," International Journal of Industrial Organization, Elsevier, vol. 28(3), pages 288-297, May.
    5. Schmidt, Robert C., 2009. "Welfare in differentiated oligopolies with more than two firms," International Journal of Industrial Organization, Elsevier, vol. 27(4), pages 501-507, July.
    6. Cremer, Helmuth & Thisse, Jacques-Francois, 1991. "Location Models of Horizontal Differentiation: A Special Case of Vertical Differentiation Models," Journal of Industrial Economics, Wiley Blackwell, vol. 39(4), pages 383-390, June.
    7. Shaked, Avner & Sutton, John, 1983. "Natural Oligopolies," Econometrica, Econometric Society, vol. 51(5), pages 1469-1483, September.
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    Cited by:

    1. CALCIANO, Filippo L., 2011. "Oligopolistic competition with general complementarities," CORE Discussion Papers 2011054, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    More about this item


    strategic complementarity; supermodular games; quasisupermodular games; fixpoint theorem; Nash equilibria;

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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