Equilibria in Large Games with Strategic Complementarities

y z x {L ukasz Balbus Pawe l Dziewulski Kevin Re ett L ukasz Wozny

March 2012

Abstract

We study the existence and computation of Nash equilibrium in large games with strategic

complementarities. Using monotone operators (in stochastic dominance orders) de ned on

the space of distributions, we rst prove existence of the greatest and least distributional

Nash equilibrium in the sense of Mas-Colell (1984) under di erent set of assumptions than

those in the existing literature. In addition, we provide results on computable monotone

distributional equilibrium comparative statics relative to ordered perturbations of the deep

parameters of our class of games. We then provide similar new results for Nash/Schmeidler

(1973) equilibria (de ned by strategies) in our large games. We conclude by discussing the

question of equilibrium uniqueness, as well as presenting applications of our results to models

of Bertrand competition, "beauty contests", and existence of equilibrium in large economies.

keywords: large games, distributional equilibria, supermodular games, games with strate-

gic complementarities, computation of equilibria

JEL codes: C72

1 Introduction and related literature

Since the seminal papers of Schmeidler (1973) and Mas-Colell (1984), an important class of

games, that have been studied in the literature, are games played by a continuum of players

indexed on a measure space. Despite obvious similarities in speci cation of their environment,

the approaches taken to de ne and verify equilibrium existence, by researchers studying large

games in the traditions of these two authors, have signi cant dierences (see e.g. discussion in

Khan, 1989). Schmeidler studies a game where players’ payo s depend on their own actions and

action pro le of all other players. That is, in his game, what matters for a player’s payo are both

her own action, and those taken by her opponents. Hence, in the Schmeidler game, one de nes an

equilibrium in strategies, i.e. functions from the set of players to their action sets. This notion of

equilibrium is di erent from the approach taken in Mas-Colell, who studies games where players’

payo s depend on their own action and the distribution on all players actions. Consequently, in

this latter tradition, one de nes an equilibrium in distributions on both players’ characteristics

We thank Ed Green, Bob Lucas, and Ed Prescott for helpful discussions during the writing of this paper.

Re ett thanks the Deans Award in Excellence Summer Grant program at the W. P. Carey School of Business

in 2011 for nancial support. Wozny thanks the Deans Grant for Young Researchers 2011 at WSE for nancial

support. All the usual caveats apply.

yInstitute of Mathematics, Wroc law University of Technology, Wroc law, Poland.

zDepartment of Economics, University of Oxford, UK.

xDepartment of Economics, Arizona State University, USA.

{Department of Theoretical and Applied Economics, Warsaw School of Economics, Warsaw, Poland. Address:

al. Niepodleg losci 162, 02-554 Warszawa, Poland. E-mail: lukasz.wozny@sgh.waw.pl.

1and actions. As a result, in this latter notion of equilibrium, the term anonymous game seems

readily justi ed, as it does not matter who chooses each action, rather only the distribution of

1actions is the object that is payo relevant . Despite these dierences, both approaches seem

like very natural generalizations of the notion of Nash equilibria for games with a nite number

of players to those situations, where the marginal in uence of any individual player’s action on

the equilibrium aggregates is insigni cant (see e.g. Horst and Scheinkman, 2009).

Regardless of the notion of an equilibrium involved, a central question that arises in this

literature concerns su cient conditions for equilibrium existence. In their seminal papers both

authors (Mas-Colell, 1984 and Schmeidler, 1973), use topological xed points theorems of Fan-

Glicksberg applied to best response maps that are continuous in appropriately chosen topologies.

Since these early results where rst presented, there has been a vast important literature that has

studied possible generalizations of existence results. Speci c existence results for both notions of

equilibria of large games have been generalized in the following works: (i) in Khan (1986, 1989);

Khan, Rath, and Sun (1997) or Balder (1999); Wiszniewska{Matyszkiel (2000), were the question

is how to allow for general spaces of players’ actions; (ii) in Rath (1996), where the question

is how to generalize the results to the case of upper semi-continuous payo s; (iii) in Balder

and Rustichini (1994) and Kim and Yannelis (1997), where the question is how to generalize the

2results to large Bayesian games ; and nally (iv) in Martins da Rocha and Topuzu (2008), where

the question is how to generalize the results to the case of non-ordered preferences. For a survey

of some recent literature concerning generalizations of existenence results for large games, we

3refer the reader to a chapter by Khan and Sun (2003) .

A second and important strand of literature in game theory that has found a large number of

applications in the economics (and operations research) has been so-called supermodular games

(or, if one prefers, games with strategic complementaries (GSC). For example, see the seminal

works of Topkis (1979), Milgrom and Roberts (1990), Veinott (1992), Zhou (1994) and more

recently Heikkila and Re ett (2006). In supermodular games, the pure strategy best response

mappings are not necessarily continuous, bur are increasing in a well-de ned set theoretic sense,

due to complementaries in the payo structure between own and other strategies. The beauty

of this approach is that one can appeal to a powerful xed point theorems of Tarski (1955) or

Veinott (1992)/Zhou (1994) for complete lattices, and a set of pure strategy Nash equilibria turns

4out to be a non-empty complete lattice Further, and equally as appealing, for GSC, one can

develop su cient conditions for equilibrium comparative statics (results not typically available,

when one uses purely topological approaches). For applications of such games in economic theory

we refer the reader to Topkis’ (1998) book.

In this paper, we integrate these two strands of literatures, and consider the existence and

characterization of equilibria of large games with strategic complementarities (large GSC). Unlike

any of the existing literature on large games, in all cases we study, our methods emphasize

monotone operators de ned in appropriate spaces of equilibrium objectives (e.g., strategies or

distributions). In doing this, we are able to link the extensive literature on large games with that

on GSC. Aside from developing a notion of a large GSC (which, itself, involves some introduction

of new structure not required in a standard GSC), our aim is to develop a toolkit that allows

1Here let us mention that anonymity can be also modeled using Schmeidler’s (1973) approach, where each

player’s payo depends on his/her own action and an aggregate (e.g. average) of players’ strategies.

2Compare also with Balder (2002) unifying approach to equilibrium existence.

3We also should mention the results in Blonski (2005) on equilibrium distributions characterization, and the

results in Rashid (1983) on approximation of equilibria by equilibria in games with nite number of players, as

these ideas are also related to the questions raised in this paper.

4A complete lattice is a partially ordered set X, which any subset has supremum and inmum in X.

2one to obtain sharp characterizations of equilibrium set (either in the sense of Schmeidler or

Mas-Colell). Finally, in all cases of equilibrium studied, per the question of existence, we are

able to relax important continuity results that are found in the existing literature. In the end,

we not only address the question of equilibrium existence, but also questions concerning both

the computation of particular Nash equilibrium, as well the question of computable equilibrium

comparative statics.

More speci cally, in the case of Mas-Colell’s game, using xed point theorem of Markowsky

5(1976) for isotone transformations of chain complete partially ordered sets , we are able to prove

existence of a distributional equilibrium under di erent set of assumptions than those studied

in the literature that has followed since Mas-Colell (1984). Further, we are able to prove these

results using constructive methods. This latter fact becomes of central importance, when we

next consider the question of equilibrium comparative statics. In particular, we are able to prove

a theorem of computable monotone comparative statics relative to ordered perturbations of the

deep parameters of the space of primitives of a game. Similarly, using our generalization of

Tarski-Kantorovitch xed point theorem (proven in theorem ?? in the appendix), our construc-

tions are able to develop explicit methods for Nash/Schmeidler equilibrium computation that

cannot be addressed directly using existing topological results. Finally, we present conditions

for existence of symmetric equilibrium and equilibrium uniqueness (which can prove useful in

applications).

An important point our paper makes is that, although the tools used for a study of equi-

librium (in an appropriately de ned sense) for standard GSC vs. large GSC are similar in a

very general methodological sense, the results one can obtain are not. For example, in a large

GSC, one can provide conditions under which the set of distributional equilibria has the great-

est and the least element, but is not a complete lattice. Similarly, this is also the case of

the Nash/Schmeidler equilibrium set for a large GSC. More speci cally, we show if the best

responses maps for each player are functions, then the set of distributional equilibria ala Mas-

Colell (respectively, Nash/Schmeidler equilibrium in strategies) is a chain complete (respectively,

countably chain complete) poset. These key di erences in the structure of the set of equilibrium

between standard GSC versus large GSC arises because of the inherent in