The nonatomic supermodular game

Games and Economic Behavior 82 (2013) 609–620

Contents lists available at ScienceDirect

Games and Economic Behavior www.elsevier.com/locate/geb

The nonatomic supermodular game Jian Yang a,∗ , Xiangtong Qi b a b

Department of Management Science and Information Systems, Rutgers University, United States Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Hong Kong

a r t i c l e

i n f o

Article history: Received 24 February 2012 Available online 14 October 2013 JEL classification: C72 Keywords: Nonatomic game Supermodular game Strong set order Stochastic dominance order Complete lattice

a b s t r a c t We introduce the nonatomic supermodular game, where no player’s action has any discernible impact on other players’ payoffs and yet strategic complementarities prevail among all players’ types and actions. For both semi-anonymous and anonymous games, we show that monotone equilibria form nonempty complete lattices and among these equilibria, the largest and smallest members vary in monotone fashions with respect to certain game-changing parameters. Results here complement existing nonatomic-game works, which focused more on pure equilibria of anonymous games where opponents’ types are not influential. They are also applicable to price competition involving diverse cost/quality parameters, as well as a slew of other situations. © 2013 Elsevier Inc. All rights reserved.

1. Introduction We are concerned with a game that is simultaneously nonatomic and supermodular. The former feature reflects that there is a continuum of actors at play, with no single player having any discernible impact on the game’s outcome. The latter feature refers to strategic complementarities that reside in interactions among a player’s type, his own action, and other players’ joint type–action distribution. For such a game, we show not only that pure equilibria exist, but also that some of them prescribe actions to types in a monotone fashion. Indeed, we know that all monotone equilibria form a complete lattice, so that any subset of monotone equilibria has both the largest and smallest elements. We rely on Zhou’s (1994) generalization of Tarski’s (1955) fixed point theorem for our derivation. This general result says that a correspondence from a complete lattice to itself would have a nonempty complete lattice as its set of fixed points, given that the correspondence always maps to a nonempty complete sublattice of the given complete lattice and the mapping is monotone in a set-theoretic sense due to Veinott (1989). To use this result, we substitute the complete lattice with our game’s set of monotone type-to-action mappings and substitute the correspondence with a construct based on the game’s best-response correspondence. Required properties on the correspondence are produced by analysis of the present game, which itself borrows from earlier results on strategic complementarities such as those of Milgrom and Roberts (1990) and Milgrom and Shannon (1994). We also establish a comparative statics result for a family of nonatomic supermodular games. Suppose games within the family are indexed. Then, we consider the family monotone when, within a game, each player’s action strategically complements the game’s index, and the type distribution of all players is increasing in the index in the stochastic dominance sense. In a monotone family, the largest and smallest equilibria of a game are themselves shown to be monotone in the game’s index. This result we achieve through a furtherance of Zhou’s analysis. Both our existence and comparative statics results on monotone equilibria can be applied to a number of situations, including that in which a continuum of infinitesimal firms

*

Corresponding author. E-mail addresses: [email protected] (J. Yang), [email protected] (X. Qi).

0899-8256/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.geb.2013.09.005

610

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

with diverse cost/quality makeups engage in price competition, as well as other settings involving work-condition choice, search, and arms races. We call a nonatomic game semi-anonymous when the payoff to a player depends on the joint type–action distribution of other players, and anonymous when the dependence reduces to one on the distribution of other players’ actions. In nonatomic-game literature, anonymous games have received far more attention than semi-anonymous ones. This can be partially attributed to the relative ease with which pure equilibria for anonymous games can be identified. For semi-anonymous games, very often less appealing mixed equilibria is the best one can hope for. The formal study of nonatomic games started with Schmeidler (1973). He showed the existence of mixed equilibria for a semi-anonymous game with a finite action space, and established the existence of pure equilibria for a corresponding anonymous one. Rashid (1983) showed that anonymous games with a bounded number of actions and huge number of players tended to have near-pure equilibria. Using an alternative representation of equilibria, Mas-Colell (1984) achieved mixed and pure equilibrium existence results for anonymous games. Khan et al. (1997) identified a certain limit to which Schmeidler’s result could be extended. Balder (2002) established pure and mixed equilibrium existence results that might be regarded as generalizations of Schmeidler’s corresponding results. With the aid of lattice-theoretic tools made relevant by our games’ strategic complementarities, however, we are able to directly tackle not only pure but also monotone equilibria of semi-anonymous games, and treat results on anonymous games as corollaries. We thus present evidence that strategic complementarities could lead to deeper understandings of competitive situations. Our formulation and approach are certainly inspired by the existing body of literature that coped with finite-player supermodular games and monotone comparative statics, a brief account of which now follows. Topkis (1978) elucidated the connection between strategic complementarities and solution comparative statics in optimization problems. Topkis (1979) also originated the study of supermodular games and the use of the fixed point theorem due to Tarski (1955) for identifying pure equilibria for such games. Milgrom and Roberts (1990) considered a supermodular game involving general complete lattice action spaces. They also provided a sufficient condition for the largest (smallest) pure equilibrium of their supermodular game to be monotone in an exogenous parameter. Vives (1990) arrived at the same pure equilibrium existence result under more flexible topologies and stronger continuity requirements and also treated Bayesian games involving strategic complementarities. In addition, one of his results (Theorem 4.1(ii)) served as a precursor to Zhou’s (1994) conclusion that the equilibrium set of a supermodular game is a complete lattice. A special case of our general version of the nonatomic supermodular game was considered by Vives (2005), who studied an anonymous game with a one-dimensional action space in which a player’s payoff depends on actions of other players through the average. Vives showed the existence of monotone equilibria and applied the results to situations involving price competition and search. In addition, a linear-normal model treated by Vives (1999) (Chapter 8) is very similar to the price competition model which we present as a major application of our theoretical developments. As is well illustrated in Radner and Rosenthal (1982), nonatomic and Bayesian games are often related. This makes van Zandt and Vives’ (2007) study of Bayesian supermodular games very relevant to our current work. These authors established the existence and monotonicity of largest and smallest equilibria and derived their monotone comparative statics results. Our order-centric approach is quite different from the earlier authors’ Tatônnement scheme, and in using it, we have avoided making extraneous topological assumptions on action spaces, or measurability and continuity assumptions on payoff functions. Our existence result concerns the set of monotone equilibria. One advantage of monotone strategies is that they seem to reflect an inherent fairness in competitive situations involving strategic complementarities, and hence are readily embraceable by diverse parties all at the same time. For instance, it is natural for competing firms to settle on equilibrium prices that are positively correlated with their costs. On the other hand, it is technically necessary that we start off with monotone strategy profiles, because these profiles form a complete lattice that naturally calls for lattice-theoretic tools. The rest of the paper is organized as follows. We present the main existence result in Section 2 and offer its proof in Section 3. We then spend Section 4 on the comparative statics result for a family of games. In Section 5, applications are made to various competitive situations. We conclude the paper in Section 6. 2. Equilibrium characterization We make extensive use of lattice-theoretic concepts like partially ordered sets and (complete) (sub)lattices. The reader may refer to Topkis (1998) and Milgrom and Roberts (1990) for a detailed description of these notions. It is known that a complete sublattice of a complete lattice is a complete lattice in its own right; also, the product of (complete) lattices over an arbitrary index set remains a (complete) lattice. We associate an arbitrary complete lattice A with its interval topology, in which sets of forms [inf A , a] and [a, sup A ] constitute a sub-basis of closed sets. We can define the Borel σ -field B ( A ) built on the interval topology. We denote by P ( A ) the space of all probability measures defined on the measurable space ( A , B ( A )). Given complete lattice B, let M( A , B ) ⊂ B A stand for the space of Borel measurable mappings from A to B. This way, a mapping g : A → B would belong to M( A , B ) if and only if g −1 ( B  ) ≡ {a ∈ A | g (a) ∈ B  } ∈ B ( A ) for any B  ∈ B ( B ). For our semi-anonymous nonatomic game Γ , player type space T is a complete lattice plus totally ordered set and player action space S is a complete lattice. The restriction on T stems from the measurability requirement on monotone mappings from T to S, a point which will be made clear by Lemma 1. For instance, T can be a compact interval of the real line .

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

611

For payoffs, we use the function f : S × P ( T × S ) × T → . For action s ∈ S, joint type–action distribution r ∈ P ( T × S ), and type θ ∈ T , the value f (s, r , θ) stands for the payoff to a type-θ player when he takes action s while facing an external environment describable by the joint type–action distribution r of other players. The game’s semi-anonymity is underscored by f ’s dependence on r rather than its projection to the action space S. The game is associated with a given type distribution p ∈ P ( T ). Any measurable type-to-action map x ∈ M( T , S ) constitutes an admissible action plan for the players. When every type-θ player adopts the action x(θ) as prescribed by x, all players will experience the same joint type–action distribution p ◦ (i T , x)−1 , where i T stands for the identity map from the type space T to itself and for every U  ∈ B ( T × S ),











p ◦ (i T , x)−1 U  = p (i T , x)−1 U 



    = p θ ∈ T  θ, x(θ) ∈ U  .

(1)

When a type-θ player takes action s and all other players adopt the action plan x, his payoff will be f (s, p ◦ (i T , x)−1 , θ). Clearly we can equate the game Γ with the tuple ( T , S , f , p ). We call x ∈ M( T , S ) a Nash equilibrium for Γ when









f x(θ), p ◦ (i T , x)−1 , θ  f s, p ◦ (i T , x)−1 , θ ,

∀θ ∈ T , s ∈ S .

(2)

Basically, an equilibrium is a player’s action response to his own type, such that no player will be tempted away from his corresponding action when all other players have adopted the said response. When p is a nonatomic distribution, rendering p ({θ}) = 0 for any θ ∈ T , the above is a clear statement that every player, by himself, is insignificant. Even when p possesses atoms, we may continue to regard each individual player as inconsequential when we attach a continuum of players to each type θ with a nonzero p ({θ}). We base our definition of supermodular games on the concepts of order upper semi-continuity, supermodularity, increasing differences, and stochastic dominance. A brief introduction to these well known notions is given in Appendix A. We deem Γ ≡ ( T , S , f , p ) supermodular when the following are true: (S1) f is order upper semi-continuous in s ∈ S for fixed (r , θ) ∈ P ( T × S ) × T ; (S2) f is supermodular in s ∈ S for fixed (r , θ) ∈ P ( T × S ) × T ; (S3) f has increasing differences in s ∈ S and (r , θ) ∈ P ( T × S ) × T when the partial order based on stochastic dominance “st ” is used for P ( T × S ). Among the three conditions, neither (S1) nor (S2) appears to be demanding. The former ensures the enclosure of each player’s best response within his own action space, and the latter expresses the complementarity between different components of a player’s own action. (S3), on the other hand, connects a player’s action with his own type as well as the environment he faces that is the making of his opponents. It explains the increased effectiveness of his efforts when things beyond his control are more enabling. For our semi-anonymous nonatomic supermodular game Γ , we concentrate on its monotone action plans. We can show not only that it possesses monotone Nash equilibria, but that these equilibria form a complete lattice. The latter would enable us to pick the highest and lowest equilibria, respectively. Theorem 1. For a semi-anonymous nonatomic supermodular game Γ , monotone Nash equilibria form a nonempty complete lattice, say X . Thus, the largest and smallest members of any subset X  ⊂ X are well defined; in particular, we can choose from X the largest and smallest monotone equilibria, say x∗ ≡ sup X and x∗ ≡ inf X , respectively. Theorem 1 is similar in spirit to Theorem 16 of van Zandt and Vives (2007) concerning Bayesian supermodular games. Still, there are appreciable differences. While we identified the non-emptiness and complete-lattice nature of the monotoneequilibrium space, van Zandt and Vives pointed out that the largest and smallest equilibria are not only in existence but also monotone. Approaches taken by the two works are complementary, with ours being order-centric and van Zandt and Vives’s relying on a Tatônnement scheme. 3. Proof of the main result Monotone equilibria for the nonatomic supermodular game Γ makes a good deal of sense. Due to the increasing differences in action s and type θ , a player’s best response would rise with his own type under any external environment. It is thus reasonable to consider monotone action plans. Meanwhile, when all other players adopt higher actions, the joint type–action distribution faced by the current player rises. The increasing differences in the player’s own action s and the external type–action distribution r would then propel the player’s own action to rise as well. Now since each player’s monotone response rises with other players’ monotone action plans, we can use Zhou’s (1994) set-valued fixed point theorem to deduce the existence and group structure of monotone equilibria. The actual proof of Theorem 1 relies on the following few lemmas, whose proofs we supply in Appendix B. Here, we use I ( A , B ) ⊂ B A to denote monotone mappings from one partially ordered set A to another partially ordered set B.

612

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

Lemma 1. Suppose A is a complete lattice plus totally ordered set and B is a complete lattice. Then, all monotone mappings from A to B are Borel measurable; that is, I ( A , B ) ⊂ M( A , B ). Lemma 2. Given partially ordered set A and complete lattice B, the set I ( A , B ) is a complete sublattice of the complete lattice B A , and hence is a complete lattice in its own right. Lemma 3. Let g 1 and g 2 be two Borel measurable functions from complete lattice A to complete lattice B. Suppose g 1  g 2 , i.e., g 1 (a)  g 2 (a) for every a ∈ A. Then, q ◦ ( g 1 )−1 st q ◦ ( g 2 )−1 for any probability measure q ∈ P ( A ). Under our setup where the type space T is a complete lattice plus totally ordered set and the action space S is a complete lattice, we can use Lemma 1 to deduce the relationship I ( T , S ) ⊂ M( T , S ), that every monotone action plan is admissible as a potential equilibrium. The restriction placed on T stems from the total ordered-ness requirement of Lemma 1, the essence of which one may learn from the counter example presented in Appendix B. In the remainder of the proof, we use the strong set order “s ” proposed by Veinott (1989) as the partial order for subsets of a given lattice A. For A 1 , A 2 ⊂ A, one has A 1 s A 2 when a1 ∧ a2 ∈ A 1 and a1 ∨ a2 ∈ A 2 for any a1 ∈ A 1 and a2 ∈ A 2 . The characterization of monotone Nash equilibria also has much to do with the following fixed point theorem of Zhou (1994), which generalizes that of Tarski (1955). Fact 1. Let A be a complete lattice and G : A ⇒ A a correspondence from A to itself. Suppose G (a) is a nonempty complete sublattice of A for every a ∈ A and G is monotone in the sense that G (a1 ) s G (a2 ) when a1  a2 . Then, G’s set of fixed points, E = {a ∈ A | a ∈ G (a)}, is a nonempty complete lattice. It should be cautioned that E is not necessarily a sublattice of A. To characterize Γ ’s set of equilibria, we use Fact 1 while substituting the complete lattice A involved in the fact with the game’s set of monotone action plans I ( T , S ). That the latter set is indeed a complete lattice has been confirmed by Lemma 2. Now we move on to identify a correspondence from I ( T , S ) to itself that would serve as G in Fact 1. To this end, we rely on two more facts. First, combining Theorems 1 and 2 of Milgrom and Roberts (1990), we can arrive at the following. Fact 2. Suppose A is a complete lattice and g : A →  is order upper semi-continuous and supermodular. Then, argmaxa∈ A g (a) is a nonempty complete sublattice of A. Second, from Milgrom and Shannon (1994), we know the following. Fact 3. Let g : A × B → , where A is a lattice and B is a partially ordered set. Suppose g (a, b) is supermodular in a ∈ A and has increasing differences in a ∈ A and b ∈ B. Then for every A  ⊂ A, argmaxa∈ A  g (a, b) is monotone in ( A  , b), in the sense that A 1 s A 2 and b1  b2 would lead to argmaxa∈ A 1 g (a, b1 ) s argmaxa∈ A 2 g (a, b2 ). Given joint type–action distribution r ∈ P ( T × S ) and player type θ ∈ T , let us define player θ ’s best-response correspondence to r as follows:

M (r , θ) = argmaxs∈ S f (s, r , θ).

(3)

Due to the fact that S is a complete lattice, as well as (S1) and (S2), we know from Fact 2 that M (r , θ) is a nonempty complete sublattice of S. From (S2), (S3), and Fact 3, we know that M (r , θ) is monotone in (r , θ). Consequently, m∗ (r , θ) = sup M (r , θ), the largest best response of player θ to the joint type–action distribution r, is well defined as a member of M (r , θ) and monotone in (r , θ). Now consider correspondence F : I ( T , S ) ⇒ I ( T , S ), defined through

    F (x) = y ∈ I ( T , S ) | y (θ) ∈ M p ◦ (i T , x)−1 , θ for every θ ∈ T ,

(4)

for every monotone type-to-action map x ∈ I ( T , S ). In essence, each F (x) contains monotone best responses to the monotone action plan x. The correspondence F is to play the role taken by G in Fact 1. First, we show F (x) = ∅ for any x ∈ I ( T , S ). For a given x ∈ I ( T , S ), let f ∗ (x) ≡ ( f ∗ (x, θ) | θ ∈ T ) be such that





f ∗ (x, θ) = m∗ p ◦ (i T , x)−1 , θ , But m∗ ’s monotonicity in

∀θ ∈ T .

(5)

guarantees that f ∗ (x) ∈ I ( T , S ). Thus having at least f ∗ (x) as its member,

θ F (x) is nonempty. Next, we show that F (x) is a complete sublattice of I ( T , S ) for any given x ∈ I ( T , S ). For a fixed x ∈ I ( T , S ), let U be an arbitrary nonempty subset of F (x). We show that sup U ∈ F (x). At every θ ∈ T , we let

  U (θ) = y  ∈ S | y  = y (θ) for some y ∈ U .

(6)

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

613

From (4), we know U (θ) is a subset of M ( p ◦ (i T , x)−1 , θ). As the latter is a nonempty complete sublattice of S, we know that





f U∗ (x, θ) ≡ sup U (θ) ∈ M p ◦ (i T , x)−1 , θ .

(7)

By (6) and the way the partial order on I ( T , S ) is defined through that on S in the component-wise fashion, (7) leads to





f U∗ (x) ≡ f U∗ (x, θ) | θ ∈ T = sup U .

(8)

According to (4), the only thing left to check is whether sup U ∈ I ( T , S ). But this is obvious as U ⊂ F (x) ⊂ I ( T , S ). We then show that the correspondence F is monotone in x. Let x1 , x2 ∈ I ( T , S ) with x1  x2 be given. By Lemma 3, we have p ◦ (i T , x1 )−1 st p ◦ (i T , x2 )−1 . Since M (r , θ) is monotone in r, we have





M p ◦ i T , x1

− 1

  − 1   , θ s M p ◦ i T , x2 , θ ,

∀θ ∈ T .

(9)

For y 1 ∈ F (x1 ) and y 2 ∈ F (x2 ), we have y 1 (θ) ∈ M ( p ◦ (i T , x1 )−1 , θ) and y 2 (θ) ∈ M ( p ◦ (i T , x2 )−1 , θ) for every θ ∈ T . By (9), it follows that





y 1 (θ) ∧ y 2 (θ) ∈ M p ◦ i T , x1

− 1

 ,θ ,

and





y 1 (θ) ∨ y 2 (θ) ∈ M p ◦ i T , x2

− 1

 ,θ ,

(10)

for every θ ∈ T . But y 1 ∧ y 2 is merely ( y 1 (θ) ∧ y 2 (θ) | θ ∈ T ), y 1 ∨ y 2 is merely ( y 1 (θ) ∨ y 2 (θ) | θ ∈ T ), and they are within I ( T , S ) because y 1 and y 2 are. Hence, we have from (4) that

 

y 1 ∧ y 2 ∈ F x1 ,

and

 

y 1 ∨ y 2 ∈ F x2 .

(11)

Now we know that F is a correspondence defined from the complete lattice I ( T , S ) to itself, that each F (x) is a nonempty complete sublattice of I ( T , S ), and that F is monotone in x. Therefore, we can use Fact 1 to reach the conclusion that the set X = {x ∈ I ( T , S ) | x ∈ F (x)} is a nonempty complete lattice. By the definitions (3) and (4), x ∈ X is paramount to each x(θ) serving as a best response for a type-θ player when all other players adhere to the monotone x as an action plan. Hence, X is the set of monotone equilibria for Γ . As X is a complete lattice, we can take from X the largest and smallest members of any subset X  of X . In particular, we can take x∗ ≡ sup X as Γ ’s largest monotone equilibrium and x∗ ≡ inf X as the game’s smallest monotone equilibrium. We have thus completed the proof of Theorem 1. On the other hand, we caution that X is not necessarily a sublattice of I ( T , S ). The above “sup” and “inf” are taken with all members of X rather than I ( T , S ). 4. Monotone comparative statics We can predict to a certain extent the movements of Γ ’s equilibria when the game varies in certain directions. Let Λ be a partially ordered set, and let (Γ (λ) | λ ∈ Λ) be a family of games. In this family, all games Γ (λ) = ( T , S , f (· | λ), p (λ)) are defined on the common type and action spaces T and S, as well as game-specific payoff functions f (· | λ) : S × P ( T × S ) × T →  and player type distributions p (λ) ∈ P ( T ). We deem the game family (Γ (λ) | λ ∈ Λ) monotone when the following are true: (M1) f (s, r , θ | λ) has increasing differences in s ∈ S and λ ∈ Λ for fixed (r , θ) ∈ P ( T × S ) × T ; (M2) p (λ) is monotone in λ in the sense that p (λ1 ) st p (λ2 ) when λ1  λ2 . In a game Γ (λ) with a higher-ranked index λ, (M1) indicates it would be easier for a player to improve his payoffs by exerting more effort, while (M2) indicates it would be more likely for a player to have a higher-ranked type. We know from Theorem 1 that, when each Γ (λ) is supermodular, its monotone equilibria form a nonempty complete lattice X (λ), from which we can choose the largest x∗ (λ) and the smallest x∗ (λ). The following further states that the two extreme monotone equilibria would move in a monotone fashion along with the index λ. Theorem 2. For a monotone family (Γ (λ) | λ ∈ Λ) of semi-anonymous nonatomic supermodular games, both its largest and smallest monotone equilibria x∗ (λ) and x∗ (λ) will rise monotonically with λ. We point out that Proposition 16 of van Zandt and Vives (2007) on Bayesian supermodular games conveys the same principle as Theorem 2. In addition, we note that a special case of the latter is one with Λ = P ( T ) and f (· | λ) = f for some f : S × P ( T × S ) × T → . For this, we can conclude from the theorem that the largest and smallest monotone equilibria of the game Γ are monotone in the game’s player type distribution p. The rise of the extreme monotone equilibria with the game index λ can be intuitively understood. Under any monotone action plan commonly adopted by all players, the joint type–action distribution also rises in concert with the type distribution. But the increasing differences in a player’s own action s and the distribution-index pair (r , λ) induces the rise of the

614

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

player’s own action. Thus the largest and smallest best-response functions will both rise. This fact, combined with the way in which fixed points of the extreme best-response maps are formed, gives rise to the monotone comparative statics result regarding the extreme monotone equilibria. To actually prove Theorem 2, we can resort to either a known result from Tarski (1955) or a comparative statics result more catered to Zhou’s setting. The known result says that if A is a complete lattice and g 1 , g 2 : A → A are monotone functions that further satisfy g 1 (a)  g 2 (a) for every a ∈ A, then the largest (smallest) fixed point of g 1 would be below the largest (smallest) fixed point of g 2 . The alternative result we use is as follows. Lemma 4. Let A be a complete lattice and B a partially ordered set. Assume, for every b ∈ B, that the correspondence G (· | b) : A ⇒ A satisfies all requirements stated in Fact 1, so that G (· | b)’s equilibria form a nonempty complete lattice E (b) in A, resulting in the presence of a largest member e ∗ (b) and smallest member e ∗ (b). Suppose further that G (a | b) is monotone in b at every a ∈ A, meaning that G (a | b1 ) s G (a | b2 ) when b1  b2 . Then, both e ∗ (b) and e ∗ (b) would rise monotonically with b. Another technical result we use is the following. Lemma 5. Let A be a complete lattice plus totally ordered set, B be a complete lattice, and q1 and q2 be two probability measures in P ( A ). Suppose q1 st q2 . Then, q1 ◦ g −1 st q2 ◦ g −1 for any g ∈ I ( A , B ). Proofs of both lemmas are provided in Appendix C. In order to use Lemma 4 in our proof, we view I ( T , S ) as A in the lemma, Λ as B, and the to-be-defined correspondence F (· | λ), whose λ-independent version has appeared in Section 3, as G (· | b). Our primary task is to prove the monotone trend of F (· | λ) in λ. Like (3), we let

M (r , θ | λ) = argmaxs∈ S f (s, r , θ | λ).

(12)

At a fixed λ ∈ Λ, we know that M (r , θ | λ) is a nonempty complete sublattice of S that is monotone in (r , θ). Like (4), we let

    F (x | λ) = y ∈ I ( T , S ) | y (θ) ∈ M p (λ) ◦ (i T , x)−1 , θ | λ for every θ ∈ T .

(13)

At a fixed λ ∈ Λ, we can show that F (x | λ) is a nonempty complete sublattice of I ( T , S ) that is monotone in x. By the same logic as used in the proof of Theorem 1, we can deduce that some x∗ (λ) and x∗ (λ) within I ( T , S ) would emerge as F (· | λ)’s extreme fixed points and thus game Γ (λ)’s extreme monotone equilibria. Let λ1 , λ2 ∈ Λ satisfying λ1  λ2 , y 1 ∈ M (r , θ | λ), and y 2 ∈ M (r , θ | λ) be given. From (M1) and (12), it follows that

















0  f y 1 ∨ y 2 , r , θ | λ2 − f y 2 , r , θ | λ2  f y 1 , r , θ | λ1 − f y 1 ∧ y 2 , r , θ | λ1  0.

(14)

The only possibility is for all inequalities to be equalities. Thus, we must have y ∧ y ∈ M (r , θ | λ ) and y ∨ y ∈ M (r , θ | λ2 ). Hence, the correspondence M (· | λ) from P ( T × S ) × T to S is monotone in λ. Meanwhile, Lemma 5 states that rankings in players’ type distributions would be carried over to their resultant type– action distributions when all players adopt the same monotone action plan. From (M2) and this lemma, we see that p (λ) ◦ (i T , x)−1 is monotone in λ. This, along with M (r , θ | λ)’s monotonicity in both λ and r, leads to M ( p (λ) ◦ (i T , x)−1 , θ | λ)’s monotonicity in λ at fixed x ∈ I ( T , S ) and θ ∈ T . But by (13), this immediately results in F (x | λ)’s monotonicity in λ at fixed x ∈ I ( T , S ). With the monotonicity of F (· | λ) in λ, we can use Lemma 4 to reach the monotonicity of x∗ (λ) and x∗ (λ) in λ. This concludes the proof of Theorem 2. We hope to ascertain whether the set of monotone equilibria X (λ) is monotone in λ in the strong set order sense. However, currently, this more general result cannot be attained. 1

2

1

1

2

5. Specialization and applications 5.1. Anonymous games The theory on anonymous games need not be separately developed. Each such game is a special case in which payoff depends only on the projection of the joint type–action distribution onto the action space. Results for this case can be treated as corollaries of Theorems 1 and 2. We leave details to Appendix D. Corollary 1. For an anonymous nonatomic supermodular game Γ  , monotone Nash equilibria form a nonempty complete lattice, say X . Thus, we can choose from X the largest and smallest monotone equilibria, say x∗ ≡ sup X and x∗ ≡ inf X , respectively. Corollary 2. For a monotone family (Γ  (λ) | λ ∈ Λ) of anonymous nonatomic supermodular games, both its largest and smallest monotone equilibria x∗ (λ) and x∗ (λ) will rise monotonically with λ.

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

615

5.2. Price competition with cost/quality considerations In this section we provide greater detail regarding the first application involving price competition, whose linear form is reminiscent of the linear-normal framework studied by Vives (1999) (Chapter 8). Suppose a, b, c, α , and β are strictly positive constants. A continuum of firms have a type distribution p, with each type θ ∈ [0, a/(2b)] signaling a firm’s unit cost as well as the quality of its product items. When a type-θ firm offers price s˜ ∈ [θ, θ + a/(2b)] and all his rivals adopt strategy x˜ = (˜x(θ  ) | θ  ∈ [0, a/(2b)]), the firm will receive payoff in the form of

 a

/(2b)        ˜f (˜s, x˜ , θ) = (˜s − θ) · a − b · (˜s − α θ) + c · x˜ θ − βθ · p dθ .

(15)

0

Here, the term (˜s − θ) represents the profit the current type-θ firm can earn for every demand unit it satisfies, and the term

a/(2b)

(˜x(θ  ) − βθ  ) · p (dθ  )] stands for the demand the firm can attract under opponents’ cost/quality 0 profile p and action profile x˜ . The latter term reflects that a lower price or higher quality at a firm helps attract customers from other firms. The b, c, α and β parameters signify the extent to which each factor attracts or repels demand, whether the factor be local price, local quality, remote price, or remote quality. We note that the current firm would never be interested in charging a price below θ . Instead of the price s˜ , we can treat the current firm’s premium s = s˜ − θ as its action. This change of decision variable would result in an easier-to-analyze nonatomic game. In the game, both type space T and action space S are [0, a/(2b)]. Corresponding to each cost/quality distribution p ∈ P ( T ), there is a nonatomic game Γ . In the game, when a θ -costing firm decides to charge premium s and opponents behave such that any θ  -costing firm is to charge premium x(θ  ), the firm will earn f (s, p ◦ (i T , x)−1 , θ). Due to (15), the latter in turn satisfies that, for any s ∈ S, r ∈ P ( T × S ), and θ ∈ T , [a − b · (˜s − α θ) + c ·









f (s, r , θ) = s · a − b · s − (α − 1) · θ + c ·

 s + (1 − β) · θ · r dθ × ds . 













(16)

T ×S

Note that

∂f (s, r , θ) = a − 2bs + c · ∂s











s + (1 − β) · θ  · r dθ  × ds + (α − 1) · bθ.

(17)

T ×S

The earlier theory can help us deal with two scenarios, (i) where α  1  β and (ii) where α  1  β . Under scenario (i), local quality is more important than local price in determining customer flows, while the opposite is true for remote qualities and prices. Under scenario (ii), local price is more important than local quality in determining customer flows, while the opposite is true for remote prices and qualities. Under scenario (i), it is easy to see that f has increasing differences in s and (r , θ). Therefore, the game is supermodular. By applying Theorems 1 and 2, Proposition 1 can be reached. Proposition 1. We can choose equilibrium premium-charging strategy x∗ for scenario (i) that is isotone in the cost/quality distribution p, such that at each fixed p, the equilibrium premium x∗ (θ) is increasing in the cost/quality parameter θ . Under scenario (ii), we can arrange for the same properties as above by defining the order for T reversely. The involved game is still supermodular. By applying Theorems 1 and 2, Proposition 2 can be reached. Proposition 2. We can choose equilibrium premium-charging strategy x∗ for scenario (ii) that is antitone in the cost/quality distribution p, such that at each fixed p, the equilibrium premium x∗ (θ) is decreasing in the cost/quality parameter θ . When β = 1, the involved games are further anonymous; hence, we can reach Propositions 1 and 2 by invoking Corollaries 1 and 2, respectively. Another example where these corollaries can be applied is scenario (iii) when α = β = 0. This is the case where firms produce indistinguishable goods even with divergent costs. In this case, we let action space S be the space [0, a/b] of prices rather than premiums. From (15), we obtain the payoff function f  , to the effect that, for s ∈ S, r  ∈ P ( S ), and θ ∈ T , 









f s, r , θ = (s − θ) · a − bs + c ·







s · r ds







.

(18)

S

When the cost distribution is some p ∈ P ( T ) and every θ  -costing firm charges some x(θ  ), a θ -costing and s-charging firm will earn f  (s, p ◦ x−1 , θ).

616

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

Now we have

∂ f   s, r , θ = a − 2bs + c · ∂s







s · r  ds + bθ.

(19)

S

As (19) is increasing in (r  , θ), we see that f  has increasing differences in s and (r  , θ). Therefore, the game above is supermodular. By applying Corollaries 1 and 2, Proposition 3 can be reached. Proposition 3. We can choose equilibrium pricing strategy x∗ for scenario (iii) that is isotone in the cost distribution p, such that at each fixed p, the equilibrium price x∗ (θ) is increasing in the cost parameter θ . 5.3. Other applications In this section, we briefly discuss three other potential applications. The first possibility is a variant of the rat-race model introduced by Akerlof (1976) to explain workers’ choice of work conditions. Here, we let action s in the discrete action space S = {0, 1, . . .} represent the work condition chosen by a worker, and we let θ stand for the innate ability of the worker. Suppose each worker’s payoff is affected by his own attributes and those of all others, in such a way that some A (θ, s, θ  , s ) stands for the pay to a θ -ability worker who has chosen condition s when all other workers are of ability θ  and have chosen condition s . When other workers’ attributes are heterogeneous, we suppose that average can be taken over opponents’ joint ability–choice distribution. Also, the cost to a θ -ability worker when he chooses condition s is C (θ, s). For this semi-anonymous model, the payoff function is











A θ, s, θ  , s · r dθ  × ds − C (θ, s).

f (s, r , θ) =

(20)

T ×S

Let B (θ, s, θ  , s ) = A (θ, s + 1, θ  , s ) − A (θ, s, θ  , s ). A game with the above payoff function would be supermodular when B (θ, s, θ  , s ) is monotone in (θ, s , θ  ) and C (θ, s) is submodular in (θ, s). Because if so, the difference











B θ, s, θ  , s · r dθ  × ds − C (θ, s + 1) + C (θ, s),

f (s + 1, r , θ) − f (s, r , θ) =

(21)

T ×S

would be monotone in (r , θ). The conditions could be satisfied, for instance, when A (θ, s, θ  , s ) = asθ s or as · (θ  + s ) and C (θ, s) = bs + c · (s − θ)2 for positive constants a, b, and c. When such is the case, we can learn from Theorem 1 that, though their abilities are not necessarily observable, the conditions the workers choose will nevertheless positively correlate with their abilities. From Theorem 2, we also know that more demanding conditions will be chosen when there is a greater supply of superior workers. ¯ , because It is interesting to note that Akerlof’s (1976) original model has the type space T being some finite {1, 2, . . . , θ} ¯ . The payoff function adopted of which r can be represented by the probability–mass vector (˜r sθ | s = 0, 1, . . . , θ = 1, 2, . . . , θ) there is

θ¯



f (s, r , θ) = θ ¯=1 θ

r˜sθ  θ 

˜

θ  =1 r s θ 

−

3 8

· (s − θ)2 .

(22)

This does not render the game supermodular. Akerlof showed the existence of the equilibrium solution x∗ with x∗ (1) = 1 and x∗ (θ) = θ + 1 for θ = 2, 3, . . . , θ¯ using specialized arguments. The second possibility is a generalization of Vives’s (2005) nonatomic-game rendering of the search model, first considered by Diamond (1982). In this anonymous game, action s represents the effort spent by a trader in looking for a partner. The payoff to a trader lies in the probability of finding a partner, which is proportional to the trader’s own effort and a function A (·) of the effort distribution r  of others. A trader’s type θ refers to the efficiency of his search. There is also a common cost function C (·) on efforts spent. For this model, the payoff function is





 

f  s, r  , θ = θ s · A r  − C (s),

(23)

so that under efficiency distribution p and search strategy x, the payoff to a θ -efficient trader who has spent an effort of s is f  (s, p ◦ x−1 , θ) = θ s · A ( p ◦ x−1 ) − C (s). A special case occurs when A (r  ) = B ( S s · r  (ds )), at which point the dependence on efforts exerted by other traders is reflected in the average only. Note that

  dC ∂ f   s, r , θ = θ · A r  − (s), ∂s ds

(24)

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

617

will increase in (r  , θ) as long as A (r  ) increases in r  . Therefore, when the success rate of a search is increasing in the effort distribution, the search game will be supermodular, and all existence and comparative statics results for monotone equilibria would be applicable to it. The third possibility is an extension of the arms race model considered in Milgrom and Roberts (1990). In this situation, players represent countries engaged in an arms race. We might use θ to stand for the technological level of a country, and let the cost of achieving a certain arms level s be some submodular C (θ, s). Submodularity means that the more technologically advanced a country is, the higher an arms level it can afford. Following Milgrom and Roberts, the benefit due to one’s own arms level s and external arms-level distribution r  can be expressed as B (s − S s · r  (ds )), where B (·) is a concave function. Thus, of most importance is the distance between a country’s own arms level and the average level of all rivals. The concavity means that ∂ B (s − S s · r  (ds ))/∂ s is increasing in r  . Hence, the anonymous game with payoff function f  (s, r  , θ) = B (s − S s · r  (ds )) − C (θ, s) is a supermodular one. Therefore, Corollaries 1 and 2 will predict the positive correlation between a country’s arms and technological levels and the rise of global arms levels as countries move up the technological ladder. Besides technological information, θ may contain other information like geography, population, and economy. The benefit function may be some more general A (s, r , θ), with r being the joint type–action distribution as opposed to the mere action distribution, and with ∂ A (s, r , θ)/∂ s increasing in (r , θ). For example, A (s, r , θ) may equal B (s − T × S W (θ  ) · s · r (dθ  × ds )) with B (·) being the same as above and W (θ  ) representing positive weights used for a type-θ  country. When W (·) is monotone, the resulting semi-anonymous game will again be supermodular. 6. Concluding remarks This paper establishes the existence, monotonicity, and comparative statics properties of pure Nash equilibria for nonatomic games involving strategic complementarities. The results cover fairly general action spaces and semi-anonymous games, where opponent types, as well as their actions, have an impact on a player’s payoff. For a monotone family of nonatomic supermodular games, however, it remains unclear whether the equilibrium set itself possesses any monotone trend. Acknowledgments The research of Jian Yang was supported by NSF Grant CMMI-0854803, and that of Xiangtong Qi was supported by the Hong Kong RGC grant GRF 618311. We would like to thank an advisory editor and a referee for their invaluable suggestions which helped improve the paper dramatically. Appendix A. Useful concepts of Section 2 Order upper semi-continuity: Given a complete lattice A, a function g : A →  is considered order upper semi-continuous, if for any totally ordered chain C ⊂ A,

lim sup

a∈C , a→inf C

g (a)  g (inf C ),

and

lim sup

a∈C , a→sup C

g (a)  g (sup C ).

Supermodularity: Given a lattice A, a function g : A →  is considered supermodular, if for any a, a ∈ A,

 









g (a) + g a  g a ∨ a + g a ∧ a . Increasing differences: Given partially ordered sets A and B, a function g : A × B →  is said to have increasing differences in a ∈ A and b ∈ B, if for any a1 , a2 ∈ A with a1  a2 and b1 , b2 ∈ B with b1  b2 ,

















g a1 , b 2 + g a2 , b 1  g a1 , b 1 + g a2 , b 2 . Stochastic dominance order: Let A be a complete lattice. For two probability measures q1 and q2 defined on the measurable space ( A , B ( A )), we say that q1 is stochastically smaller than q2 , written as q1 st q2 , when





g (a) · q1 (da)  A

g (a) · q2 (da), A

for every bounded and monotone function g ∈ M( A , ). This stochastic dominance relationship offers a partial order for the space P ( A ) of all probability measures on ( A , B ( A )). According to Shaked and Shanthikumar (2007) (p. 266; their n can be extended to the current complete lattice A), q1 st q2 is indeed equivalent to

q1 (U )  q2 (U ),

for any upper set U ∈ B ( A ),

where an upper set U is such that a ∈ U whenever a ∈ U and a  a.

618

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

Appendix B. Technical details of Section 3 Proof of Lemma 1. Let us fix a monotone mapping g ∈ I ( A , B ). For any b ∈ B, let the set A b ⊂ A be defined by

    A b = g −1 [b, sup B ] ≡ a ∈ A | g (a)  b .

(25)

By the monotonicity of g, we know that

Ab =





{a} =

a∈ A b

[a, sup A ].

(26)

a∈ A b

As A is a complete lattice, inf A b is well defined in A. Due to (26),

Ab =



[a, sup A ] ⊂ [inf A b , sup A ].

(27)

a∈ A b

When inf A b ∈ A b , we certainly have A b = [inf A b , sup A ], a closed interval in A. Suppose, on the other hand, inf A b ∈ / A b . The meaning of infimum dictates that, for any a  inf A b such that a = inf A b , there is a ∈ A b such that a  a is not true. Because A is totally ordered, we have a  a. Therefore,



Ab =

[a, sup A ] = (inf A b , sup A ] ≡ A \ [inf A , inf A b ].

(28)

ainf A b but a=inf A b

Taken together, A b is either a closed interval or the complement of a closed interval in A. On the other hand, B ( A ) is the smallest σ -field that contains all closed intervals of A, and B ( B ) is the smallest σ -field that contains all intervals of the type [b, sup B ]. This means that the smallest σ -field that contains all A b ’s is exactly the pre-image of B ( B ) through g. But the above also indicates that the pre-image is a subset of B ( A ). Therefore, g −1 ( B  ) ∈ B ( A ) for any B  ∈ B ( B ). Hence, g ∈ M( A , B ). 2 A monotone function from [0, 1]2 to [0, 1] that is not Borel-measurable: Let Z be any subset of [0, 1] that is not Borelmeasurable. Define f : [0, 1]2 → [0, 1] so that f (x1 , x2 ) = 0 when x1 < 1 − x2 and f (x1 , x2 ) = 1 when x1 > 1 − x2 . When x1 = x = 1 − x2 , however, we let f (x, 1 − x) = 1 or 0 according as whether x ∈ Z or not. This function is apparently monotone. Consider f −1 ([1/2, 1]), which is the union of A ≡ {(x1 , x2 ) ∈ [0, 1]2 | x1 > 1 − x2 } and B ≡ {(x, 1 − x) | x ∈ Z }. If f −1 ([1/2, 1]) were Borel-measurable, then its intersection with the Borel-measurable line {(x, 1 − x) | x ∈ [0, 1]2 }, which is merely the set B, would also be measurable. But we have g −1 ( B ) = Z for continuous function g : [0, 1] → [0, 1]2 satisfying g (x) = (x, 1 − x). As Z is not Borel-measurable, B is not either, and hence neither is f −1 ([1/2, 1]). Therefore, f is not Borel-measurable. We note that this counter example is facilitated by peculiarities of the Borel measure. Indeed, any monotone function from [0, 1]n to [0, 1] is known to be Lebesgue-measurable. Unfortunately, we do not know how to get around the Borel measure built on the interval topology in this study. So the restriction on the type space T remains. Proof of Lemma 2. Let Q be a subset of I ( A , B ). For each a ∈ A, we define the projection set Q(a) ⊂ B, so that

  Q(a) = b ∈ B | g (a) = b for some g ∈ Q .

(29)

Construct z so that z(a) = sup Q(a) for every a ∈ A. It is easy to show that z exists in B , and that it fits the definition for sup Q. Now the key is to show that z ∈ I ( A , B ). To this end, let a1 , a2 ∈ A satisfy a1  a2 . Then, for any g ∈ Q, we must have g (a1 )  g (a2 ) because Q ⊂ I ( A , B ). In view of the definition of z(a2 ), this leads to A

 

 

 

z a2  g a2  g a1 .

(30)

As (30) is true for every g ∈ Q, we see that z(a2 )  b for any b ∈ Q(a1 ). However, this means that z(a2 )  z(a1 ). This being true for any arbitrary pair (a1 , a2 ) with a1  a2 entails that z ∈ I ( A , B ). 2 Proof of Lemma 3. For any bounded and monotone function h in M( B , ), we have





1 −1

h(b) · q ◦ ( g )





B









h g 1 (a) · q(da)

(db) = A





h g 2 (a) · q(da) =



B

A

because g 1  g 2 and h is monotone.

2





h(b) · q ◦ g 2

− 1 

(db),

(31)

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

619

Appendix C. Technical details of Section 4 Proof of Lemma 4. According to Zhou (1994),

e ∗ (b) = sup A ∗ (b),





where A ∗ (b) = a ∈ A | ∃c ∈ G (a | b) with c  a .

(32)

We first show that A ∗ (b) is “half-monotone” in b, in a sense that will soon be made clear. Let b1 , b2 ∈ B with b1  b2 , and let a1 ∈ A ∗ (b1 ) and a2 ∈ A ∗ (b2 ). By (32), we know there are c 1 ∈ G (a1 | b1 ) with c 1  a1 and c 2 ∈ G (a2 | b2 ) with c 2  a2 .

Since G (a1 ∨ a2 | b2 ) is nonempty, we can suppose that some c ∈ G (a1 ∨ a2 | b2 ). Now consider c ∗ = c ∨ c 2 . Note that  c 2 and a1 ∨ a2  a2 . Hence, by G (a | b2 )’s monotonicity in a and the fact that c 2 ∈ G (a2 | b2 ), we have c ∗ ∈ G (a1 ∨ a2 | b2 ). Consider further c ∗∗ = c ∗ ∨ c 1 . Note that c ∗∗  c 1 , b2  b1 , and a1 ∨ a2  a1 . Hence, by G (a | b)’s monotonicity in both a and b, as well as the fact that c 1 ∈ G (a1 | b1 ), we conclude that c ∗∗ ∈ G (a1 ∨ a2 | b2 ). It is also clear that c ∗∗  c 1 ∨ c 2  a1 ∨ a2 . Thus, a1 ∨ a2 ∈ A ∗ (b2 ). We do not claim to know about the other “half”, that a1 ∧ a2 ∈ A ∗ (b1 ). However, the one half we do know would suffice. Let a1 , a2 ∈ A with a1  a2 be given. From Zhou, we know E (b) ⊂ A ∗ (b) for any b ∈ B. Since the former is nonempty, the latter must be nonempty as well. Therefore, there is some a2 ∈ A ∗ (b2 ). Now for any a1 ∈ A ∗ (b1 ), we know from the above that a1 ∨ a2 ∈ A ∗ (b2 ). Hence,

c∗

 

sup A ∗ b2  a1 ∨ a2  a1 ,

  ∀a1 ∈ A ∗ b1 .

(33)

This leads to sup A ∗ (b2 )  sup A ∗ (b1 ), and by (32), e ∗ (b2 )  e ∗ (b1 ). Symmetrically, we can show that e ∗ (b) is monotone in b as well. For that matter, we can resort to A ∗ (b) as defined in





A ∗ (b) = a ∈ A | ∃c ∈ G (a | b) with c  a ,

(34)

and the property that a1 ∧ a2 ∈ A ∗ (b1 ) when b1  b2 , a1 ∈ A ∗ (b1 ), and a2 ∈ A ∗ (b2 ).

2

Note that neither A ∗ (b) nor A ∗ (b) enjoys its corresponding other half of the monotone trend, and that E (b) in general is not simply A ∗ (b) ∩ A ∗ (b). We thus conjecture that, in general E (b) is not monotone in b. Proof of Lemma 5. For any bounded and monotone function h in M( B , ), we have







h(b) · q1 ◦ g −1 (db) =

B











h g (a) · q1 (da) A





h g (a) · q2 (da) =







h(b) · q2 ◦ g −1 (db),

B

A

simply because q st q and that the monotonicities of h and g lead to the monotonicity of h( g (·)). 1

(35)

2

2

Appendix D. Development of the anonymous case in Section 5 Let A and B be two complete lattices and q a probability measure in P ( A × B ). We use q | A to denote q’s marginal distribution on A, i.e., q | A ∈ P ( A ) which satisfies

 





q | A A = q A × B ,

∀ A  ∈ B( A ).

A semi-anonymous nonatomic game Γ such that

f (s, r , θ) = f  (s, r | S , θ),

(36)

= ( T , S , f , p ) is further deemed anonymous when there exists f 

∀s ∈ S , r ∈ P ( T × S ), θ ∈ T .

: S × P(S) × T →  (37)

That is, the game is anonymous when other players affect a given player’s payoff only through their action distribution rather than through the joint type–action distribution. For convenience, we use Γ  ≡ ( T , S , f  , p ) to denote this anonymous game. We can easily show that





p ◦ (i T , x)−1  S = p ◦ x−1 ,

as for any S 



∈ B ( S ),

  



∀ p ∈ P ( T ), x ∈ M( T , S ), 





(38)



p ◦ (i T , x)−1  S S  = p ◦ (i T , x)−1 T × S  = p (i T , x)−1 T × S 



       = p θ ∈ T  θ, x(θ) ∈ T × S  = p θ ∈ T  x(θ) ∈ S        = p x−1 S  = p ◦ x−1 S  .

(39)

620

J. Yang, X. Qi / Games and Economic Behavior 82 (2013) 609–620

Thus, in view of (2) and (37), we call x ∈ M( T , S ) a Nash equilibrium for Γ  when









f  x(θ), p ◦ x−1 , θ  f  s, p ◦ x−1 , θ ,

∀θ ∈ T , s ∈ S .

(40)

We deem Γ  supermodular when the following are true: (S 1) f  is order upper semi-continuous in s ∈ S for fixed (r  , θ) ∈ P ( S ) × T ; (S 2) f  is supermodular in s ∈ S for fixed (r  , θ) ∈ P ( S ) × T ; (S 3) f  has increasing differences in s ∈ S and (r  , θ) ∈ P ( S ) × T when the partial order based on stochastic dominance “st ” is used for P ( S ). Clearly, the satisfactions of (S 1) and (S 2) by Γ  would lead to the respective satisfactions of (S1) and (S2) by the original Γ defined through f . The analogy with (S3) depends on the marginal operator’s order preservation property. Lemma 6. Let A and B be complete lattices and q1 and q2 be two probability measures in P ( A × B ). Then q1 st q2 would lead to q1 | A st q2 | A . Proof. Let h : A →  be a bounded and monotone function. Define h : A × B →  by letting h (a, b) = h(a) for every a ∈ A and b ∈ B. Note that h is a bounded and monotone function as well. Since q1 st q2 , we know that





h(a) · q1 | A (da) =



A×B

A

h (a, b) · q1 (da × db)

h(a) · q1 (da × db) =



A×B





2

h (a, b) · q (da × db) =

 A×B

2

h(a) · q2 | A (da).

h(a) · q (da × db) = A×B

As h is arbitrarily bounded and monotone, we see that q1 | A st q2 | A .

(41)

A

2

Due to Lemma 6, we also know Γ  ’s satisfaction of (S 3) would result in Γ ’s satisfaction of (S3). Taken together, all of the above lead to Corollary 1 of Theorem 1. We now come to the corresponding monotone comparative statics result. Let Λ be a partially ordered set, and let (Γ  (λ) | λ ∈ Λ) be a family of anonymous nonatomic games, where each Γ  (λ) = ( T , S , f  (· | λ), p (λ)). We deem the game family monotone when the (M2) already stated in Section 4, along with the following, are true: (M 1) f  (s, r  , θ | λ) has increasing differences in s ∈ S and λ ∈ Λ for fixed (r  , θ) ∈ P ( S ) × T . Clearly, the satisfaction of (M 1) by (Γ  (λ) | λ ∈ Λ) would entail the satisfaction of (M1) by the game family (Γ (λ) | λ ∈ Λ) where each Γ (λ) is defined through f (· | λ), the λ-dependent version of f as defined in (37). These together lead to Corollary 2 of Theorem 2. Like before, a special case is one with Λ = P ( T ) and f  (· | λ) = f  for some f  : S × P ( S ) × T → . For this case, we can conclude from Corollary 2 that the largest and smallest monotone equilibria of the game Γ  are monotone in its player type distribution p. References Akerlof, G., 1976. The economics of caste and of the rat race and other woeful tales. Quart. J. Econ. 90, 599–617. Balder, E.J., 2002. A unifying pair of Cournot–Nash equilibrium existence results. J. Econ. Theory 102, 437–470. Diamond, P.A., 1982. Aggregate demand management in search equilibrium. J. Polit. Economy 91, 881–894. Khan, M.A., Rath, K.P., Sun, Y., 1997. On the existence of pure strategy equilibrium in games with a continuum of players. J. Econ. Theory 76, 13–46. Mas-Colell, A., 1984. On a theorem of Schmeidler. J. Math. Econ. 13, 201–206. Milgrom, P.R., Roberts, J., 1990. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58, 1255–1277. Milgrom, P.R., Shannon, C., 1994. Monotone comparative statics. Econometrica 62, 157–180. Radner, R., Rosenthal, R., 1982. Private information and pure-strategy equilibria. Math. Oper. Res. 7, 401–409. Rashid, S., 1983. Equilibrium points of non-atomic games: asymptotic results. Econ. Letters 12, 7–10. Schmeidler, D., 1973. Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Tarski, A., 1955. A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285–309. Topkis, D.M., 1978. Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321. Topkis, D.M., 1979. Equilibrium points in nonzero-sum n-person submodular games. SIAM J. Control Optim. 17, 773–787. Topkis, D.M., 1998. Supermodularity and Complementarity. Princeton University Press, Princeton, New Jersey. Van Zandt, T., Vives, X., 2007. Monotone equilibrium in Bayesian games of strategic complementarities. J. Econ. Theory 134, 339–360. Veinott, A.F., 1989. Lattice programming. Unpublished notes from lectures delivered at Johns Hopkins University. Vives, X., 1990. Nash equilibrium with strategic complementarities. J. Math. Econ. 19, 305–321. Vives, X., 1999. Oligopoly Pricing. MIT Press, Cambridge, Massachusetts. Vives, X., 2005. Complementarities and games: new developments. J. Econ. Lit. 43, 437–479. Zhou, L., 1994. The set of Nash equilibria of a supermodular game is a complete lattice. Games Econ. Behav. 7, 295–300.