Rationalizability, Adaptive Dynamics, and the

Correspondence Principle in Games with Strategic

Substitutes

Sunanda Roy Tarun SabarwalDepartment of Economics Department of EconomicsIowa State University University of KansasAmes IA, 50011, USA Lawrence KS, 66045, USAsunanda@iastate.edu sabarwal@ku.edu

Abstract

New insights into the theory of games with strategic substitutes (GSS) are developed. Thesegames possess extremal serially undominated strategies that provide bounds on predictedbehavior and on limiting behavior of adaptive dynamics, similar to games with strategiccomplements (GSC). In parameterized GSS, monotone equilibrium selections are dynami-cally stable under natural conditions, as in parameterized GSC. Dominance solvability inGSS is not equivalent to uniqueness of Nash equilibrium, but is equivalent to uniqueness ofsimply rationalizable strategies. Convergence of best response dynamics in GSS is equiva-lent to global convergence of adaptive dynamics, is equivalent to dominance solvability, andimplies uniqueness of equilibrium, all in contrast to GSC. In particular, Cournot stabilityis equivalent to dominance solvability in GSS. The results shed light on predicted behav-ior, learning, global stability, uniqueness of equilibrium, and dynamic stability of monotonecomparative statics in GSS. Several examples are provided.

JEL Numbers: C70, C72, C62Keywords: Rationalizability, Adaptive Dynamics, Correspondence Principle, Strategic Sub-stitutes, Learning, Dominance Solvability, Global Stability, Monotone comparative statics

First Draft: July 2008This Version: February 2, 2010

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1 Introduction

Games with strategic substitutes (GSS) and games with strategic complements (GSC) for-malize two basic economic interactions.1

In GSS, if one player takes a higher (or more aggressive, or more intensive) action, agiven player’s best-response is to take a lower action. That is, best-response of each player isweakly decreasing in actions of the other players. For example, consider a simple, textbookCournot duopoly, where a given firm’s profit-maximizing output is lower when the other firmincreases its output.

In GSC, if one player takes a higher action, a given player’s best response is to take ahigher action, too. That is, best-response of each player is weakly increasing in actions of theother players. For example, consider a game of network externalities, where a given player’smarginal benefit from adopting a technology is increasing as more other players adopt thesame technology.

Versions of such games arise in diverse economic environments, including competitivestrategy, public goods, industrial organization, natural resource utilization, manufacturinganalysis, team management, tournaments, resource allocation, business portfolio develop-ment, principal-agent modeling, multi-lateral contracting, auctions, technological innovation,behavioral economics, and others.

Research on GSS is not as well-developed as that for GSC,2 partly because uncoveringresults of the same generality has proved to be harder, and indeed, may be impossible insome cases.3 In this paper, we develop the theory of GSS in several directions.

We show that in GSS, there always exist a smallest and a largest profile of seriallyundominated (and rationalizable) strategies, analogous to the result due to Milgrom andRoberts (1990) for GSC. As serially undominated strategies include the solution conceptsof pure strategy Nash equilibrium, mixed strategy Nash equilibrium, correlated equilibrium,rationalizable strategies, and the dominance solution, these extremal serially undominatedstrategies provide bounds that are robust to a variety of predicted economic behaviors.

These extremal strategy profiles need not necessarily be Nash equilibria, but they dosatisfy a behavioral condition we term ‘simply rationalizable’; intuitively, strategy profiles

1Such games are defined in Bulow, Geanakoplos, and Klemperer (1985), and as they show, models ofstrategic investment, entry deterrence, technological innovation, dumping in international trade, naturalresource extraction, business portfolio selection, and others can be viewed in a more unifying frameworkaccording as the variables under consideration are strategic complements or strategic substitutes. Earlierdevelopments are provided in Topkis (1978) and Topkis (1979).

2The main properties of GSC are relatively well understood. Some of this work can be seen in Topkis(1978), Topkis (1979), Lippman, Mamer, and McCardle (1987), Sobel (1988), Milgrom and Roberts (1990),Vives (1990), Zhou (1994), Milgrom and Shannon (1994), Milgrom and Roberts (1994), Shannon (1995),Villas-Boas (1997), Edlin and Shannon (1998), Echenique (2002), Echenique and Sabarwal (2003), and Quah(2007), among others. Extensive bibliographies are available in Topkis (1998), in Vives (1999), and in Vives(2005).

3See, for example, Amir (1996), Villas-Boas (1997), Zimper (2007), Roy and Sabarwal (2008), Roy andSabarwal (2009), Jensen (2010), among others.

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that can be simultaneously rationalized using no more than two iterations of conjecturesabout play by competitors. Mathematically, these are fixed points of the second iterate ofthe (joint) best response correspondence. In this terminology, a GSS is dominance solvable,if, and only if it has a unique profile of simply rationalizable strategies. This generalizes andprovides an economic foundation for a result due to Zimper (2007). These results are appliedto Cournot oligopolies, common pool resource games, games with linear best responses, andtournament games.

In GSS, these extremal serially undominated strategies provide bounds on the limits oflearning behavior in the broad class of learning dynamics formalized by adaptive dynamics,analogous to the result due to Milgrom and Roberts (1990) for GSC.

Convergence of best response dynamics (starting at either the infimum or the supremumof the strategy space) in GSS is equivalent to global convergence of adaptive dynamics, isequivalent to dominance solvability, and implies uniqueness of equilibrium, all in contrast toGSC.

In particular, convergence of best-response dynamics implies uniqueness and global sta-bility of equilibrium in GSS. This provides a converse to the well-known result due to Moulin(1984) that in strategic games, dominance solvability implies Cournot stability. For GSS,Cournot stability is equivalent to dominance solvability. These results provide a new per-spective on global stability, as compared to, for example, Al-Nowaihi and Levine (1985) andOkuguchi and Yamazaki (2008). In GSS, global stability can be analyzed using best responsedynamics as an alternative to the traditional eigen-value approach. The results are appliedto games with linear best responses and tournaments.

In parameterized GSS, monotone selections of equilibria are dynamically stable, analo-gous to results due to Echenique (2002) for parameterized GSC. We show (approximately)that in parameterized GSS, under natural conditions, if an equilibrium selection is mono-tone increasing, then it is dynamically stable under adaptive dynamics, and if it is nowhereincreasing, then it is not stable under adaptive dynamics. Thus, when considering dynami-cally stable equilibria (as proposed by Samuelson’s correspondence principle), we may expectmonotone selections of equilibria to arise naturally in GSS. Echenique (2002) shows theseresults for parameterized GSC by implicitly requiring a type of strict single-crossing prop-erty. Our condition focuses on an appropriate tradeoff between the direct parameter effectand the indirect strategic substitute effect. In our case, a strict single-crossing property isnot needed.

The paper proceeds as follows. Section 2 presents results on existence of serially undom-inated strategies in GSS and implications of such extremal strategies. Section 3 investigatesbounds on limits of adaptive dynamics in GSS, and implications of such bounds for globalstability and uniqueness of equilibrium. Section 4 shows the dynamic stability of monotoneequilibrium selections in parameterized GSS.

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2 Rationalizability

Let I be a non-empty set of players, and for each player i, a strategy space that is a partiallyordered set (X i,�i), and a real-valued payoff function, denoted f i(xi, x−i). As usual, thedomain of each f i is the product of the strategy spaces, (X,�) endowed with the productorder.4 The strategic game Γ = {I, (X i,�i, f i)i∈I} is a game with strategic substitutes

if for every player i,

1. X i is a complete lattice,

2. For every x−i, f i is order upper semi-continous in xi, and for every xi, f i is ordercontinuous in x−i,

3. For every fixed x−i, f i is quasi-supermodular in xi,5 and

4. f i satisfies the decreasing single-crossing property in (xi; x−i).6

As compared to Milgrom and Roberts (1990), we have replaced supermodular with itsordinal generalization, quasi-supermodular, and replaced increasing differences with the de-creasing single-crossing property, an ordinal generalization of decreasing differences. Thedecreasing single-crossing property captures the idea of strategic substitutes, just as thesingle-crossing property formalizes the idea of strategic complements.

For each player i, the best response of player i is denoted gi(x−i). As is well-known,for each player i, the best response of player i, gi(x−i), is a non-empty and complete sub-lattice of X i.7 As usual, let gi(x−i) = sup gi(x−i) and gi(x−i) = inf gi(x−i) be the extremalbest responses. As shown in Roy and Sabarwal (2009), for each player i, and for eachprofile of other player strategies x−i, gi(x−i) is nonincreasing in x−i,

8 and therefore, for eachplayer i, both gi(x−i) and gi(x−i) are nonincreasing functions.9 Let g : X � X, g(x) =(gi(x−i))i∈I , denote the joint best-response correspondence. Then the correspondenceg is nonincreasing, and the functions g(x) = inf g(x) and g(x) = sup g(x) are nonincreasing.

As usual,10 a pure strategy xi ∈ X i is strongly dominated, if there exists x̂i ∈ X i

such that for every x−i, f i(xi, x−i) < f i(x̂i, x−i). For a given set of strategy profiles X̂ ⊂ X,

4For notational convenience, we shall sometimes drop the index i from the notation for the partial order.5As in Milgrom and Shannon (1994), a function f : X → R (where X is a lattice) is quasi-supermodular

if (1) f(x) ≥ f(x ∧ y) =⇒ f(x ∨ y) ≥ f(y), and (2) f(x) > f(x ∧ y) =⇒ f(x ∨ y) > f(y).6 A function f : X × T → R (where X is a lattice and T is a partially ordered set) satisfies decreasing

single-crossing property in (x; t) if for every x′ > x′′ and t′ > t′′, (1) f(x′, t′′) ≤ f(x′′, t′′) =⇒ f(x′, t′) ≤f(x′′, t′), and (2) f(x′, t′′) < f(x′′, t′′) =⇒ f(x′, t′) < f(x′′, t′). This property is discussed in some detail inRoy and Sabarwal (2009). Amir (1996) terms this property the dual single-crossing property.

7Confer Milgrom and Shannon (1994), Milgrom and Roberts (1990), Vives (1990), Zhou (1994).8For every x−i and x′

−i, if x−i � x′

−ithen gi(x′

−i) vi gi(x−i), where the order on nonempty subsets of

X i is the standard (induced) set order used in the literature. That is, for non-empty subsets A, B of X i,A vi B if for every a ∈ A, and for every b ∈ B, a∧ b ∈ A, and a∨ b ∈ B, where the operations ∧,∨ are withrespect to �i.

9For every x−i and x′

−i, if x−i � x′

−ithen gi(x′

−i) � gi(x−i) and gi(x′

−i) � gi(x−i).

10Following Milgrom and Roberts (1990).

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player i’s undominated responses to X̂, is the set

Ui(X̂) ={

xi ∈ X i|∀x′

i ∈ X i, ∃x̂ ∈ X̂, f i(xi, x̂−i) ≥ f i(x′

i, x̂−i)}

.

Let U(X̂) = (Ui(X̂)i∈I) denote the collection of undominated responses to X̂, one for eachplayer, and let U(X̂) = [inf U(X̂), sup U(X̂)] be the smallest order interval containing U(X̂).

As usual, the (pure strategy) Nash equilibria of the game are given by E = {x ∈ X|x ∈ g(x)}.Recall that the solution concepts Nash equilbria, mixed strategy Nash equilbria, correlatedequilibria, and rationalizable strategies are all included in the set of serially undominatedstrategies. The following lemma is helpful to show existence of extremal serially undominatedstrategies in GSS.

Lemma 1. Let Γ be a game with strategic substitutes.For every a, b ∈ X such that a � b, U [a, b] = [g(b), g(a)].

Proof. Let’s first see that U [a, b] ⊂ [g(b), g(a)]. Consider the contrapositive. Supposey 6∈ [g(b), g(a)]. Then either, y 6� g(a) or y 6� g(b). Suppose y 6� g(a). In particular, considerplayer i such that yi 6� gi(a−i). Then yi ∧ gi(a−i) dominates yi, as follows. Indeed, for everyx ∈ [a, b],

f i(yi ∨ gi(a−i), a−i) − f i(gi(a−i), a−i) < 0=⇒ f i(yi, a−i) − f i(yi ∧ gi(a−i), a−i) < 0=⇒ f i(yi, x−i) − f i(yi ∧ gi(a−i), x−i) < 0,

where the first inquality follows from the definition of gi(a), the first implication follows fromquasi-supermodularity, and the second implication follows from decreasing single-crossingproperty. The case y 6� g(b) follows similarly. Therefore, y 6∈ U [a, b]. Consequently, U [a, b] ⊂[g(b), g(a)].

This shows that U [a, b] ⊂ [g(b), g(a)]. Moreover, as g is a best response, g(b) and g(a)

are in U [a, b], whence [g(b), g(a)] ⊂ U [a, b].

Theorem 1. Let Γ be a game with strategic substitutes. For each player i, there existsmallest and largest serially undominated strategies, denoted xi and xi, respectively.

Proof. Let x0 = inf X, and x0 = sup X, and consider the sequences defined as follows. Fork ≥ 1, let xk = g(xk−1) and xk = g(xk−1). Notice that for every k, xk � xk. This holds

trivially for k = 0. Suppose xk � xk. Then xk+1 = g(xk) � g(xk) � g(xk) = xk+1.

Let U0(X) = X, and for k ≥ 1, let Uk(X) = U(Uk−1(X)), where, as above, U(S) is thecollection of undominated responses to S. Thus, serially undominated strategies are given

by the set∞⋂

k=0

Uk(X). Notice that U is monotone nondecreasing; that is, S ⊂ S ′ ⇒ U(S) ⊂U(S ′).

It can be shown by induction that for k ≥ 0, Uk(X) ⊂ [xk, xk], as follows. This holdstrivially for k = 0. Suppose it holds for k − 1. Then for k,

Uk(X) = U(Uk−1(X)) ⊂ U [xk−1, xk−1] ⊂ [g(xk−1), g(xk−1)] = [xk, xk],

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where the first inclusion follows from the inductive hypothesis and monotonicity of U , andthe second inclusion follows from the previous result.

Moreover, notice that the sequence (xk) is nondecreasing, and the sequence (xk) is non-increasing, as follows. If xk � xk+1, then xk+1 = g(xk) � g(xk+1) = xk+2, and similarly, ifxk � xk+1, then xk+1 � xk+2. Thus, the sequence (xk) is nondecreasing, and the sequence(xk) is nonincreasing, if x0 � x1, and x0 � x1. But this is true, because x0 = inf X, andx0 = sup X.

As X is complete, let x = limk xk and x = limk xk. Consequently,∞⋂

k=0

Uk(X) ⊂ [x, x].

That is, the set of serially undominated strategies is contained in the order interval given by[x, x].

Notice now that x is a best response to x, and x is a best response to x. That is,x ∈ g(x) and x ∈ g(x), as follows. Suppose x 6∈ g(x). Then there is i, and xi such thatf i(xi, x−i) − f i(xi, x−i) > 0. But then, by upper semi-continuity in the i variable, andcontinuity in the −i variables, for all k sufficiently large, f i(xi, x

k−i) − f i(xk+1

i , xk−i) > 0,

contradicting the optimality of xk+1i . Similarly, x ∈ g(x).

Finally, note that x and x are in∞⋂

k=0

Uk(X), as follows. Trivially, x and x are in U0(X).

Suppose x and x are in Uk(X). Then x ∈ Uk+1(X), because x ∈ Uk(X) and x is abest response to x, and x ∈ Uk+1(X), because x ∈ Uk(X) and x is a best response to x.Thus, serially undominated strategies lie in [x, x], and the end points are extremal seriallyundominated strategies.

Notice that in GSS (in contrast to GSC), extremal serially undominated strategies maynot necessarily be Nash equilibria. For example, consider a standard 3-firm Cournot oligopolywith linear inverse demand given by p = a− b(x1 + x2 + x3), with constant marginal cost, c,and with production capacity constrained to [0, xmax] for each firm. In this case, the uniqueNash equilibrium is given by x1 = x2 = x3 = a−c

4b, but as shown in more detail in example

1 below, (0, 0, 0) is the smallest serially undominated strategy profile, and (xmax, xmax, xmax)is the largest.

Moreover, notice that when best responses are functions, the construction of the sequences(xk) and (xk) above is the same as in Milgrom and Roberts (1990). The difference emergeswhen best responses may be correspondences. In this case, for GSS, a direct application ofthe proof in Milgrom and Roberts (1990) may be problematic, as follows.

Suppose, as in Milgrom and Roberts (1990), that y0 = inf X, and z0 = sup X, and fork ≥ 1, yk = g(yk−1), and zk = g(zk−1). For GSC, this construction yields the following useful

facts: (yk) is monotone nondecreasing, (zk) is monotone nonincreasing, and these sequencesare comparable all along; that is, for every k, yk � zk. It is then shown that (yk) converges tothe smallest profile of serially undominated strategy, and (zk) converges to the largest profileof serially undominated strategies. These facts turn out to be very useful in analyzing GSC.

For GSS, this construction does not get us very far. In this case, y0 � z0, and theny0 � y1, but then y2 � y1, and then y2 � y3, and y4 � y3, and so on. A monotonic

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relationship in the progression of the elements in the sequence does not emerge. Similarly,the sequence (zk) is less amenable to analysis. Another argument sometimes put forthis to consider the second-iterate of the analogous construction for games with strategiccomplements. In this case, it is true that y0 � y2, and inductively, for every k, y2k � y2k+2,and therefore, the sequence (y2k) is nondecreasing, and similarly, (z2k) is nonincreasing. Butwith nonincreasing g, it does not follow that in general, g ◦g(y0) � g ◦g(z0),11 and therefore,

a clear comparison across the sequences (yk) and (zk) does not emerge.

The lemma above indicates a fruitful construction. Notice that U0(X) ⊂ [y0, z0] = X.

The lemma shows that U1(X) ⊂[

g(z0), g(y0)]

, and U1(X) =

[

g(z0), g(y0)]

. Thus, at thefirst iteration, it is useful to transform y0 = inf X and z0 = sup X to y1 = g(z0) andz1 = g(y0). Similarly, another iteration of the lemma yields

U2(X) = U(U(X)) ⊂ U([

z1, y1]

) ⊂ U([

z1, y1]

) =[

g(y1), g(z1)]

.

Thus, at the second iteration, it is useful to transform y1 to y2 = g(y1), and z1 to z2 = g(z1).The process continues inductively. This is the construction followed in the proof of thetheorem, and it turns out be very useful in what follows.

We can relate this construction to the idea of a “smallest” and a “largest” best responsedynamic, as follows. Consider the sequences (xk) and (xk) in the proof of the theoremabove. That is, x0 = inf X, x0 = sup X, and for k ≥ 1, xk = g(xk−1) and xk = g(xk−1). The

(simultaneous) best response dynamic starting at inf X is the sequence (yk)∞k=0 givenby yk = xk if k is even, and xk if k is odd. Notice that when g is a best-response function,(yk) is the standard simultaneous best-response dynamic starting at inf X, as follows. Noticethat y0 = x0 = inf X, and y1 = x1 = g(x0) = g(y0). Suppose yk = g(yk−1). Then if k iseven, yk+1 = xk+1 = g(xk) = g(yk), and if k is odd, then yk+1 = xk+1 = g(xk) = g(yk).In either case, yk+1 = g(yk). When g is a correspondence, we view (yk) as an analogue ofthe “smallest” best response dynamic used in Milgrom and Roberts (1990) and Echenique(2002).

Similarly, the (simultaneous) best response dynamic starting at sup X is the se-quence (zk)∞k=0 given by zk = xk if k is odd, and xk if k is even. Again, when g is abest-response function, (zk) is the standard simultaneous best-response dynamic startingat sup X. When g is a correspondence,(zk) is an analogue of the “largest” best responsedynamic used in Milgrom and Roberts (1990) and in Echenique (2002).12

Recall that Nash equilibria (that is, fixed points of g) are included in the set of seriallyundominated strategies. The following result shows that the set of fixed points of g ◦ g arealso included in the set of serially undominated strategies. This observation is useful tocharacterize dominance solvability.

Corollary 1. Let Γ be a game with strategic substitutes, x and x be the extremal serially

11It can be concluded that g ◦ g(y0) � g ◦ g(z0).12In the section on the correspondence principle below, we generalize these dynamics to start from arbitrary

points in the strategy space.

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undominated strategies, and FP (g ◦ g) be fixed points of g ◦ g. In this case,

FP (g ◦ g) ⊂∞⋂

k=0

Uk(X) ⊂ [x, x].

Moreover, x is the smallest fixed point of g ◦ g and x is the largest fixed point of g ◦ g.

Proof. Recall that g ◦ g : X � X is defined as g ◦ g(x) =⋃

y∈g(x)

g(y). For the first inclusion,

consider an arbitrary x ∈ g ◦ g(x). Let y ∈ g(x) be such that x ∈ g(y). Then, by induction,

x and y are in∞⋂

k=0

Uk(X), as follows. Trivially, x and y are in U0(X). Suppose x and y

are in Uk(X). Then x ∈ Uk+1(X), because y ∈ Uk(X) and x is a best response to y, andy ∈ Uk+1(X), because x ∈ Uk(X) and y is a best response to x. The second inclusion isfrom the theorem above. For the second statement, it is easy to check that x ∈ g(x) andx ∈ g(x) imply that both x and x are fixed points of g ◦ g.

Corollary 2. Let Γ be a game with strategic substitutes, x and x be the extremal seriallyundominated strategies, and g be the joint best-response correspondence. The following areequivalent.

1. Γ is dominance solvable

2. x = x

3. g ◦ g has a unique fixed point

Proof. Notice that

E ⊂ FP (g ◦ g) ⊂∞⋂

k=0

Uk(X) ⊂ [x, x],

where the first inclusion is obvious, and the remaining statement is from the corollary above.With this chain of inclusions, (3) implies (2) implies (1) follows trivially, because both x andx are fixed points of g ◦ g. For (1) implies (3), notice that if the game is dominance solvable,then the serially undominated strategies x and x are the same, hence the chain of inclusionsabove shows that g ◦ g has a unique fixed point.

This corollary generalizes a result due to Zimper (2007), and presents an alternative proof.The model here is more general, using ordinal versions of the cardinal assumptions used inMilgrom and Roberts (1990) and in Zimper (2007).13 Moreover, Zimper (2007) assumes thatbest-response functions are order-continuous; we make assumptions on the primitive payofffunctions. Furthermore, Zimper (2007) presents results for best response functions (althoughhe mentions that the results go through for correspondences); our results are presented for

13We use the ordinal decreasing single-crossing property versus the cardinal property of decreasing differ-ences, and the ordinal quasi-supermodularity versus the cardinal supermodularity. Use of the ordinal versionsis facilitated by a recent characterization of nonincreasing best responses in Roy and Sabarwal (2009), basedon an extension of the monotonicity theorem in Milgrom and Shannon (1994).

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correspondences. The result here is also based on a different proof. Zimper (2007) does notprove the lemma above, and takes a different route.14 Our proof follows the steps of Milgromand Roberts (1990) more closely by first proving the lemma above, which yields a valuableinsight into the design of the appropriate best response dynamics. Indeed, this insight shallbe very useful in deriving additional results in this paper.

Fixed points of g◦g have a behavioral interpretation in terms of strategy profiles that arerationalizable with short cycles of justification. That is, suppose x ∈ g ◦ g(x). In this case,let y ∈ g(x) such that x ∈ g(y). Then the profile of strategies x is rationalizable with thefollowing cycle of conjectures. For each i, player i plays xi because she believes her opponentsshall play y−i, because each of her opponents j further believes that his opponents shall playx−j . Say that a profile of strategies x is simply rationalizable, if there is a strategy profiley such that for every player i, xi can be justified by such a short cycle of conjectures. Thereasoning above shows that if a profile of strategies x is a fixed point of g ◦ g, then it issimply rationalizable. In the other direction, it is easy to check that if for each i, playeri plays xi because that is a best response to her belief that her opponents shall play y−i,because each of her opponents j best responds with yj based on his further belief that hisopponents shall play x−j , then the profile of strategies x is a (joint) best response to theprofile of strategies y, and y is a best response to x, whence x is a fixed point of g ◦ g. Thus,an economic foundation for fixed points of g ◦ g can be provided in the behavioral terms ofsimply rationalizable strategies.

Simply rationalizable strategies do not rely on high orders of deduction. Nash equilibriaare simply rationalizable (x and y are the same), but in general, simply rationalizable strate-gies may include more strategies than Nash strategies (confer the 3-firm Cournot oligopolyexample above with details provided in example 1 below). Moreover, simply rationalizablestrategies may form only a strict subset of all rationalizable strategies, because rationaliz-able strategies include conjectural cycles of all orders. Indeed, in the original example inBernheim (1984), there are rationalizable strategies that are not simply rationalizable.

With these ideas in place, the following corollary presents another characterizaton ofdominance solvability in GSS.

Corollary 3. Let Γ be a game with strategic substitutes.Γ is dominance solvable iff there is a unique profile of simply rationalizable strategies.

Essentially, this result shows that in GSS that have simply rationalizable strategies, ifmultiplicity of simply rationalizable strategies is ruled out, then multiplicity of all rational-izable strategies is ruled out, and serially undominated strategies correspond to the uniqueNash equilibrium in the game.

As in GSC, the bounds provided by extremal serially undominated strategies in GSSmay not always be tight, but in some cases, they provide sharp predictions. The followingexamples provide some insight.

14He shows that at each iterative step, the largest and smallest rationalizable strategies are also the largestand smallest undominated strategies, respectively. He then works directly with the best-response aspect ofthese extremal rationalizable strategies.

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Example 1. Consider a 3-firm Cournot oligopoly with linear inverse demand, p = a −b(x1 + x2 + x3), constant marginal cost, c, and with production capacity constrained to[0, xmax] for each firm. In this case, the joint best response function is given by g(x1, x2, x3) =

(a−c−b(x2+x3)2b

, a−c−b(x1+x3)2b

, a−c−b(x1+x2)2b

), and the unique Nash equilibrium is given by (x1, x2, x3) =(a−c

4b, a−c

4b, a−c

4b). Moreover, g ◦ g(x1, x2, x3) = (2x1+x2+x3

4, x1+2x2+x3

4, x1+x2+2x3

4), and it is easy

to see that every point on the diagonal of [0, xmax]3 is a fixed point of g ◦ g. In particular,(0, 0, 0) is the smallest serially undominated strategy and (xmax, xmax, xmax) is the largest.Thus, this game is not dominance solvable, and serially undominated strategies provide nohelp in narrowing the range of predicted outcomes.

Notice that in this example, the (joint) best response function is linear. As linear bestresponses arise in other contexts, too, the next example provides a general result for thiscase.

Example 2 (Linear best responses). Consider a GSS with finitely many players, N ,where each X i = [0, xmax

i ] ⊂ R, and X = [0, xmax] ⊂ RN .15 Suppose the joint best response

function is linear; that is g : X → X such that g(x) = a +Bx, where each component of theN×N matrix B is nonpositive. Thus, g is nonincreasing. Moreover, for range of g to be in X,suppose −Bxmax ≤ a ≤ xmax. Notice that the game has a unique equilibrium, if, and only if,the matrix (I−B) is invertible. In this case, the unique equilibrium is x∗ = (I−B)−1a. Theiterated best response is given by g ◦ g(x) = a + Ba + B2x, and therefore, g ◦ g has a uniquefixed point, if, and only if, the matrix (I −B2) is invertible. As determinant calcuations areeasy to make, this result is useful in applications. Consider the following simple economicapplications.

Example 2-1. The 3-firm Cournot duopoly in example 1 is a special case of example 2,

with B =

0 −12

−12

−12

0 −12

−12

−12

0

. It is easy to show that I − B is invertible, but I − B2 is not.

Thus, the game has a unique Nash equilibrium, but is not dominance solvable.

Example 2-2. Consider a 3-firm Cournot oligopoly with linear inverse demand, as in ex-ample 1, but with quadratic costs, cx2

i , and with production capacity constrained to [0, xmax]for each firm. In this case, the joint best-response function is given by g(x1, x2, x3) =

(a−b(x2+x3)2b+2c

, a−b(x1+x3)2b+2c

, a−b(x1+x2)2b+2c

). Therefore, B =

0 − b2b+2c

− b2b+2c

− b2b+2c

0 − b2b+2c

− b2b+2c

− b2b+2c

0

. It is easy

to find values of a, b, and c that makes this a special case of example 2 such that both I −Band I −B2 are invertible.16 For example, let b = c = 1, and xmax

2≤ a

4≤ xmax. Consequently,

such an oligopoly is dominance solvable, and serially undominated strategies (and severalother solution concepts) predict a unique outcome.

15Here, xmax is the vector with i-th component xmax

i.

16Of course, if I − B2 is invertible, then so is I − B.

9

Example 2-3. Another game that shares the same structure is the common-pool resourcegame.17 Consider a 3-player common-pool resource game. Each player has an endowmentw > 0. There are two investment options – a common resource (such as a fishery) that ex-hibits diminishing marginal return, and an outside option with diminishing marginal return.If player i invests an amount xi ≤ w of his endowment into the common resource, he receivesa proportional share of the total output xi

x1+x2+x3

(a(x1 + x2 + x3) − b(x1 + x2 + x3)2), and

he receives r(wi−xi)−s(w−xi)2 on the outside investment w−xi.

18 Thus, payoff to playeri is

f i(x1, x2, x3) = r(wi − xi)− s(w − xi)2 +

xi

x1 + x2 + x3

(

a(x1 + x2 + x3) − b(x1 + x2 + x3)2)

,

if x1 + x2 + x3 > 0, and rw − sw2, otherwise. Notice that best response of player i isgiven by gi(xj , xk) = a−r+2sw

2b+2s− b

2b+2s(xj + xk). For range of gi to lie in [0, w], we assume

a−r2b

≤ w ≤ a−r2(b−s)

.19 The matrix B is B =

0 − b2b+2s

− b2b+2s

− b2b+2s

0 − b2b+2s

− b2b+2s

− b2b+2s

0

. It is easy to choose

b > s such that I −B2 (and hence I −B) is invertible. For example, consider b = 1, s = 1/2.In this case, the game is dominance solvable, and a unique outcome is predicted.

Example 3 (Tournaments). A game with non-linear best responses is that of tourna-ments.20 Suppose a tournament has 3 players, where a reward r > 0 is shared by the playerswho succeed in the tournament. If one player succeeds, he gets r for sure, if two playerssucceed, each gets r with probability one-half, and if all players succeed, each gets r withprobability one-third. Each player chooses effort xi ∈ [0, 1] with probability of success xi.Expected reward per unit for player i is

πi(xi, xj , xk) = xi(1 − xj)(1 − xk) +1

2xixj(1 − xk) +

1

2xixk(1 − xj) +

1

3xixjxk.

The quadratic cost of effort xi is cx2i . The payoff to player i is expected reward minus

cost of effort. That is, f i(xi, xj , xk, t) = rπi(xi, xj, xk) − cx2i . It is easy to calculate that

best response of player i is given by gi(xj , xk) = r2c

(1 − 12(xj + xk) + 1

3xjxk). Suppose, for

convenience, r = 2c.

This game is dominance solvable, as follows. Let x = (x1, x2, x3) be the smallest fixedpoint of g ◦ g, and x = (x1, x2, x3) the largest. Then x ≤ x, and moreover, x = g(x) andx = g(x). Using these in the best-response function and simplifying terms yields x1x2 +x2x3 + x1x3 = x1x2 + x2x3 + x1x3.

Suppose x1 > x1. Suppose further, as case 1, x2 = x3 = 0. Then x2 = x3 = 0, and usingthese values in x = g(x) yields 1 = x1 > x1 = 2, a contradiction. Suppose, as case 2, that

17See, for example, Ostrom, Gardner, and Walker (1994). Additional analysis of this game as a GSS ispresented in Roy and Sabarwal (2009).

18Here, a, b, r, s > 0.19Hence, we need a > r and b > s.20This version is based on Dubey, Haimanko, and Zapechelnyuk (2006). Additional analysis of this game

as a GSS is presented in Roy and Sabarwal (2009).

10

x2 > 0 or x3 > 0. Then, x1x2 +x2x3 +x1x3 > x1x2 +x2x3 +x1x3, a contradiction. Therefore,x1 = x1. Similarly, x2 = x2 and x3 = x3, and consequently, g ◦ g has at most one fixed point.Moreover, it is easy to check that (3 −

√6, 3 −

√6, 3 −

√6) is a Nash equilibrium, hence a

fixed point of g and of g ◦ g. Thus, the game is dominance solvable.

For two-player games, dominance solvability is equivalent to uniqueness of Nash equilib-rium, similar to the result for GSC.

Corollary 4. Let Γ be a two-player game with strategic substitutes.Γ is dominance solvable iff Γ has a unique Nash equilibrium.

Proof. Sufficiency follows from the definition of dominance solvability. For necessity, sup-pose Γ has a unique Nash equilibrium. Let (x1, x2) be the profile of smallest serially un-dominated strategies, and (x1, x2) be the profile of largest serially undominated strategies.By the theorem above, (x1, x2) ∈ g(x1, x2) = g1(x2) × g2(x1). That is, x1 ∈ g1(x2) andx2 ∈ g2(x1). Similarly, x1 ∈ g1(x2) and x2 ∈ g2(x1). Consider the profile (x1, x2). Then(x1, x2) ∈ g(x1, x2), whence (x1, x2) is a Nash equilibrium. Similarly, (x1, x2) is a Nashequilibrium. By uniqueness, (x1, x2) = (x1, x2), and therefore, Γ is dominance solvable.

As is well-known, this result also follows from the theory of GSS, by reversing the orderon the strategy space of one of the players. The approach here keeps as fixed the naturalorder in the game. As shown above, this corollary does not extend to games with more thantwo players; the 3-firm Cournot oligopoly with linear demand and constant marginal costprovides a counter-example.

3 Adaptive Dynamics

Building on ideas in the previous section, this section investigates limiting behavior of learn-ing processes formalized by the broad class of adaptive dynamics. The “extremal” seriallyundominated strategy profiles shown above provide bounds on limits of learning behaviorin the class of adaptive dynamics, analogous to results for GSC. Additional investigationuncovers a new characterization of dominance solvability in terms of global stability. Theresults highlight differences between GSS and GSC, and provide connections to the literatureon global stability and uniqueness of equilibrium.

For a game Γ, consider a process (x(t))t∈T̂ , where t ∈ T̂ is a time parameter that maybe discrete or continuous, and for each t, x(t) ∈ X. For a process (x(t)), denote the set ofpast play from T to t by P (T, t) = {x(s)|T ≤ s < t}. As in Milgrom and Roberts (1990), aprocess (x(t)) is an adaptive dynamic in Γ if for every T , there is T ′ such that for everyt ≥ T ′, x(t) ∈ U [inf P (T, t), sup P (T, t)]. We have the following results.

Theorem 2. Let Γ be a game with strategic substitutes, and (x(t))t∈T̂ be an adaptive dynamicin Γ. For every k, there is Tk such that for all t ≥ Tk, x(t) ∈ [xk, xk], where x0 = inf X,x0 = sup X, and for k ≥ 1, xk = g(xk−1) and xk = g(xk−1).

Proof. The statement holds trivially for k = 0. Suppose there is Tk−1 such that forall t ≥ Tk−1, x(t) ∈ [xk−1, xk−1]. Then for all t ≥ Tk−1, [inf P (Tk−1, t), sup P (Tk−1, t)] ⊂

11

[xk−1, xk−1]. Now, by definition of an adaptive dynamic, let Tk be such that for all t ≥ Tk,x(t) ∈ U [inf P (Tk−1, t), sup P (Tk−1, t)], and consequently, for all t ≥ Tk,

x(t) ∈ U [inf P (Tk−1, t), sup P (Tk−1, t)] ⊂ U [xk−1, xk−1] = [g(xk−1), g(xk−1)] = [xk, xk],

where the inclusion follows from the monotonicity of U , and the first equality follows fromthe previous theorem.

Corollary 5. Let Γ be a game with strategic substitutes, (x(t))t∈T̂ be an adaptive dynamicin Γ, and x and x be the extremal serially undominated strategies. In this case,

x � lim inf x(t) � lim sup x(t) � x.

Moreover, if each player has a finite strategy space, then there is T ∗ such that for everyt ≥ T ∗, x � x(t) � x.

This result shows a similarity between GSS and GSC. In both cases, extremal seriallyundominated strategies provide bounds on limiting behavior of learning processes in thebroad class of adaptive dynamics. The insights that lead to the results in the previoussection yield the benefit that the proofs here follow easily from similar steps in Milgrom andRoberts (1990). The following corollary presents some differences.

Corollary 6. Let Γ be a game with strategic substitutes, and x and x be the extremal seriallyundominated strategies. The following are equivalent.

1. Simultaneous best response dynamic starting at inf X converges to a Nash equilibrium

2. Simultaneous best response dynamic starting at sup X converges to a Nash equilibrium

3. Every adaptive dynamic converges to a Nash equilibrium

4. x = x

5. Γ is dominance solvable

Moreover, in each of the above cases, the game has a unique Nash equilibrium.

Proof. The implications (4) implies (3) implies (1), and (3) implies (2) are obvious. For(1) implies (4), let (yk) be the best response dynamic starting at inf X. Then (y2k) is asubsequence of the convergent sequence (xk) and (y2k+1) is a subsequence of the convergentsequence (xk), and therefore, if (yk) converges, then x = x. Similarly, (2) implies (4). Theequivalence of (4) and (5) is established earlier.

This corollary presents a new characterization of dominance solvability in GSS, andhighlights some differences between GSC and GSS. In GSC, convergence of best responsedynamics (either (1) or (2)) does not necessarily imply global stability in terms of adaptivedynamics, (3); in GSS, these are equivalent. In GSC, convergence of best response dynamicsdoes not necessarily imply dominance solvability; in GSS, these are equivalent. In GSC,

12

convergence of best response dynamics does not necessarily imply uniqueness of equilibrium;in GSS, it does.21

In particular, this corollary provides connections to the literature on global stability anduniqueness of equilibrium. It shows that convergence of best-response dynamics impliesuniqueness and global stability of equilibrium. Consequently, it provides a converse to thestandard result that dominance solvability implies Cournot stability (in terms of convergenceof best response dynamics), as in Moulin (1984). For GSS, Cournot stability implies dom-inance solvability as well. Additionally, this result provides another perspective on globalstability and uniqueness of equilibrium, as compared to the traditional eigen-value analysisseen, for example, in Al-Nowaihi and Levine (1985), and in Okuguchi and Yamazaki (2008).

Example 2, continued As shown above, for a GSS with linear best-responses of the formg(x) = a + Bx, invertibility of I − B2 is equivalent to dominance solvability, and therefore,invertibility of I − B2 is equivalent to global stability and uniqueness of equilibrium. Thiscondition is easy to check, relying only on a determinant calculation; confer examples 2-1,2-2, and 2-3 above.

More generally, as best response dynamics are amenable to computation, this resultpresents a potentially powerful tool to check uniqueness and global stability of Nash equi-librium in GSS. As an example, let’s revisit the game of tournaments.

Example 3, continued. Consider the 3-player game of tournaments, with best responseof player i given by gi(xj , xk) = 1− 1

2(xj +xk)+ 1

3xjxk. Notice that if we start best response

dynamics anywhere on the diagonal in [0, 1]3, then we remain on the diagonal. Therefore,the best response dynamic (xk)∞k=0 starting at (0, 0, 0) remains on the diagonal. Moreover,our earlier result shows that the subsequence (x2k) is nondecreasing and converges to x, thesubsequence (x2k+1) is nonincreasing and converges to x, and x and x are fixed points ofg ◦ g. Furthermore, as the diagonal is closed, x and x are on the diagonal. Let’s look atfixed points of g ◦ g on the diagonal. Using x1 = x2 = x3 = x, say, it is easy to calculatethat these fixed points are given by roots of the equation (x2 + 3)(x2 − 6x + 3) = 0. Theunique root in [0, 1] is x = 3−

√6. Thus the simultaneous best response dynamic converges

to (3 −√

6, 3 −√

6, 3 −√

6). Therefore, every adaptive dynamic converges to this uniqueNash equilibrium, and the game is globally stable in terms of adaptive dynamics.

4 Correspondence Principle

This section shows the dynamic stability of monotone selections of equilibria in parameterizedgames with strategic substitutes, analogous to results for parameterized GSC. As is well-known, there is a long list of results showing the existence of monotone selections of equilibiria(as the parameter increases) in parameterized GSC, but before Echenique (2002), therewas no convincing argument why such monotone selections are reasonable predictions in a

21Notice that these results are not due to the particular definition of the best response dynamics whenworking with correspondences, because the results still hold when best responses are functions, and in thiscase, best response dynamics are exactly the standard ones.

13

game. Applying the idea of Samuelson’s correspondence principle to GSC, Echenique (2002)provides a dynamic stability argument by showing (approximately) that if an equilibriumselection is monotone increasing, then it is dynamically stable under adaptive dynamics, andif it is nowhere increasing, then it is not stable under adaptive dynamics. We provide ananalogous result for GSS.

In this section, as in Echenique (2002), it will be technically useful to work with discretetime processes. Similar to Echenique (2002), for a game Γ, a process (x(k))∞k=0 is an adaptive

dynamic in Γ if there is γ > 0 such that for all k ≥ 0, x(k) ∈ U [inf P (k − γ, k), sup P (k −γ, k)], where as earlier, P (k − γ, k) is the history of past play from k − γ to k; that is,P (k − γ, k) = {x(k − γ), x(k − γ + 1), . . . , x(k − 1)}. By convention, when γ ≥ k, we setk − γ = 0.

Similar to Echenique (2002), in order to talk about dynamic stability, it is helpful to havea notion of best response dynamics starting from arbitrary points in the strategy space. LetΓ be a game with strategic substitutes, and y � z be elements of X. The (simultaneous)best response dynamic-1 starting at y is the sequence (yk)∞k=0 where y0 = y, and fork ≥ 1, yk = g(yk−1) if n is even, and yk = g(yk−1) if n is odd.22 Notice that when g is a best-

response function, (yk) is the standard simultaneous best-response dynamic starting at y.When g is a correspondence, best response dynamic-1 is an analogue to the “smallest” bestresponse dynamic used above, and in Echenique (2002). Similarly, the (simultaneous) best

response dynamic-2 starting at z is the sequence (zk)∞k=0 where z0 = z, and for k ≥ 1,zk = g(zk−1) if n is even, and zk = g(zk−1) if n is odd.23 Again, when g is a best-response

function, (zk) is the standard simultaneous best-response dynamic starting at z. When gis a correspondence, best response dynamic-2 is an analogue to the “largest” best responsedynamic used above, and in Echenique (2002). Moreover, notice that both best responsedynamics coincide when g is a function and y = z.

Lemma 2. Let Γ be a game with strategic substitutes, y � z be elements of X, and (yk)∞k=0

and (zk)∞k=0 be best response dynamics 1 and 2, respectively. For every n ≥ 0,

U [yn, zn] = [zn+1, yn+1] if n is even, andU [zn, yn] = [yn+1, zn+1] if n is odd.

Proof. Notice that by a previous lemma, U [y0, z0] = [g(z0), g(y0)] = [z1, y1]. Suppose n is

even and U [yn, zn] = [zn+1, yn+1]. Then U [zn+1, yn+1] = [g(yn+1), g(zn+1)] = [yn+2, zn+2] asdesired. Similarly, if n is odd.

Theorem 3. Let Γ be a game with strategic substitutes, y � z be elements of X, and (yk)∞k=0

and (zk)∞k=0 be best response dynamics 1 and 2, respectively.For every x0 ∈ [y, z], and for every adaptive dynamic (x(k)) starting at x0, the following istrue.

For every N, there is KN , such that for all k ≥ KN , x(k) ∈{

[yN , zN ] if N is even, and[

zN , yN]

if N is odd.

22When y = inf X , this is the best response dynamic starting at inf X , as defined above.23When z = supX , this is the best response dynamic starting at supX , as defined above.

14

Proof. Consider N = 0. Let K0 = 0. Notice that x(0) = x0 ∈ [y0, z0], by assumption.Suppose for 0 ≤ k ≤ k̂ − 1, x(k) ∈ [y0, z0]. Then P (0, k̂) ⊂ [y0, z0], whence

x(k̂) ∈ U [inf P (k̂ − γ, k̂), sup P (k̂ − γ, k̂)]

⊂ U [inf P (0, k̂), sup P (0, k̂)]

⊂ U [y0, z0] ⊂ [y0, z0],

where membership follows from definition of an adaptive dynamic, the first inclusion followsfrom P (k̂ − γ, k̂) ⊂ P (0, k̂) and monotonicity of U , the second inclusion follows from theinductive hypothesis and monotonicity of U , and the last inclusion follows trivially. Thus,for all k ≥ 0, x(k) ∈ [y0, z0].

Suppose the statement is true for N −1. Let KN−1 be given by the inductive hypothesis.Let KN = KN−1 +γ, where γ is from the definition of adaptive dynamic. Suppose N is even.Fix k̂ ≥ KN = KN−1 + γ. Then

x(k̂) ∈ U [inf P (k̂ − γ, k̂), sup P (k̂ − γ, k̂)]

⊂ U [inf P (KN−1, k̂), sup P (KN−1, k̂)]

⊂ U [zN−1, yN−1] = [yN , zN ],

where membership follows from definition of an adaptive dynamic, the first inclusion followsfrom P (k̂ − γ, k̂) ⊂ P (KN−1, k̂) and monotonicity of U , the second inclusion follows fromthe inductive hypothesis and monotonicity of U , and the equality follows from the lemmaabove. Thus, for all k ≥ KN , x(k) ∈ [yN , zN ]. The case where N is odd follows similarly.

This theorem shows that every adaptive dynamic that starts in the order interval [y, z] ⊂X is appropriately bounded by the best response dynamics 1 (starting at y) and 2 (starting atz). Notice that monotone subsequences of best response dynamics 1 and 2 is not necessarilyimplied in the above proof, but it can be guaranteed if an initial monotonicity condition issatisfied, as shown in the following lemma.

Lemma 3. Let Γ be a game with strategic substitutes, y � z be elements of X, and (yk)∞k=0

and (zk)∞k=0 be best response dynamics 1 and 2, respectively.

(1) If y0 � y2, then (y2k) is nondecreasing, and there is y such that y2k → y, and(y2k−1) is nonincreasing, and there is y such that y2k−1 → y.Moreover, y, y ∈ FP (g ◦ g), and hence are serially undominated.

(2) If z2 � z0, then (z2k) is nonincreasing, and there is z such that z2k → z, and(z2k−1) is nondecreasing, and there is z such that z2k−1 → z.Moreover, z, z ∈ FP (g ◦ g), and hence are serially undominated.

Proof. Consider statement (1). Notice that the sequence (y2k) is nondecreasing, as follows.By assumption, y0 � y2. Suppose, y2k−2 � y2k. Then y2k = g ◦g(y2k−2) � g ◦g(y2k) = y2k+2,

because g ◦ g is nondecreasing. Therefore, by completeness, there is y such that y2k → y.

Similarly, the sequence (y2k−1) is nondecreasing, because y0 � y2 ⇒ y3 = g(y2) � g(y0) = y1.Suppose y2k+1 � y2k−1. Then y2k+3 = g ◦ g(y2k+1) � g ◦ g(y2k−1) = y2k+1. By completeness,

there is y such that y2k−1 → y.

15

The statement y, y ∈ FP (g ◦ g) follows from the observation that y ∈ g(y), and y ∈ g(y).Suppose y 6∈ g(y). Then there is i, and xi such that f i(xi, y−i) − f i(y

i, y

−i) > 0. Butthen, by upper semi-continuity in the i variable, and continuity in the −i variables, for allk sufficiently large, f i(xi, y

2k−1−i ) − f i(y2k

i , y2k−1−i ) > 0, contradicting the optimality of y2k

i .Similarly, y ∈ g(y). Consequently, y, y ∈ FP (g ◦ g). Statement (2) is proved similarly.

This lemma shows that if an initial monotonicity condition is satisfied, then particularsubsequences of best response dynamics 1 and 2 are monotone convergent and converge tosimply rationalizable strategies. (Recall that in the previous section, when y = inf X andz = sup X, monotonicity of these subsequences holds automatically, because in that case,y0 = y = inf X � y2 holds trivially, and similarly, z2 � z0 holds trivially.) Combining thesetwo results gives the following bounds on limiting behavior of every adaptive dynamic thatstarts in the order interval [y, z] ⊂ X.

Corollary 7. Let Γ be a game with strategic substitutes, y � z be elements of X, and (yk)∞k=0

and (zk)∞k=0 be best response dynamics 1 and 2, respectively. For every x0 ∈ [y, z], and forevery adaptive dynamic (x(k)) starting at x0, the following is true.(1) If y0 � y2, then there exist serially undominated y, y such that

x � y � lim inf x(k) � lim sup x(k) � y � x,

where x and x are the extremal serially undominated strategies in Γ.(2) If z2 � z0, then there exist serially undominated z, z such that

x � z � lim inf x(k) � lim sup x(k) � z � x.

Proof. Consider an arbitrary x0 ∈ [y, z], and an arbitrary adaptive dynamic (x(k)) startingat x0. To see that (1) holds, consider an arbitrary convergent subsequence (x(kl)) of (x(k)).By the previous theorem, for N = 0, there is K0 such that for all kl ≥ K0, y0 � x(kl),whence y0 � liml x(kl). For N = 2, there is K2 such that for all kl ≥ K2, y2 � x(kl), whencey2 � liml x(kl). And by induction, for 2N , there is K2N such that for all kl ≥ K2N , y2N �x(kl), whence y2N � liml x(kl). Consequently, using the previous lemma, y � liml x(kl), andy is serially undominated. As (x(kl)) is an arbitrary convergent subsequence, it follows thaty � lim inf x(k). Similarly, lim sup x(k) � y.

Moreover, x0 = inf X � y implies that for every k ≥ 1, x2k = g◦g(x2k−2) � g◦g(y2k−2) =

y2k, whence x � y. Similarly, y � x. Statement (2) follows similarly.

This corollary generalizes corollary 5, and is useful in proving dynamic stability of mono-tone equilibrium selections, as shown next.

Parameterized Games With Strategic Substitutes

As earlier, consider a set of players I, and for each player i, a partially ordered strategyspace (X i,�i). The overall strategy space is the product of X i, denoted X, and endowedwith the product order.24 Moreover, consider a partially ordered set of parameters, T .25

24The topology on X i is the standard order interval topology, and the topology on X is the producttopology.

25For convenience, the partial order on T is denoted by the same symbol, �, and T is assumed to have

16

Each player i has a payoff function, f i : X × T → R, denoted f i(xi, x−i, t). The collectionΓ = (I, T, (X i,�i, f i)i∈I) is a parameterized game with strategic substitutes, if forevery player i,

• (X i,�i) is a complete lattice,

• For every (x−i, t), f i is order upper semi-continous in xi, and for every xi, f i is ordercontinuous in (x−i, t),

• For every (x−i, t), fi is quasi-supermodular in xi,

• For every x−i, f i satisfies single-crossing property in (xi; t), and

• For every t, f i satisfies decreasing single-crossing property in (xi; x−i).

As usual, single-crossing property in (xi; t) implies that each player’s best response,gi(x−i, t) is nondecreasing in the parameter, a standard assumption in GSC and in GSS.As earlier, each player’s best response is nonincreasing in other player strategies. Thus, thejoint best response, g(x, t) is nondecreasing in t and nonincreasing in x.

As usual, for each t ∈ T , a parameterized GSS, Γ, naturally defines a GSS, Γ(t), withthe same strategy spaces as Γ and with appropriate sections of the payoff functions. Thedefinition of an adaptive dynamic in Γ(t) remains the same as above. Let E(t) denote theset of (pure strategy) Nash equilibria in Γ(t). An equilibrium selection is a functione : T → X such that for every t, e(t) ∈ E(t).

As is well-known, parameterized GSS do not necessarily exhibit monotone comparativestatics, and therefore, nondecreasing equilibrium selections do not necessarily exist in suchgames. Roy and Sabarwal (2009) provide intuitive conditions that guarantee monotonecomparative statics in parameterized GSS. Similar conditions turn out to be useful in showingdynamic stability of monotone equilibrium selections, as follows. Let Γ be a parameterizedGSS. An equilibrium selection e : T → X satisfies condition 1 on

[

t, t]

, if for every t0, t̂[

t, t]

such that t0 � t̂, e(t0) � g(g(e(t0), t̂), t̂). An equilibrium selection e : T → X satisfies

condition 2 on[

t, t]

, if for every t̂, t1 in[

t, t]

such that t̂ � t1, g(g(e(t1), t̂), t̂) � e(t1).

Notice that when best responses are unique, that is, when g is a function, conditions 1and 2 follow from a monotonicity condition in t, as follows. Suppose that

For each x in the range of e, g(g(x, t), t) is nondecreasing in t.

In this case, it is easy to check that conditions 1 and 2 are both satisfied. As discussedin detail in Roy and Sabarwal (2009), in games with strategic substitutes, conditions 1and 2 present a natural tradeoff between a direct parameter effect and an indirect strategicsubstitute effect, as follows. Consider a game with two players, i = 1, 2. Starting from anexisting equilibrium, e(t0) = (x∗

1, x∗

2) at t = t0, an increase in t to t̂ has two effects on player1’s best response function, g1(·, ·). One effect is an increase in g1, because best-response is

the standard order interval topology.

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nondecreasing in t. (This is the direct parameter effect.) The other effect is a decrease in g1,because an increase in t increases g2(x∗

1, t), and x1 and x2 are strategic substitutes. (This isthe indirect strategic substitute effect arising from player 1’s response to player 2’s responseto an increase in t.) Similar statements are valid for player 2 as well. Taken together,e(t0) � g(g(e(t0), t̂), t̂) in condition 1 says that for each player, the indirect strategicsubstitute effect does not dominate the direct parameter effect when the parameter goes up.Condition 2 makes the analogous statement when the parameter goes down. The intuitionfor the general case is similar.26 Notice that in games with strategic complements, botheffects work in the same direction. Therefore, once the direct parameter effect is assumed tobe favorable, (as formalized, for example, by a strict single crossing property in (xi; t),) theindirect strategic complement effect serves to reinforce the direct effect, and the conditionsabove are satisfied automatically.

When each player’s strategies are real-valued and payoff functions are strictly quasi-concave (in own variable) and twice continuously differentiable, Roy and Sabarwal (2009)present transparent and easy-to-use conditions on payoff functions that guarantee the con-ditons above. They also present conditions for general payoff functions.

In this section, conditions 1 and 2 are useful as follows. Consider t0 � t̂. If condition 1 issatisfied, then the best response dynamic-1 given by y0 = e(t0), and for k ≥ 1, yk = g(yk−1, t̂)

if n is even, and yk = g(yk−1, t̂) if n is odd has the feature that y0 � y2. Similarly, considert̂ � t1. If condition 2 is satisfied then the best response dynamic-2 given by z0 = e(t1), andfor k ≥ 1, zk = g(zk−1, t̂) if n is odd, and zk = g(zk−1, t̂) if n is even has the feature thatz2 � z0.

To state and prove an analogue of Echenique’s correspondence principle for GSS, thefollowing concepts are useful. Let Γ be a parameterized GSS and t ∈ T . A point x̂ ∈ X isweakly stable at t, if there is a neighborhood V of x̂ such that for every x ∈ V , there isan adaptive dynamic (x(k)) in Γ(t) that starts at x and converges to x̂. A point x̂ ∈ X isstrongly stable at t, if there is a neighborhood V of x̂ such that for every x ∈ V , everyadaptive dynamic (x(k)) in Γ(t) that starts at x converges to x̂. An equilibrium selectione : T → X is nowhere weakly increasing on

[

t, t]

, if for every t0, t1 ∈[

t, t]

, t0 ≺ t1implies e(t0) 6� e(t1).

27 An equilibrium selection e : T → X, is strictly increasing if it isnondecreasing and for every t0 ≺ t̂ ≺ t1, [e(t0), e(t1)] is a neighborhood of e(t̂) in X.28 Fornotational convenience, we sometimes denote g(·, t) as gt(·).

For the following two results, we restrict the parameter space to satisfy a basic “density”property. A parameter space T is admissible if for every order interval [t, t] in T andfor every t̂ such that t ≺ t̂ ≺ t, every neighborhood of t̂ contains t0, t1 ∈ [t, t] such thatt0 ≺ t̂ ≺ t1. Notice that this property is fairly basic. In particular, T ⊂ R

n is convex, asassumed in Echenique (2002), is admissible. This property rules out parameter spaces where

26Roy and Sabarwal (2009) use versions of such conditions to guarantee monotone comparative statics inparameterized GSS.

27As described in Echenique (2002), this is stronger than the negation of weakly increasing.28Notice that as in Echenique (2002), when X is in some finite dimensional Euclidean space, this definition

is equivalent to t0 ≺ t1 ⇒ e(t0) � e(t1). Another relevant case is when X is a subset of a Banach latticethat has a positive cone with a nonempty interior.

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order intervals contain isolated points.

Theorem 4. (Correspondence Principle 1) Let Γ be a parameterized game with strategicsubstitutes, T be admissible, and e be a continuous equilibrium selection.If e is nowhere weakly increasing and satisfies condition 1 on

[

t, t]

, then for every t̂ such

that t ≺ t̂ ≺ t, e(t̂) is not weakly stable at t̂.

Proof. Fix t̂ such that t ≺ t̂ ≺ t. Consider e(t̂), and an arbitrary neighborhood V of e(t̂).By continuity of e, let t0 be such that t � t0 ≺ t̂ and e(t0) ∈ V . Then, by nowhere weaklyincreasing, e(t0) 6� e(t̂). Consider an arbitrary adaptive dynamic (x(k)) in Γ(t̂) starting atx(0) = e(t0). Let y0 = e(t0) and for k ≥ 1, yk = g

t̂(yk−1) if n is even, and yk = gt̂(y

k−1) if n is

odd. By condition 1, y0 � y2. Therefore, by the previous corollary, e(t0) � y � lim inf x(k),

whence x(k) 6→ e(t̂).

Theorem 5. (Correspondence Principle 2) Let Γ be a parameterized game with strategicsubstitutes, T be admissible, and e be a continuous equilibrium selection.If e is strictly increasing and satisfies conditions 1 and 2 on

[

t, t]

, then for every t̂ such that

t ≺ t̂ ≺ t and e(t̂) is an isolated fixed point of gt̂ ◦ gt̂, e(t̂) is strongly stable at t̂.

Proof. Fix t̂ such that t ≺ t̂ ≺ t and e(t̂) is an isolated fixed point of gt̂ ◦ gt̂. Let N be aneighborhood of e(t̂) such that N ∩ E(t̂) =

{

e(t̂)}

. As e is continuous, let t0, t1 ∈[

t, t]

be

such that t0 ≺ t̂ ≺ t1, and e(t0) and e(t1) are in N . As e is strictly increasing, [e(t0), e(t1)]is a neighborhood of e(t̂). Consequently, V = [e(t0), e(t1)]∩N is a neighborhood of e(t̂) ande(t̂) is the only fixed point of gt̂ ◦ gt̂ in V .

Fix x0 ∈ V arbitrarily, and let (x(k)) be an arbitrary adaptive dynamic in Γ(t̂) startingat x0. Let (yk) and (zk) be best response dynamics 1 and 2, respectively, with y0 = e(t0)and z0 = e(t1). Using conditions 1 and 2, it follows that

e(t0) = y0 � y � lim inf x(k) � lim sup x(k) � z � z0 = e(t1),

whence y and z are in [e(t0), e(t1)]. As y and z are fixed points of gt̂ ◦ gt̂, by local isolation,

y = z = e(t̂). Thus, x(k) → e(t̂), as desired.

Echenique (2002) shows these results for parameterized GSC without using conditions 1and 2, but implicitly assuming a strict single-crossing property (correspondences are assumedto be strongly increasing in t). The results here show the importance of focusing on anappropriate tradeoff between the direct parameter effect and the indirect strategic effectrather than on a strict single-crossing property. As described above, for parameterizedGSC, a strict single-crossing property leads to a strong direct parameter effect, and theindirect effect does not matter anymore, because it serves to reinforce the direct effect. Forparameterized GSS, it is precisely the reversed nature of the indirect strategic substituteeffect that requires conditions 1 and 2 to be useful in proving the analogous results. Indeed,in this case, we do not require a strict single-crossing property and correspondences are notassumed to be strongly increasing in t.

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