Nash equilibrium with strategic complementarities.

*(English)*Zbl 0708.90094The 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
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##### References:

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