Fixed point theorems and their applications to theory of Nash equilibria

Heikkilä, Seppo; Reffett, Kevin

doi:10.1016/j.na.2005.06.043

Cited by 26 publications

(17 citation statements)

References 12 publications

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“…In fact, the set is a σ -complete lattice, which means that only countable subsets of F (see Example 2.1 in Heikkilä and Reffett 2006). Therefore, the set is a countable chain complete.…”

Section: Remarkmentioning

confidence: 99%

A qualitative theory of large games with strategic complementarities

et al. 2017

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We study the existence and computation of equilibrium in large games with strategic complementarities. Using monotone operators defined on the space of distributions partially ordered with respect to the first-order stochastic dominance, we prove existence of a greatest and least distributional Nash equilibrium. In particular, we obtain our results under a different set of conditions than those in the existing literature. Moreover, we provide computable monotone distributional equilibrium comparative statics with respect to the parameters of the game. Finally, we apply our results to models of social distance, large stopping games, keeping up with the Joneses, as well as a general class of linear non-atomic games.

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“…In fact, the set is a σ -complete lattice, which means that only countable subsets of F (see Example 2.1 in Heikkilä and Reffett 2006). Therefore, the set is a countable chain complete.…”

Section: Remarkmentioning

confidence: 99%

A qualitative theory of large games with strategic complementarities

et al. 2017

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“…In Smithson (1971), Heikkilä and Reffett (2006), and Veinott (1992), various set relations for ascending correspondences have been proposed. For example, for Y * = 2 Y \∅, and A, B ∈ 2 Y \∅, we say B ↑ A in the weak upward set relation (respectively, weak downward set relation denoted by ↓ ) if for all x 1 ∈ A, there exists x 2 ∈ B such that x 1 ≤ x 2 (respectively, if for all x 2 ∈ B, there exists x 1 ∈ A such that x 1 ≤ x 2 ).…”

Section: Useful Mathematical Terminologymentioning

confidence: 99%

Differential information in large games with strategic complementarities

et al. 2014

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We study equilibrium in large games of strategic complementarities (GSC) with differential information. We define an appropriate notion of distributional Bayesian Nash equilibrium and prove its existence. Furthermore, we characterize order-theoretic properties of the equilibrium set, provide monotone comparative statics for ordered perturbations of the space of games, and provide explicit algorithms for computing extremal equilibria. We complement the paper with new results on the existence of Bayesian Nash equilibrium in the sense of Balder and Rustichini (J Econ Theory 62(2):385-393, 1994) or Kim and Yannelis (J Econ Theory 77(2):330-353, 1997) for large GSC and provide an analogous characterization of the equilibrium set as in the case of distributional Bayesian Nash equilibrium. Finally, we apply our results to riot games, beauty contests, and common value auctions. In all cases, standard existence and comparative statics tools in the theory of supermodular games for finite numbers of agents do not apply in general, and new constructions are required.

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“…In [9] existence and comparison results are derived for games Γ with arbitrary nonempty set I of players, and when the strategy set of player i depends on the strategy vector s −i of other players and the utility functions are vector-valued.…”

Section: Special Casesmentioning

confidence: 99%

Existence and comparison results for fixed points of multifunctions with applications to normal-form games

Heikkilä

2007

Journal of Mathematical Analysis and Applications

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In this paper we apply generalized iteration methods to prove comparison results which show how fixed points of a multifunction can be bounded by least and greatest fixed points of single-valued functions. As an application we prove existence and comparison results for fixed points of multifunctions. These results are applied to normal-form games, by proving existence and comparison results for pure and mixed Nash equilibria and their utilities.

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Fixed point theorems and their applications to theory of Nash equilibria

Cited by 26 publications

References 12 publications

A qualitative theory of large games with strategic complementarities

A qualitative theory of large games with strategic complementarities

Differential information in large games with strategic complementarities

Existence and comparison results for fixed points of multifunctions with applications to normal-form games

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