##
**Nash equilibrium with strategic complementarities.**
*(English)*
Zbl 0708.90094

The existence of Nash equilibrium in non-cooperative games is usually established under the assumption that payoff functions are quasiconcave. This paper starts with a different framework: more precisely, it focuses on games where action sets are lattices and payoff functions have monotonicity properties (they are “supermodular” and have “increasing differences” in the appropriate variables). This model was introduced by D. M. Topkis [SIAM J. Control Optimization 17, 773-787 (1979; Zbl 0433.90091)]. Tarski’s fixed point theorem is a basic tool for the analysis.

The results do not only concern the existence of Nash equilibria in supermodular games but also the order structure of the equilibrium set and the stability of solutions (with respect to “Cournot tĂ˘tonnement”). Some aspects of the analysis are extended to Bayesian games.

The previous approach applies in particular to games “with strategic complementarities”. This class of games is relevant to economic applications, as illustrated in the paper.

The results do not only concern the existence of Nash equilibria in supermodular games but also the order structure of the equilibrium set and the stability of solutions (with respect to “Cournot tĂ˘tonnement”). Some aspects of the analysis are extended to Bayesian games.

The previous approach applies in particular to games “with strategic complementarities”. This class of games is relevant to economic applications, as illustrated in the paper.

Reviewer: F.Forges

### MSC:

90C10 | Integer programming |

91A24 | Positional games (pursuit and evasion, etc.) |

91B24 | Microeconomic theory (price theory and economic markets) |

### Keywords:

lattice; function with increasing differences; Nash equilibrium; supermodular games; stability of solutions; Bayesian games### Citations:

Zbl 0433.90091
Full Text:
DOI

### References:

[1] | Ash, R., Real analysis and probability, (1972), Academic Press New York · Zbl 1381.28001 |

[2] | Bamon, Ramon; Fraysse, Jean, Existence of Cournot equilibrium in large markets, Econometrica, 53, 587-597, (1985), May · Zbl 0573.90014 |

[3] | Birkhoff, Garrett, Lattice theory, () · Zbl 0063.00402 |

[4] | Bulow, Jeremy; Geanakoplos, John; Klemperer, Paul, Multimarket oligopoly: strategic substitutes and complements, Journal of political economy, 93, 488-511, (1983) |

[5] | Cooper, R.; John, A., Coordinating coordination failures in Keynesian models, Cowles discussion paper no. 745R, (1985), Cowles Foundation, Yale University New Haven, CT · Zbl 0647.90014 |

[6] | Dasgupta, Partha; Maskin, Eric, The existence of equilibria in discontinuous economic games, Review of economic studies, 53, 1-26, (1986), Jan. · Zbl 0578.90098 |

[7] | Friedman, James W., Oligopoly theory, (1983), Cambridge Surveys of Economic Literature · Zbl 0522.90011 |

[8] | Heller, Walter P., Coordination failure under complete markets with applications to effective demand, Working paper, (1985), University of California San Diego, CA |

[9] | Hildenbrand, Werner, Core and equilibria of a large economy, (1976), Princeton University Press Princeton, NJ · Zbl 0351.90012 |

[10] | Hirsch, Morris, Systems of differential equations that are competitive or cooperative II: convergence almost everywhere, SIAM journal of mathematical analysis, (1985) · Zbl 0658.34023 |

[11] | McManus, M., Equilibrium, numbers and size in Cournot oligopoly, Yorkshire bulletin of social and economic research, 16, (1964) |

[12] | Milgrom, Paul; Weber, Robert, Distributional strategies for games with incomplete information, Mathematics of operations research, 10, no. 4, 619-632, (1985), Nov. · Zbl 0582.90106 |

[13] | Nishimura, K.; Friedman, J., Existence of Nash equilibrium in n-person games without quasiconcavity, International economic review, 637-648, (1981), Oct. · Zbl 0478.90086 |

[14] | Novshek, William, On the existence of Cournot equilibrium, Review of economic studies, 52, no. 1, 85-98, (1985) · Zbl 0547.90011 |

[15] | Radner, Ray; Rosenthal, Robert, Private information and pure-strategy equilibria, Mathematical operations research, 7, no. 3, (1982) · Zbl 0512.90096 |

[16] | Rand, David, Exotic phenomena in games and duopoly models, Journal of mathematical economics, 5, no. 2, 173-184, (1978) · Zbl 0393.90014 |

[17] | Roberts, John; Sonnenschein, Hugo, On the existence of Cournot equilibrium without concave profit functions, Journal of economic theory, 13, (1976) · Zbl 0341.90011 |

[18] | Roberts, John; Sonnenschein, Hugo, On the foundations of the theory of monopolistic competition, Econometrica, 45, 101-113, (1977) · Zbl 0352.90019 |

[19] | Spady, Robert, Noncooperative price setting by asymmetric multiproduct firms, Mimeo, (1984), Bellcore |

[20] | Spence, Michael, Product selection, fixed costs and monopolistic competition, Review of economic studies, 43, (1976), June · Zbl 0362.90013 |

[21] | Tarski, Alfred, A lattice-theoretical fixpoint theorem and its applications, Pacific journal of mathematics, 5, 285-308, (1955) · Zbl 0064.26004 |

[22] | Topkis, Donald, Minimizing a submodular function on a lattice, Operations research, 26, no. 2, (1978) · Zbl 0379.90089 |

[23] | Topkis, Donald, Equilibrium points in nonzero-sum n-person submodular games, SIAM journal of control and optimization, 17, 773-787, (1979), Nov. · Zbl 0433.90091 |

[24] | Vives, Xavier, On the efficiency of Cournot and bertrand equilibria with product differentiation, Journal of economy theory, 36, 166-175, (1985), June · Zbl 0596.90017 |

[25] | Vives, Xavier, Nash equilibrium in oligopoly games with monotone best responses, CARESS working paper no. 85-10, (1985), University of Pennsylvania Philadelphia |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.