DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
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
