On strategic complementarities in discontinuous games with totally ordered strategies

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Abstract

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.

Introduction

Upper semicontinuity, quasisupermodularity, and Milgrom and Shannon’s (1994) single crossing are sufficient for a normal-form game where the strategy sets are compact lattices in Euclidean spaces to have a pure strategy Nash equilibrium.1 In many economics-related games, the strategy sets are totally ordered. In such cases, both upper semicontinuity and single crossing are excessively demanding. The focus of this paper is on relaxing both of the two conditions.

In games with strategic complementarities, the best-reply correspondences are, usually, assumed to be nonempty-valued and subcomplete-sublattice-valued. These propitious properties of the best-reply correspondences are achieved by making a not entirely innocuous assumption, namely that each payoff function is upper semicontinuous in own strategy, which noticeably narrows the class of games in which equilibrium existence can be studied with the aid of lattice-theoretic tools. In this paper, upper semicontinuity is replaced with one of the following pairs of conditions: either with Tian and Zhou’s (1995) transfer weak upper continuity and directional upper semicontinuity or with Reny’s (1999) better-reply security and directional upper semicontinuity, thereby making it possible to cover new classes of games to which the seminal contributions by Vives (1990), Milgrom and Shannon (1994), and Reny (1999) cannot be applied.

For games where the payoff functions are transfer weakly upper continuous in own strategies, single crossing is generalized to directional transfer single crossing. The word ‘directional’ means that single crossing is divided into upward single crossing and downward single crossing, and the word ‘transfer’ reflects the fact that, in this paper, the notion of an increasing correspondences is understood in Smithson’s (1971) and Fujimoto’s (1984) sense. We illustrate the interplay of the different notions with the aid of a partnership game (Example 4) and a war of attrition game (Example 5). In the latter, player 1’s payoff function satisfies upward single crossing and player 2’s payoff function satisfies downward single crossing when the natural order relation on player 2’s strategy set is reversed.

The lattice-theoretic approach covers a large number of oligopoly models (see, e.g.  Roberts and Sonnenschein, 1976, Vives, 1990, Amir, 1996, Vives, 1999, Amir and De Castro, 2015). However, the classic Bertrand oligopoly model with homogeneous products is not one of them since its payoff functions are too discontinuous. At the same time, in the two-firm case, for example, if, initially, the profit-maximizing firms charge prices exceeding the unit cost of production, then any of them has no incentive to lower its price in reaction to an increase in the price charged by its rival. On the other hand, if the demand curve has a conventional convex shape, the quasiconcavity of the Bertrand duopoly game tends to fail, and, consequently, it might be impossible to apply Reny’s (1999) equilibrium existence theorem and any of its generalizations (see, e.g.  McLennan et al., 2011, Barelli and Meneghel, 2013, Carmona, 2016, Carmona and Podczeck, 2016, Reny, 2016).

We handle the equilibrium existence problem in games where the best-reply correspondences are not necessarily nonempty-valued everywhere in two steps. The first step employs lattice-theoretic tools and directional upper semicontinuity to investigate the existence of ε-equilibria, and the second step relies on the fact that, in the better-reply secure games, the cluster points of a sequence of ε-Nash equilibria are Nash equilibria. In order to express strategic complementarities in terms of ε-best-reply correspondences, two more directional modifications of single crossing are introduced. The proposed equilibrium existence conditions are illustrated on a nonquasiconcave Bertrand duopoly model with homogeneous products (Example 6).

The structure of the paper is as follows. Section  2 contains some theoretical underpinnings necessary for studying strategic complementarities in discontinuous games. The main results of the paper are presented in Section  3, and the illustrating examples are provided in Section  4.

Section snippets

Preliminaries

This section provides some lattice-theoretic and topological definitions and auxiliary results.

Equilibrium existence results

This section begins with Theorem 1, an equilibrium existence result for games where each payoff function is transfer weakly upper continuous in own strategy. Then, Theorem 2 provides a set of sufficient conditions for the existence of an equilibrium in better-reply secure games.

Let I={1,,n}. If, for each iI,Xi is a partially ordered set with the binary relation i, then X=iIXi and Xi=jI{i}Xj are posets with the corresponding product relations; that is, for example, xy in X if xiiyi

Applications

This section explains, with the aid of economics-related examples, the paper’s major contributions. The partnership game studied in Example 4 illustrates the strengths of the generalized upper semicontinuity conditions used in Theorem 1. In the game, the players’ payoff functions are not upper semicontinuous in own strategies, and, moreover, their best-reply correspondences are neither Veinott-increasing upward nor increasing downward.

In order to apply Theorem 1 to the war of attrition game

Conclusions

Lattice-theoretic tools can be used to study equilibrium existence in strategic games with totally ordered strategy sets where the payoff functions are not upper semicontinuous in own strategies and do not satisfy the single-crossing property. If the payoff functions are transfer weakly upper continuous in own strategies, the existence of a Nash equilibrium follows from directional upper semicontinuity and directional transfer single crossing. In games where best-reply correspondences are not

Acknowledgments

We are grateful to Rabah Amir, Luciano De Castro, John Quah and an anonymous referee for helpful comments.

References (39)

  • Amir, R., De Castro, L., 2015. Nash equilibrium in games with quasi-monotonic best responses, Working Paper, University...
  • S. Athey

    Single crossing properties and the existence of pure strategy equilibria in games of incomplete information

    Econometrica

    (2001)
  • P. Barelli et al.

    A note on the equilibrium existence problem in discontinuous games

    Econometrica

    (2013)
  • G. Birkhoff

    Lattice Theory

    (1967)
  • S. Carl et al.

    Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory

    (2011)
  • G. Carmona

    Reducible equilibrium properties: comments on recent existence results

    Econom. Theory

    (2016)
  • G. Carmona et al.

    Existence of Nash equilibrium in ordinal games with discontinuous preferences

    Econom. Theory

    (2016)
  • A.S. Edlin et al.

    Strict single crossing and the strict Spence-Mirrlees condition: a comment on monotone comparative statics

    Econometrica

    (1998)
  • T. Fujimoto

    An extension of Tarski’s fixed point theorem and its application to isotone complementarity problems

    Math. Program.

    (1984)
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