A Characterization of Strategic Complementarities

TL;DR: It is shown that generic 2 X 2 games either have no pure-strategy equilibria, or are GSC, a negative result because it implies that the predictive power of complementarities alone is very weak.

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.

Summary

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.
• I present an example where no order that makes the game a GSC, avoids strictly dominated strategies in the interval prediction—this is further evidence that the interval prediction is problematic.
• In Section 2 I give some preliminary definitions and present the model that the results apply to.

2.2 The Model

• The reader should think of the results as dealing with the best-response function, or correspondence, of a game.
• The set of fixed points of φ—or, if best-responses are unique, of f—coincides with the set of pure-strategy Nash equilibria of Γ. I shall only deal with pure strategies and pure-strategy equilibria, but one can apply my results to the best-response correspondence of the mixed extension of a game.
• If players meet to play Γ over and over again, and if they, in each round of 4I focus on pure strategies because the GSC literature considers usually orders on pure strategies.
• Absence of cycles means that no Cournot best-response dynamics will get caught in a cycle.
• And global stability means that players that engage in Cournot best-response dynamics will eventually approach Nash equilibrium play.

3 Unique Equilibrium

• The same number of iterates away from e, which is slightly cumbersome but easy (see the proof of Theorem 3 for the details, and for the proof when X is infinite).
• Theorem 2 may have some mathematical interest, independently of the application to game theory that I emphasize.

3.1 Examples

• One implication of these results is that a game that satisfies the dominant diagonal condition in Gabay and Moulin (1980), and has thus a unique, globally stable, equilibrium, is a GSC.
• 6See section 3.3 for the difference between lattice and totally ordered strategy spaces.
• The intuitive idea that zero-sum games and GSC are different is therefore false.

3.2 Non-unique best-responses

• Theorem 2 for functions can be generalized to correspondences with finite values, but the characterization is not completely tight.
• Absence of cycles implies the existence of a total order that makes the correspondence weakly monotone, but a weakly monotone correspondence may have cycles, as weak monotonicity does not control all selections from the correspondence.
• I need the hypothesis of a correspondence with finite values to use non-standard analysis the way I do.
• If Γ is a game with unique best-responses, of course the bestresponses have finite values, and both weakly increasing and increasing in the strong set order coincide with monotone increasing.
• I have not been able to extend the characterization in Theorem 3 to arbitrary games with infinite strategy spaces and non-unique best-responses.

3.3 Totally ordered strategy space vs. a lattice strategy space,

• If a game’s best-response function does not have cycles, then the game is a GSC because there is a total order on strategies such that best responses are monotone increasing, and a totally ordered set is a lattice.
• In GSC where strategy spaces are non-complete lattices, best-responses may have cycles.
• In the usual definition of GSC—supermodular games—there is a link between the order and the topology on the strategy spaces that, among other things, ensures the existence of equilibria (by Tarski’s Theorem), and that if equilibrium is unique it must be globally stable.
• In infinite games, though, there may be GSC without equilibria, and GSC with a unique equilibrium that is not globally stable.
• I wish to emphasize that, despite this technical problem, for games with multiple equilibria, all of Vives’s, Topkis’s and Milgrom and Roberts’s results hold trivially with the order that I construct in section 4.

4 Multiple Equilibria

• I work out the details in Theorem 5, but besides answering the question in this paper, such a coarse order is uninteresting.
• Theorem 7 below says that, if a function f on a finite set has at least two different fixed points, then there is a lattice order such that f is monotone increasing, the interval prediction is the order interval between the fixed points, and any other lattice order that does the same work must have a larger interval prediction.
• The usual definitions of GSC ensure that best-responses are increasing in the strong set order, and that φ takes subcomplete- sublattice-values.
• The order that I construct in Theorem 5 does not guarantee that φ has these properties (unless, of course, φ is a function), but Topkis’s, Vives’s, Milgrom and Roberts’s, and Milgrom and Shannon’s results are true with the constructed order.
• Topkis, Vives and Zhou present examples where it is not a sublattice (see Echenique (2001), though).

5 Discussion of the interval prediction.

• I show that there are games for which the interval prediction is unavoidably vacuous, and that there are games for which the interval prediction must contain strictly dominated strategies.
• Theorem 7 delivers an order with the best possible interval prediction, but the construction of this order is in general quite cumbersome.
• To see this, suppose, for example, that (β, α) is the smallest strategy profile.
• All strategies in the coordination games are rationalizable.

6 Comparative Statics

• In a parameterized GSC, if a parameter t is complementary to players’ choices, there are selections of equilibria that are monotone increasing in t (Lippman, Mamer, and McCardle 1987, Sobel 1988, Villas-Boas 1997, Milgrom and Roberts 1990, Milgrom and Roberts 1994, Milgrom and Shannon 1994, Echenique 2002, Echenique and Sabarwal 2000).
• Using the framework in Milgrom and Roberts (1994), the question can be phrased as follows.
• I will rule out all possible cycles in best-response dynamics, except for the cycle involving (β, β).
• The authors have shown that all 2X2 games with a unique, strict Nash equilibrium are GSC.

8 Application 2: Ordinal Potential Games

• Potential games were studied in detail by Monderer and Shapley (1996) (see their paper for references to earlier work on potential games).
• My results shed some light on the relation between GSC and ordinal potential games.
• Let Γ be a finite two-player game with unique best responses.
• I do not discuss this here, the counterexamples are very simple.
• An ordinal potential game cannot have an infinite improvement path.

9.1 Proof of Theorem 3

• Steps 4 and 5 prove the first statement for arbitrary X by non-standard analysis methods.
• The idea is to embed X in a hyperfinite set, apply the result for finite sets to get an order that works in the hyperfinite set, and then restrict the order to X. Step 1.

9.2 Proof of Theorem 7

• In any case, x ∈ C∪A∪Be ∪Be, which establishes the claim.
• Be into Be, has exactly one fixed point, and no cycles.
• I now prove that (X,≤) is a complete lattice (steps 1 and 2), that f is monotone increasing (step 3), and that any other lattice order that makes f monotone increasing must have a larger interval prediction (step 4).
• I shall need the following two facts, the facts are immediate from the definition of ≤.
• So, f monotone increasing implies that f nM(x∗) fnM(z).

Figures

Figure 1: A coordination game.

Full text

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

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