FROM SELF TO OTHER: ESOTERICISM AS DISCOURSE OF ...

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 sen