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A CHARACTERIZATION OF STRATEGIC COMPLEMENTARITIES

Federico Echenique

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SOCIAL SCIENCE WORKING PAPER 1142

October 2002

A Characterization of Strategic Complementarities

Federico Echenique

Abstract

I characterize games for which there is an order on strategies such that the game has

strategic complementarities. I prove that, with some qualiﬁcations, games with a unique

equilibrium have complementarities if and only if Cournot best-response dynamics has

no cycles; and that all games with multiple equilibria have complementarities.

As applications of my results, I show: 1. That generic 2X2 games either have no

pure-strategy equilibria, or have complementarities. 2. That generic two-player ﬁnite

ordinal potential games have complementarities.

JEL classiﬁcation numbers: C62, C72

A Characterization of Strategic Complementarities

Federico Echenique

∗

1 Introduction

A game has strategic complementarities (Topkis 1979, Vives 1990) if, given an order on

players’ strategies, an increase in one player’s strategy makes the other players want to

increase their strategies. For example, the players could be ﬁrms in price competition; if

each ﬁrm’s optimal price is an increasing function of the prices set by their opponents, the

game has strategic complementarities. In games of strategic complementarities (GSC),

Nash equilibria have a certain order structure; in particular, there is a smallest and

largest equilibrium (Topkis, Vives, Zhou (1994)). Further, the set of all rationalizable

strategies, and the set of limits of adaptive learning, is bounded below by the smallest

equilibrium and above by the largest equilibrium (Vives 1990, Milgrom and Roberts 1990,

Milgrom and Shannon 1994). GSC are a well-behaved class of games, and a useful tool

for economists.

Consider the coordination game in Figure 1, is it a GSC? That is, is there an order on

players’ strategies so that best-responses are monotone increasing? Yes, let α be smaller

than β. Then, player 2’s best response to α is α and to β is β. So, when 1 increases her

strategy from α to β, 2’s best response increases from α to β. Similarly for 1. So, with

this order the coordination game is a GSC.

The question I address is: what does this depend on? That is, when can we ﬁnd an

order on strategies so that a game is a GSC? Note that how strategies are ordered is not

∗

I thank an associate editor and a referee for their comments. I also thank Elvio Accinelli, Bob

Anderson, Juan Dubra, Paul Milgrom, Stephen Morris, Charles P´ugh, Ilya Segal, Chris Shannon, Xavier

Vives, and seminar participants at Arizona State and Stanford Universities. A conversation with Ted

O’Donoghue and Clara Wang prompted me to work on the research presented here. The non-standard

proof of Theorem 3 owes a great deal to Bob Anderson, I am deeply grateful for his help. I worked

out the results in Section 8 in response to Stephen Morris’s very stimulating questions. Finally, part

of this paper was written while I visited UC Berkeley’s Economics Department, I appreciate Berkeley’s

hospitality. Any errors are my responsibility.

α β

α 2, 2 0, 0

β 0, 0 1, 1

Figure 1: A coordination game.

part of the description of a game—it does not aﬀect the available strategies or payoﬀs.

We as analysts use the order as a tool, therefore we are justiﬁed in choosing the order to

conform to our theory. Often, results on GSC are used by coming up with a clever order

on strategies that makes a situation into a GSC, this paper is concerned with how often

such orders exists.

We normally introduce strategic complementarities by assuming supermodular pay-

oﬀs. The crucial feature for most results on GSC is that the game’s best-response corresp-

ondence is monotone increasing, and the assumption of supermodular payoﬀs is suﬃcient

for monotone increasing best-responses.

1

I shall argue that it is enough, for the purpose

of this paper, to deﬁne a GSC as a game for which there is a partial order on strategies

so that best-responses are monotone increasing (and such that strategies have a lattice

structure).

My results are:

1. With some qualiﬁcations, a game with a unique pure-strategy Nash equilibrium is

a GSC if and only if Cournot best-response dynamics has no cycles except for the

equilibrium.

2. A game with two or more pure-strategy Nash equilibria is always a GSC.

I illustrate my results with two applications: Generically, 2X2 games are either GSC

or have no pure-strategy equilibria. And, generically, a ﬁnite two-player ordinal potential

game is a GSC, but ordinal potential games with more than two players need not be GSC.

I now discuss my results.

1. With some qualiﬁcations, in games with a unique Nash equilibrium, strategic

complementarities is equivalent to the absence of cycles in Cournot best-response dynam-

ics. If b is the game’s best-response function (the product of the players’ best-response

1

In fact, the ﬁrst paper using lattice-programming techniques in economics, Vives (1985), deﬁned

GSC like I do here.

2

functions) then Cournot best-response dynamics starting at x is deﬁned by x

0

= x,

x

n

= b(x

n−1

), n = 1, 2, . . .. Absence of cycles means that, if x is not an equilibrium,

Cournot best-response dynamics starting at x never returns to x, i.e. x

n

6= x for all

n ≥ 1, or, equivalently, that x 6= b

n

(x) for all n ≥ 1. In ﬁnite games, absence of cycles is

equivalent to global stability, that is that b

n

(x) converges to the equilibrium for all x. In

inﬁnite games, absence of cycles is a weaker condition than global stability—so I show

that a game with a unique, globally stable, equilibrium is a GSC.

2. A game with two or more pure-strategy equilibria is a GSC, so the vast majority

of games that we encounter in applied work are GSC. The order in the coordination

game of Figure 1 that makes its best-response function increasing involves making one

Nash equilibrium the smallest point in the joint strategy space, and the other equilib-

rium the largest point in the strategy space. I show that, if a game has at least two

equilibria, the same trick always works; we can order the strategies such that one equi-

librium is the largest strategy proﬁle and the other equilibrium is the smallest, and such

that best-responses are monotone increasing. This result has implications for the use of

complementarities to obtain predictions in games.

The literature on GSC has developed a set-valued prediction concept: the set of

strategies that are larger than the smallest Nash equilibrium and smaller than the largest

Nash equilibrium. This “interval prediction” contains all rationalizable strategies, and

all strategies that are limits of adaptive learning. Is the interval prediction in general a

sharp prediction? Milgrom and Roberts suggest that the answer may be negative:

Indeed, for some games, these bounds are so wide that our result is of little

help: it is even possible that these bounds are so wide that the minimum

and maximum elements of the strategy space are equilibria. (Milgrom and

Roberts 1990, p. 1258)

Milgrom and Roberts go on to argue that, in some models, “the bounds are quite narrow.”

They present as examples an arms-race game, and a class of Bertrand oligopoly models,

where there is a unique equilibrium.

My results imply that this is generally the case: if a game does not have a unique

equilibrium, the interval prediction is essentially vacuous, as all games with multiple

equilibria are GSC where the smallest and largest equilibria are the smallest and largest

strategy proﬁles. This happens because games with multiple equilibria always involve a

3