Existence of Nash equilibrium in Discontinuous

Subaggregative Games ∗

Fereshteh Mohseninejad † Kevin Reffett ‡

Preliminary Draft: July, 2018

∗We thank David Ahn, M. Ali Khan, Alejandro Manelli, Ed Prescott, and especially Martin Kaae Jensen for very helpful conversations on the preliminary draft of this paper. This is a preliminary draft. Comments are welcome, and will be incorporated into the next revision of the paper. Please send these comments to the corresponding author. The usual caveats apply. †Department of Economics, Queen Mary University of London, School of Economics. Corresponding author: [email protected] ‡Department of Economics, Arizona State University, Tempe AZ. USA

1 Abstract

We prove new results on existence of Nash equilibrium in a class of aggregative games with infinite di- mensional action spaces and discontinuous payoffs. In addition, we also provide new equilibrium aggregate comparative statics results within a subclass of these aggregative games. In particular, we study aggregative games with separable aggregators. We prove aggregative games with weakly separable aggregators admit a subaggregative representations. When these aggregators are additionally additively separable, we provide suffi cient conditions on the normal-form aggregative game such that they admit subaggregative representa- tions that are linear "subaggregative games of strategic substitutes" (Sa-GSS) (resp., linear "subaggregate games of strategy complements" (Sa-GSC)). We are then able to prove the existence of pure-strategy Nash in subaggregates in both linear Sa-GSS and linear Sa-GSC, which imply pure strategy Nash equilibrium in normal-form. We are also able to extend the class of aggregate comparative statics results to a larger class of aggregative games than currently in the literature. As a Sa-GSS (resp., Sa-GSC) need not be a normal-form

Game of Strategic Complements (GSC) or a normal-form Game of Strategic Substitutes (GSS), our results are not contained in the existing literature on existence in GSC or GSS. Relative to the existing literature, per the literature on existence of pure-strategy Nash equilibrium games with discontinuous payoffs, our re- sults allow action spaces to be infinite-dimensional (e.g., compact Hausdorff lattices), and do not involve any convexity considerations.

2 I. Introduction

The literature on aggregative games in economics is extensive. Such games arise in numerous fields in economics, including industrial organization, political economy, public finance, contests, models of conflict, macroeconomics, and network games, among many other fields. Aggregative games are games where each players has payoffs that depend only on their own individual actions, and an "aggregate" of the other player’s actions, and can arise in settings of complete or incomplete information. The semi-anonymous structure of such games allow one to exploit their aggregative structure of simplify proofs of existence and/or characterization of Nash equilibria.

Canonical examples of aggregative games in the economics literature include models Cournot and

Bertrand competition (both with and without product differentiation), joint production games, network games with spillovers and/or endogenous network formation, strategic models of collective household choice, rent-seeking/Tulluck contests, games studying the provision of public goods, beauty contests and models of

fighting/conflict, environmental control, fishwars/common resource games, patent races, R&D models and models of innovation, models with demand externalities, Bayesian persuasion and signaling games, intrinsic common agency problems with adverse selection or moral hazard, distributional aggregative games/supply- function games, among others.

Additionally, the literature on the existence of Nash equilibrium in games of complete and incomplete information with discontinuous payoffs is equally as voluminous. The most recent literature begins, perhaps, with the important work of Dasgupta and Maskin ([25]), and includes a number of important early papers on the subject. (e.g., Simon ([65]), and Baye, Tian, and Zhou ([11]). More recent contributions to this literature on games with complete information include Barelli and Meneghal ([10]), Carmona ( [18], [19]),

Carbonell-Nicolau ([17]), McLennan, Monteiro, and Tourky ([48]), Reny ([58], [60]), Prokopovych ([55]), and

Nessah and Tian ([51]), among many others.

As Reny ([60]) points out, in general, there are two approaches to such questions, some focusing on the more "local" properties of payoffs, and others more "global" in nature. Per local approaches, one typically requires the convexity/linearity of the underlying action space (e.g., to state suffi cient conditions on ordinal representations of preferences in terms of conditions such as "better-reply security", or "point-security with respect the a subset of players", etc.). Per the latter global approach, order and "payoff complementarities" are used as a critical dimension to resolve the non-existence problem. For example, in an important recent paper, Prokopovych and Yannelis ([57]) proposed an extension of the tools of Topkis ([72] [73]), Vives

([76]), Veinott ([75]), Li Calzi and Veinott ([44]), Shannon ([64]), and Milgrom and Shannon ([49]) where strategy sets are chains. In their paper, upper semicontinuity conditions are replaced by "directional upper

1 semicontinuity", which is a generalization of Veinott’s idea of upper chain subcompleteness quasi-lattice function (also Veinott ([75], Chapter 3), or Milgrom and Shannon ([49]) notion of upper order semicontinuity).

Further, in this work, the notion of an "increasing correspondence" in Veinott’s strong set ordering best replies is replaced with "directed upward" or "directed downward". This latter idea has a long history in the mathematics literature dating back to at least Smithson ([66]), with more recent treatments applied to the question of existence of pure-strategy Nash equilibria where action sets are not sublattices in Heikkila and Reffett ([33]).

Another important recent paper very close in spirit to ours is KuKushkin ([42]), where he adapts and extends ideas on the better-reply dynamics literature and potential games in some of his previous work in Kukuskin ([40], [41]). Like ours, his methods are also global, but exploit order theoretic/strategic complementarity constructions at the normal-form level of the game. Our methods only use this structure at the "subaggregative" level of the game.

In this paper, we take an approach in the spirit of this latter recent literature by using order theoretic methods to resolve the non-existence issues in discontinuous games, but our approach is based on "aggre- gation". That is, we study a new class of normal-form aggregative games of complete information which we refer to as a "subaggregative game". These games are very general aggregative games at the level of normal-form strategies, but with additively separable aggregators (e.g., see Jensen ([35]) and Acemoglu and

Jensen ([2]) for a nice discussion of generalized aggregators). In particular, under the additional assumptions of aggregate separability, by exploiting the separable structure of generalized aggregators, we can define a

"subaggregative game", where normal-form actions only need to be "subaggregate consistent" with subag- gregate Nash equilibria. We then ask the question: when do aggregative games that admit subaggregative representations have pure strategy Nash equilibrium?

To answer this question, we recast the characterization of substitutes and complementarities in subag- gregative games in the language of aggregate and subaggregate lattice programming, and ask the question when normal-form strategies can be "aggregated" in a manner that yields a subaggregative representation of the aggregative game where that subaggregative representation is a linear GSS or linear GSC in subaggre- gates? That is, we are able to define the notion of a "subaggregative games of strategic substitutes" (Sa-GSS) and "subaggregative games of strategic complements" (Sa-GSC), as well as provide suffi cient conditions on the primitives of the aggregative game that imply under subaggregation that the subaggregative game is a linear Sa-GSS (resp, linear Sa-GSC). Hence, such aggregative games have pure-strategy Nash equilibrium in subaggregates. In the case of linear Sa-GSC, although in this paper we do not consider this case, in principle we can easily extend the results ot Bayesian aggregative games that admit subaggregative representations that are linear Sa-GSC.. Nash equilibria in subaggregates, in turn, then imply immediately the existence of

2 normal-form Nash equilibrium in the original aggregative game (as normal-form strategies are "subaggregate consistent").

Critically, the aggregative games we study in normal-form are not necessarily either GSS or GSC in normal-form. We provide a simple example, but our class of games viewed from the level of normal-form is substantial broader than those considered in the existing literature. In particular, we do require each player to have payoffs that are supermodular in own actions; but this does not imply that the Sa-GSC is a GSC

(as we do not require increasing "subaggregate" functions). In particular, as subaggregate functions are only required to be upper semicontinuous, a Sa-GSC needed not have increasing differences between normal formal actions. Importantly, as a direct by-product of our lattice programming construction, for the case of Sa-GSC, we are also able to prove a new equilibrium aggregate (and subaggregative) comparative statics result for a class of aggregative games which are not necessarily GSC.1

It is well-known that lattice programming tools are powerful per existence questions in games with discontinuous payoffs. In GSC, action spaces are non-convex, continuity is only required to by in order topologies, and action spaces are allowed to be infinite dimensional. These methods were originally developed in seminal work in (a) operations research by Veinott ( his body of results best presented in [75]), Topkis

([71] [72] [73]), and Li Calzi and Veinott ([44]), and (b) in economics by Milgrom and Shannon ([49]), and provide one with not only a systematic method for verifying existence of Nash equilibria in discontinuous and/or nonconvex games, but also for obtaining "robust" comparisons of equilibrium objects. (e.g., this is the subject of our companion paper in Jensen, Mohseninejad, and Reffett ([37]). In addition, in many settings, GSS also have equilibrium (e.g., aggregative games. See Kukushin ([39] [42], Dubay, Hainmanko, and Zapehelnyuk ([27], and Jensen ([35]). In many cases, such results can be tied to the literature on potential games (e.g., Voorneveld ([78] and Dubay, Hainmanko, and Zapehelnyuk ([27]).

Separability assumptions on the aggregator in the game are typical in both the theoretical and applied work in the literature. Given separability, we define the notion of a "subaggregative game", and then rewrite our class of aggregative games as a linear aggregative games in subaggregates. 2 As the subaggregates are typically elements of chains (e.g., compact subsets of the real-line), the resulting normal-form subaggregative game in subaggregates has a simple structure, many powerful tools in the theory of linear aggregative games can be brought to bear on the problem (for both existence of Nash equilibria, as well as aggregate and subaggregate equilibrium characterization). Further, as most aggregative games in the applied literature are

1 We should mention, in a companion paper Jensen, Mohseninejad, and Reffett ([37[), we also use our methods to provide robust comparative statics results in Sa-GSS. As the methods are somewhat different than those here, we refer the reader to this paper for those results. 2 A "linear" aggregative game is an aggregative game that has an aggregator that is linear in "normal-form" choices. In our setting, under separability conditions per the aggregator, we map the general aggregative game (with a very general normal-form action space) into a linear aggregative game in subaggregates (the relevant "normal-form choices".)

3 linear aggregative games, we are able to unify the theory of aggregative games with general normal-form action spaces, to the theory of linear aggregative games.

A key focal point of the literature we address in the paper is series of papers by Acemoglu and Jensen

([1] [2] [3]). In particular, in Acemoglu and Jensen ([2]), they consider finite player aggregative games with finite dimensional action spaces with "generalized aggregators" (i.e., aggregators with an additively separable structure). We show in the paper this assumption of the existence of "generalized aggregators" in the aggregative game allows one to construct a very sharp lattice programming characterization of extremal equilibrium best reply maps, which in turn leads to the ability to develop a systematic approach to equilibrium aggregate and subaggregate comparative statics. This, in turn allows us to recast existence questions relative to subaggregates (as opposed to normal-form actions). One important class of games where results for aggregate games are known are aggregative games that are GSS in normal-form. (e.g., Acemoglu and Jensen

([2], section 3).3 For this subclass of games, we are able to provide sharp (and weaker) suffi cient conditions for existence of Nash equilibria. In particular, our subaggregative games are not GSS.

Per questions of existence of pure strategy Nash equilibrium in normal-form games, our aggregate and subaggregate lattice programming approach to aggregating subaggregative games to obtain the existence of Nash equilibrium in normal-form games also contributes to the existing large and important literature potential games (e.g., (Monderer and Shapley ([50]), Voorneveld ([78]), Dubay, Hainmanko, and Zapehel- nyuk ([27]), Jensen ([35]), among others). In particular, we are able to provide new suffi cient conditions for the existence of pure-strategy Nash equilibrium in normal-form games that are Sa-GSS using the results in potential games for GSS (e.g., Dubay, Hainmanko, and Zapehelnyuk ([27]), Jensen ([35]), and Acemoglu and Jensen ([2]) More specifically, for the class of aggregative games with strategic substitutes in aggre- gates/subaggregates, we establish the existence of pure-strategy Nash equilibrium relative to reduced-form

"linear subaggregative game with strategic substitutes" using standard results in best-reply potential games

(e.g., Jensen ([35]). What is interesting, though, is in the games where our results apply, when viewed from the vantage point of their normal-form structure, we allow for (a) infinite-dimensional action spaces (i.e., compact Hausdorff topological lattices), and (b) are not necessarily game of strategic substitutes in normal- form. So our approach extends the scope of results of the literature on potential games to a potentially much broader class of normal-form games.4

The literature on aggregative games with strategic substitutes in economics that is included within our framework is extensive. Examples of such aggregative games in economics including Cournot games with multidimensional choices (see Novshek ( [53]), KuKushkin ([39]), Amir ([4]), Amir and Lambson ([5],

3 There is an interesting literature on the comparative statics of GSS (e.g., see Roy and Saberwal ([61]) [62]). 4 Also, see Martimort and Stoles ([46] [47]) for a nice discussion per the latter fact, with examples of simple aggregative games that are not potential games.

4 Einy, Haimanko, Moreno, Shitovitz ([28]), and Ewerhart ([29], among many others5 ), aggregative games on networks with strategic substitutes or complements (see Galeotti, Goyal, Jackson, Vega-Redondo, and

Yariv ([31]), Amir and Lazzati ([6]), Jackson and Zenou ([38]), models of peer effects in education (see Calvo-

Armengol, Patacchini, and Zenou ([16]), models of social interactions and conformity (see Blume, Brock,

Durlauf, and Jayaraman ([14]), the theory of public goods provision (see Bergstrom, Blume, and Varian ([12] and Cornes and Hartley ([23]), and environmental games (Cornes ([22]).

Our work also allows one to address the question of aggregate and subaggregate equilibrium comparative statics. An important early paper on comparative statics in aggregative games is Corchon ([21]). In his paper, he studies suffi cient conditions for aggregate comparative statics in strongly concave games. Nit ([54]) studies the equilibrium comparative statics of aggregates in a class of contests. introduce complementarities and/or substitutes into aggregative games. A systematic approach to such problems has been proposed in series of recent papers by Acemoglu and Jensen ([1] [2] [3]), where the authors provide a collection of new results on the comparative statics of aggregates in a very general collection of environments. Unlike our approach to the problem, they do not approach to the problem from the viewpoint of self-generating aggregate and subaggregate lattice programming.

In our companion paper (e.g., Jensen, Mohseninejad, and Reffett ([37]) we address exactly these issues of equilibrium aggregate and subaggregate comparative statics in a systematic manner. The point is such equilibrium aggregate and subaggregate comparative statics results (especially for Sa-GSS) is very subtle, and relies upon implicit aggregate and subaggregate lattice programming techniques. For these methods to not be vacuous, the question of equilibrium existence must be addressed. That is, as they are essentially

"self-generation" techniques (i.e., equilibrium programming), we must know pure-strategy Nash equilibria exist. These discussions are beyond the scope of this paper, and we refer the reader to this companion paper for discussion. Still, in this paper, we are able to provide a new result on equilibrium aggregate and subaggregative comparative statics for Sa-GSCs.

In the next section of this paper, we define the class of subaggregative games with complementarities and substitutes. These games are allowed to have infinite dimensional action spaces. They also involve no convexity considerations. Then, Section 3 contains our existence of pure-strategy Nash equilibrium in these games. The appendix includes all the mathematical terminology used in the paper, as well as statements of a a few critical theorems in extend-real valued supermodular lattice programming we use repeatedly in the proofs.

5 See also the monograph of Vives ([77]).

5 II. Subaggregative Games with Complements and/or Substitutes

We begin by defining a general class of aggregative games. To sharpen our characterizations of aggregate and subaggregate comparative statics, as is typical in the literature, we assume a separable structure for the aggregator in the game. Separability assumptions (e.g., weak separability) on aggregators imposes a hierarchical structure in the aggregative game. This hierachical structure is particular sharp if an aggregator

φ(x) is additively separable in each normal-form strategy xi Xi of player i N, it admits a "linear" ∈ ∈ representation on subaggregates qi given by

yˆ = φ(x) = H( gi(xi)) = H( qi) i i X X where qi = gi(xi) is the i subaggregate, and g = (g1, ..., gn) = (q1, ..., qn) as the vector of continuous sub- − aggregate mappings, with H(y) a strictly increasing and continuous function of its argument. 6 Indeed the existence of weakly separable aggregator will allow the aggregative game under study to admit a "subaggre- gate representation". This subaggregate representation will be very tractable in the case that the aggregator is additionally additively separable, with each subaggregate function having order preserving, order reversing, or "mixed-monotone" structure (i.e., increasing in some arguments, decreasing in others). In this paper, we focus on just "two stages" of disaggregation; but as will be clear in the sequel, using simple recursive/dynamic programming argument, we can embed any arbitrary number of "substages" into the subaggregation of the game. This observation could be important, for example, when taking the model to data, each of these levels of "subaggregation" levels will (a) have (testable) equilibrium comparative statics predictions, and (b) represent cross-sectional observations of "subunit’s"in the game.

Aggregative games. We begin by defining an aggregative game. The player set is denoted by

N = 1, 2, ..., n , a typical player i N having a nonempty action compact space Xi Xi, Xi is a Hausdorff { } ∈ ⊂ topological space. Let the joint strategy profiles be denoted by X = Xi X, X endowed with the i N ∈ ⊂ product topology, x X denoting a typical element of X. By (xi, x iY) X, we denote the decomposition ∈ − ∈ of the joint strategy profile x X, where player i plays component xi Xi, and the remaining players play ∈ ∈ the profile x i = (x1, x2, .., xi 1, xi+1, .., xn) X1 X2 ... Xi 1 Xi+1 ...Xn = X i. − − ∈ × × × − × × − i Assume each player i has an real-valued payoff function U : Xi Y T R, that is upper semi- × × → continuous in its first argument, continuous in its second and third argument, where the parameter space

T is a topological space, (xi, y, t) Xi Y T , Y E (where E denotes the space (R, ) with the ∈ × × ⊂ ≥ ≥ 6 The value of the aggregator is yˆ = H( i gi(xi)). We will incorporate the function H(ˆy) into the payoff structure, so the appropriate notion of an "aggregate" in our games is y = gi P i P

6 7 standard Euclidean order). Then, when each player i N chooses xi Xi, given a strategy profile for the ∈ ∈ other players x i X i, the resulting joint strategy profile x X induces a realization of the aggregate − ∈ − ∈ y Y, where the aggregator is given by a continuous function φ(x) = y Y E. 8 . In an aggregative ∈ ∈ ⊂ game, we shall assume that aggregator φ(x) exists such that each player i N has a reduced-form payoff ∈ i i for a normal-form Nash game for player i N is given by U (xi, φ(xi, x i), t), where U (xi, φ(x), t) is the ∈ − i aggregate payoff U (xi, y, t) evaluated at the aggregator y = φ(x).

i An aggregative Nash game is a normal-form game defined by the tuple (U ,Xi)i N , φ, t T . The { ∈ ∈ } best reply correspondence in this game for player i N defined as follows: given x i X i, solve ∈ − ∈ −

i i V (x i, t) = max U (xi, φ(xi; x i), t) − xi Xi − ∈ with the associated best reply map for player i N given by: ∈

i Xi∗(x i, t) = arg max U (xi, φ(xi; x i), t) − xi Xi − ∈

Under our standard continuity/compactness conditions on the game, each of these programs is well-defined, with each player’sbest reply map nonempty and compact-valued.

i A pure-strategy Nash equilibrium in the aggregative game (U ,Xi)i N , φ, t T at parameter value { ∈ ∈ } t T is then a joint profile vector x¯(t) X, such that for all i N, x¯i(t) Xi satisfies: ∈ ∈ ∈ ∈

i i (1) U (¯xi(t), φ(¯x(t), t), t) U (xi, φ(xi;x ¯ i(t), t)) xi Xi ≥ − ∀ ∈

We denote the set of pure-strategy Nash equilibrium at parameter t T by the correspondence. ∈

(2) Ψx(t) = x¯(t) X (1) holds at t T { ∈ | ∈ }

Any pure-strategy Nash equilibrium x¯(t) Ψx(t), x¯(t) necessarily induces a corresponding equilibrium ∈ 7 We assume the action spaces are compact Hausdorfftopological spaces as we need the best reply maps to be compact-valued. To guarantee this, we assume payoff functions upper semicontinuous in own actions, continuous in the opponents strategies (to apply the generalized maximum theorem of Ausubel and Deneckere ([8], Theorem 2). If we assume payoffs are continuous, we can work in arbitrary compact topological spaces (as Berge’stheorem will apply). Given that our action spaces are not convex, even in this case, our results in that case would extend the existing results in the literature on discontinuous games. 8 We shall eventually deviate from the literature and not require the aggregator to be increasing. That is, in some cases, we will require the aggregator to be "mixed monotone" (i.e., increasing in some actions, and decreasing in others). This will be a critical distinction, as the resulting game viewed as a normal-form game will be neither a game of strategic substitutes or game of strategie complements. This will in part distinguish our results from those in the existing literature in important ways (e.g., Jensen ([35]), Acemoglu and Jensen ([2], section 3). See Example 7. later in the paper.

7 aggregate y¯(t) = φ(¯x(t)). We denote the set of equilibrium aggregates in the game by the correspondence:

(3) Ψy(t) = y¯(t) y¯(t) = φ(¯x(t)), x¯(t) Ψx(t) { | ∈ }

Notice, if the set of pure-strategy Nash equilibria Ψx(t) is non-empty compact-valued, under our continuity conditions on that aggregator φ(x), the set Ψy(t) is non-empty and compact-valued. Alternatively, if the set

9 of Nash equilibria Ψx(t) is chain-complete, the set of Nash equilibrium aggregates Ψy(t) is chain complete.

In either case, if Y E is a chain, the elements Ψy(t) and Ψy(t) are then well-defined equilibrium ⊂ ∨ ∧ 10 i aggregates. An aggregative game (U ,Xi)i N , φ, t T is proper if the set of equilibrium aggregates { ∈ ∈ } Ψy(t) is (a) nonempty, and (b) has least and greatest elements Ψy(t) and Ψy(t), both elements of Ψy(t). ∧ ∨ For any aggregator φ(x), define the following correspondence for player i N : ∈

1 Φi− (y; x i) = xi Xi y = φ(x), x i X i − { ∈ | − ∈ − }

1 where Φi− (y; x i) defines the set actions xi Xi for player i that "implement" aggregate y = φ(x) when − ∈ her opponents play x i X i. We say an aggregator φ(x) is pairwise-injective if for all players i N, − ∈ − ∈ 1 given x i X i, the mapping φ− : Y Xi is a function. An aggregator is pairwise-bijective if it is − ∈ − → pairwise-injective, and Y = φ(Xi; x i) for all i N. − ∈

Subaggregative Games. As in Acemoglu and Jensen ([2]), our focus in the paper in the end is on aggregative games where the aggregators φ(x) are additively separable in x (see Gorman ([32] and Blackorby,

Primont, and Russell ([9]) for a discussion of additive separability). Still, in this section, we shall also allow weaker forms of separability, and discuss the structure of "subaggregative representations" of aggregative games. Also, assume Y E (i.e., real-valued aggregators). ⊂ Separability assumptions on aggregators will imply our aggregative games shall always admit subaggre- gate representations (i.e., "subaggregative games"). Further, it will mean that when considering existence questions, we shall always be able to reduce the associated fixed point problem to a mapping that is defined only in terms of "subaggregates" (i.e., study the fixed points of an "reduced-form subaggregative game").

The former idea will be a game where the actual primitive data of the game can be stated in terms of only subaggregates; the later representation of the game states the game in terms of the original primitives of the

9 Note, when applying order theoretic arguments to characterize the nonemptiness of Ψx(t), compactness here could be in an appropriate order topology (i.e., the interval topology). As we remark later, in Hausdorff spaces, there are important relationships between complete lattices and compact sets. Recall, the result of Frink ([30], Theorem 9) discussing the relationship between compactness in the interval topology and the completeness of a lattice. 10 We mention further, per characterizing the continuity properties of least and greatest equilibrium aggregates and/or subag- gregates, by an important result in Acemoglu and Jensen ([2], Theorem 1), if T is an ordered topological space (e.g., a Hausdorff space), as the aggregator φ(x) is continuous, the least (resp., greatest) aggregate (and subaggregates if φ is additively separable) will exist, with the least (resp., greatest) aggregate lower semicontinuous (resp., uppersemicontinuous) on T.

8 aggregative game, only constructing existence of pure strategy Nash equilibria via a fixed point problem in subaggregates.

Further, if aggregators are real-valued and increasing in ssubaggregates (but not necessarily in normal- form actions), and subaggregates also real-valued, aggregators in subaggregates will always be at least pairwise-injective in subaggregates.11 This in some cases be helpful in characterizing equilibria via lattice programming arguments. For example, this fact will allow us to obtain lattice programming representations of self-generation optimization problems that can be used characterize the relationship between the pure- strategy Nash equilibrium sets Ψx(t) and equilibrium aggregate comparative statics associated set of Nash equilibrium aggregates Ψy(t), but it also allows us to relate the equilibrium set of aggregates Ψy(t) to the set of Nash equilibrium subaggregates Ψq(t) = qi(t) qi(t) = gi(¯xi(t)), q(t) = (q1(t), ..., qn(t)), x(t) Ψx(t) for { | ∈ } each t N. Additionally, the subaggregate structure proves critical when considering suffi cient conditions ∈ for the existence of pure strategy Nash equilibrium in aggregative games with discontinuous normal-form payoffs.

We should also remark, although not really the focus of this paper, when aggregative games admit ag- gregators that are additionally pairwise-bijective, the game will have many very desirably properties that are not present in the case aggregators are only pairwise-injective. In particular, per aggregate and subaggregate lattice programming constructions, choice sets will often be sublattices. In such cases, additive separability is not as important. So whenever we do not state to the contrary, always assume our generalized aggregators are only pairwise injective in subaggregates.

We begin with the following definitions. We first define some useful properties of aggregator. We should remark, the idea of a generalized aggregator was introduced in Jensen ([35]), and used exclusively in for the aggregative games studied in Acemoglu and Jensen ([2]).

Definition 1. An aggregator φ(x) is weakly separable if φ(x) = H(g1(x1), g2(x2), ..., gn(xn)), where H :

n Y Y is continuous and strictly increasing, and each gi(xi) is continuous. The aggregator φ(x) will → → be a generalized aggregator if φ(x) = H( gi(xi)), g = (g1, ..., gn), with H : Y E continuous and strictly i → increasing in y = gi(xi), and each subaggregateP gi : Xi Y is continuous in xi Xi. If φ(x) is a i → ∈ generalized aggregator,P with each gi(xi) linear for i N, we say φ(x) is a linear aggregator. An generalized ∈ aggregator is a linear aggregator in subaggregates if φ(y) = H( qi), for qi Qi for all i = 1, 2, ..., n , Qi i ∈ { } E non-empty and compact,H : Y E continuous and strictlyP increasing in y = qi. An aggregative ⊂ → i game with a linear aggregator will be referred to as a linear aggregative game. P

11 As will be clear, when aggregators are additively separable, it is very natural to assume aggregators are only increasing in subaggregate values; not necessarily the subaggregate functions in their normal-form strategies. In essence, the subaggregate function serves like a subaggregate payoff; its value is what is critical when considering the structure of strategic complemen- tarities or substitutes.

9 When studying the subaggregate structure of our games, we need to specify the relationship between

the normal-form strategy space Xi and the "range" of the collection of subaggregates qi = gi(xi). To do this, we need lower and upper bounds for the values for each of the i-subaggregates. These bounds are given

by the elements gi = minxi Xi gi(xi) and gi = maxxi Xi gi(xi) which as the subaggregate functions are ∧ ∈ ∨ ∈ continuous, and action spaces are nonempty and compact, are both are well-defined by Berge’s Maximum

Theorem.

Define the space Qi = qi qi [ gi, gi], qi = gi(xi), xi Xi E. Notice, the set Qi is generated { | ∈ ∧ ∨ ∈ } ⊂ from the set Xi via player i N subaggregate function gi(xi). Define the set Q = Qi, and give this ∈ i Y space its product order. Note, as each action space Xi is compact, and gi(xi) is continuous on Xi, the set

n q = (q1, q2, .., qn) Q is a chain-complete in E . In any case, each Qi is a subchain of [ gi, gi]. If the game ∈ ∧ ∨ has a pairwise-bijective aggregator in subaggregates, we can additionally take qi Qi = Xi for each ∈ i player i N. P P ∈ A key construction in this paper is the concept of a subaggregative representation of an aggregative

game.

i Definition 2. An aggregative game (U ,Xi)i N , φ, t T admits a subaggregative representation if there { ∈ ∈ } exists a weakly separable aggregator φ(x) = H(g1(x1), g2(x2), ..., gn(xn)) such that (a) for each player i ∈ i i N, her payoff is Πq(xi, gi(xi), q i, t) = U (xi,H(gi(xi), q i), t), where q i = (q1, q2, ..., qi 1, qi+1, .., qn) is − − − −

the vector of opponent subaggregate strategies, qj = gj(xj), xj Xj, and (b) each player i N has a ∈ ∈ subaggregate best reply correspondence

(4) Qi∗(q i, t) = qi∗(q i, t) qi∗(q i,t) = gi(xi∗(q i, t)), xi∗(q i, t) Xi∗(q i, t) − { − | − − − ∈ − }

where

i (5) Xi∗(q i, t) = arg max Πq(xi, gi(xi), q i, t) − xi Xi − ∈

12 i is player i N normal-form best reply correspondence. We say an aggregative game (U ,Xi)i N , φ, t T ∈ { ∈ ∈ } has a Nash equilibrium in subaggregates if the aggregative game admits a subaggregative representation, and

the joint subaggregate profile mapping in subaggregates is given by a correspondence Q∗ = (Q1∗, .., Qn∗ ),

where Q∗(q, t): Q T Q has a fixed point q∗(t) Ψq(t) Q. × → ∈ ⊂ Clearly, when the aggregator φ(x) in an aggregative game is weakly separable, the aggregative game

12 It bears mentioning, as the action spaces Xi are each compact Hausdorff spaces, therefore T4 (and hence, regular topological spaces), and the payoffs are upper semicontinuous in own actions, continuous in opponents strategies, by Ausubel and Deneckere ([8], Theorem 2), Xi∗(q i, t) is nonempty, upper hemicontinuous, and compact-valued. −

10 i (U ,Xi)i N , φ, t T admits a subaggregate representation. This observation is critical to all the results { ∈ ∈ } in this paper. We shall be particularly interested in aggregative games with generalized aggregators. These games will not only admit a subaggregative representation, but that subaggregative representation will be a linear aggregate game.

The following lemma provides a characterization of the set of normal-form Nash equilibrium in an

i aggregative game (U ,Xi)i N , φ, t T with a weakly separable aggregator in terms of Nash equilibrium in { ∈ ∈ } subaggregates. In particular, a joint strategy profile x∗(t) Ψx(t) X is a pure-strategy Nash equilibrium ∈ ⊂ i in the normal-form aggregative game (U ,Xi)i N , φ, t T iff the joint strategy profile in subaggregates { ∈ ∈ } q∗(t) = gi(x∗(t)) Ψq(t) Q is Nash equilibrium in subaggregates. It bears mentioning, therefore, that ∈ ⊂ this lemma is closely related to the principle of "aggregate concurrence" introduced in Martimort and Stole

([47]). That is, in our case, the lemma defines the notion of "subaggregate concurrence" for normal-form

i aggregative games (U ,Xi)i N , φ, t T with weakly separable aggregate structure: { ∈ ∈ }

i Lemma 3. Let (U ,Xi)i N , φ, t T be an aggregative game with a weakly separable generalized aggre- { ∈ ∈ } gator φ(x) = H(g1(x1), g2(x2), ..., gn(xn)). Then, q∗(t) Ψq(t) Q is a pure-strategy Nash equilibrium ∈ ⊂ in subaggregates if and only x∗(t) Ψx(t) X∗ is a pure strategy normal-form Nash equilibrium in the ∈ ⊂ i aggregative game (U ,Xi)i N , φ, t T { ∈ ∈ }

i Proof. By the definition of a subaggregate representation of the game (U ,Xi)i N , φ, t T , the joint { ∈ ∈ } best reply correspondence in aggregative game in normal-form can be written as follows: for any q =

(g1(x1), ..., gn(xn)) = g(x), x X, we have the joint best reply: ∈

X∗(q, t) = X∗(q, t)

= (X1∗(g 1(x 1), t), ...., Xn∗(g n(x n), t)) − − − −

= X∗(x, t) where

X∗ : X T X × ⇒

Then, x∗(t) Ψx(t) X is a pure strategy Nash equilibrium iff its a fixed point of X∗(x, t) for each t T. ∈ ⊂ ∈ The associated subaggregative representation of the aggregative game with joint best reply correspon-

11 dence X∗(g(x)) is:

Q∗(q, t) = (Q1∗(q 1), t), ..., Qn∗ (qn(q n), t)) − −

= Q∗(g(x), t) where

Q∗ : Q T Q × →

Therefore, q∗(t) Ψq(t) Q is a pure strategy Nash equilibrium in subaggregates iff q∗(t) is a fixed point ∈ ⊂ of the mapping Q∗(q, t) for each t T. ∈ Say q∗(t) is a fixed point of Q∗(q, t) at t T. For any q Q, define the set: ⇒ ∈ ∈

1 g− (q) = x X q = g(x) X { ∈ | } ⊂

Then, as q∗(t) is a fixed point of Q, we have

q∗(t) Q(q∗(t), t)) ∈ 1 = Q(g(x∗(t)), t) for any x∗(t) g− (q∗(t)) ∈

1 1 Using the definition of g− (q), as for x∗(t) g− (q∗(t)), we have q∗(t) = g(x∗(t)) Q∗(q∗(t), t) is a fixed ∈ ∈ 1 point of Q∗. Therefore, x∗(t) g (q∗(t)) x∗(t) X∗(x∗(t), t). Hence, x∗(t) is a fixed point of X∗(x∗(t), t), ∈ ⇒ ∈ and therefore a Nash equilibrium in the normal-form aggregative game.

let x∗(t) Ψx(t) X. Then, for x∗(t), we have q∗(t) = g(x∗(t)). As x∗(t) is a Nash equilibrium, we: ⇐ ∈ ⊂

x∗(t) X∗(x∗(t), t)) ∈

= X∗(q∗(t), t)

Then, x∗(t) X∗(q∗(t), t), by the definition of Q∗, q∗(t) = g(x∗), with ∈

q∗(t) Q∗(q∗(t), t) ∈

= Q∗(x∗(t), t))

hence, the subaggregate q∗(t) is a fixed point of Q∗(q, t) (and therefore Nash equilibrium in subaggregates).

12 Subaggregate Games with Complements and Substitutes. Although we shall be interested primarily in this paper with aggregative games that have aggregative and/or subaggregative substitute structure between individual actions and subaggregate, we will define similar games with complementarities also. Such games will be very simply to characterizing using well-known results in the literature. So, in some sense, the strategic substitutes case is more interesting. Regardless, all of these aggregative games are lattice games; i.e., all action spaces are sublattices.

i We now provide some additional definitions. Let (U ,Xi)i N , φ, t T be an aggregative in normal- { ∈ ∈ } form aggregative, φ its aggregator. For the rest of the paper, we shall assume each player’s action space

Xi Xi is additionally a sublattice for all i N, with Xi a lattice, the space of joint strategy profiles ⊂ ∈ denoted by X = Xi X endowed with its relative product order (also a sublattice), X endowed with i N ∈ ⊂ the product orderY (also a lattice), T a partially ordered set (Poset). Given the compactness requirements on action spaces Xi already imposed, each Xi and X are also now both compact and regular in the interval topology. Further, in this context, as each of these spaces is a compact Hausdorff space, each are complete lattices in the interval topology. This implies they will be Hausdorff in its order topology (e.g., Atsumi

([7],Theorem 3 ).13

As we shall be consider both complementarity and substitutes between aggregates and subaggregates in the games themselves. For this, its best to define the games with complementarities/substitutes for a fixed parameter t T, as comparative statics are not the focal point of this note. ∈

i Definition 4. An aggregative game (U ,Xi)i N , φ, t T is a Aggregative Game of Strategic Complements { ∈ ∈ } (GSC ) (resp., Aggregative Game of Strategic Substitutes (GSS)) if there exists an aggregator φ : X Y such → i i that each player i N has a reduced-form normal-form real-valued payoff Π (xi, x i; t) =U (xi, φ(x); t), with ∈ − i action space Xi a sublattice, X = Xi the joint action space, X given its product order, where Π (xi; x i, t) − 14 is (i) supermodular in xi Xi, andY (ii) has increasing (resp., decreasing) differences between (xi; x i), for ∈ − each t T. ∈ Next, we now define subaggregative games of strategic substitutes/strategic complements.

i Definition 5. An aggregative game (U ,Xi, gi)i N , t T is a Subaggregative Game of Strategic Comple- { ∈ ∈ } ments (Sa-GSC) (resp., Subaggregative Game of Strategic Substitutes (Sa-GSS)) if it admits a subaggregative

13 A few remarks on lattices with Hausdorff topologies. Isnel ([34]) shows a necessary and suffi cient condition for a lattice to be Hausdorff in its complete topology is every net in the lattice has an order convergent subnet. Atsumi ([7]) shows a necessary and suffi cient for a lattice to be compact and Hausdorff in its interval topology. This therefore implies that if a complete lattice is Hausdorff in its interval topology, the lattice also compact and Hausdorff in its order topology (and hence, the two order topologies coincide). Frink ([30]) studies the relationship between completeness and compactness in its interval topology and order topologies. For excellent discussions of intrinsic topologies in topological lattices and semilattices, see Choe ([20]) and Lawson ([43]). 14 In the appendix, we define extended real-valued supermodular functions. See that discussion. These are super functions from a lattice to a posemigroup (P, , ) where is properly increasing. ∗ ≥ ∗ ∗

13 representation with a joint subaggregate best reply correspondence Q∗(q; t) = (Q1∗(q 1; t), ..., Qn∗ (q n; t) hav- − − ing Q∗(q; t) and Q∗(q; t) decreasing (resp., increasing) selections. A Subaggregative Game of Strategic ∨ ∧ Complements (Sa-GSC) (resp., Subaggregative Game of Strategic Substitutes (Sa-GSS)) that is a linear aggregative game in subaggregates if for each player i N, each t T (i.e., the subaggregate best reply ∈ ∈ correspondence Qi∗(q i; t) = Qi∗(zi; t) where zi = qj) will be referred to as a linear Sa-GSS (resp., linear − j=i Sa-GSC ) X6

For the rest of the paper, we shall focus on linear Sa-GSS (resp., linear Sa-GSC). That is, we shall assume a generalized aggregator exists in the game. As will be clear in the next lemma, a linear Sa-GSS (resp. linear

Sa-GSC) need not be a GSS (resp., GSC) in normal-form. The following lemma provides suffi cient conditions

i on the primitives of the aggregative game (U ,Xi)i N , φ, t T such that the aggregative game admits a { ∈ ∈ } subaggregative representation that is is a GSS (resp., GSC) in subaggregates q Q : ∈

i Lemma 6. Let (U ,Xi)i N , φ, t T be an aggregative game with generalized aggregator φ(x) = H( qi)), { ∈ ∈ } i i qi = gi(xi) for all i N, such that (a) each player i N has a payoff Π (xi, gi(xi)+zi, t)= U (xi,H(gPi(xi)+ ∈ ∈ q i zi, t), where zi = j=i qj, where Πq(xi, gi(xi) + zi, t) is supermodular in xi Xi. Assume additionally one 6 ∈ of the following twoP conditions for each player i N : (b.i) each aggregator gi(xi) is monotone increas- ∈ i ing, and Πq(xi, gi(xi) + zi, t) has decreasing (resp., increasing) differences between (xi; zi), or (b.ii) for the a b a B a b partition of individual action space xi = (xi , xi ), the aggregator gi(xi , xi ) is mixed-monotone in (xi , xi ), i a b a b a and Πq(xi , xi , gi(xi , xi ) + zi, t) has decreasing (resp., increasing) differences in (xi ; zi), and has increasing b (resp., decreasing differences) in (x ; zi), or (b.iii) players j = 1, 2, 3, .., k = N1 satisfy (b.i) and the re- i { } i maining players j = k + 1, ..., n = N2 satisfy (b.ii) for 1 < k < n. Then, (U ,Xi)i N , φ, t T admits a { } { ∈ ∈ } subaggregative representation that is a linear Sa-GSS (resp., linear Sa-GSC) in subaggregates q Q. ∈

Proof. We prove the case of a linear Sa-GSS. The proof for a linear Sa-GSC is simply an order dual.

Assume first conditions (a) and (b.i) hold. For each player i N, given qj Qj for j = i (hence, zi), ∈ ∈ 6 as the payoffs are upper semicontinuous in xi, continuous in zi, and the action space is compact, by the generalized maximum theorem (Ausubel and Deneckere ([8], Theorem 2), her best reply correspondence is

i Xi∗(zi, t) = arg max Πq(xi, gi(xi) + zi, t) xi Xi ∈ is nonempty, uhc, and compact-valued for each (q i, t) Q i T. Further, as the action space is a sublattice, − ∈ − × i and Πq(xi, gi(xi) + zi, t) is (i) supermodular in xi, and (ii) has decreasing differences in (xi; zi), each t, by 15 Topkis’theorem (e.g., Topkis ([73], Theorem 2.8.2), Xi∗(zi; t) is strong set order descending in zi, such

15 Strong set order "descending" means strong set order ascending in the dual order on zi Zi,Zi = zi qj gj (xj ), xj Xi, ∈ { | − ∈

14 that X∗(zi; t) and X∗(zi; t) are decreasing selections. ∨ i ∧ i Define the subaggregate best reply correspondence

Q∗(zi; t) = q∗(zi; t) q∗(zi; t) = gi(x∗(zi; t)), x∗(zi; t) X∗(zi; t) i { i | i i i ∈ i }

As the subaggregate function gi(xi) is continuous and increasing, and X∗(zi; t) and X∗(zi; t) are decreas- ∨ i ∧ i ing selections of X∗(zi; t), and the subaggregate function gi(xi) is increasing, Q∗(zi; t) and Q∗(zi; t) are i ∨ i ∧ i decreasing selections of zi, each t. Finally, as the subaggregate best reply only depends on zi = j=i qj, the 6 Sa-GSS is a linear aggregative game in subaggregates. That proves the lemma for the conditionsP (a) and (b.i).

Next, assume conditions (a) and (b.ii) hold. For each player i N, given qj Qj for j = i, her best ∈ ∈ 6 reply correspondence is

a b i Xi∗(zi, t) = (Xi ∗,Xi ∗)(zi; t) = arg max Πq(xi, gi(xi) + zi, t) xi Xi ∈ which is again by the generalized maximum theorem is nonempty and compact-valued for each (q i, t) − ∈ i Q i T. Further, as the action space is a sublattice, and Πq(xi, gi(xi) + zi, t) is (i) supermodular in xi, (ii) − × a b has decreasing differences in (xi ; zi), and has (iii) increasing differences in (xi ; zi) each t, again by Topkis’ a b theorem, Xi ∗(zi; t) (resp., Xi ∗(zi; t)) is strong set order descending (resp., strong set order ascending) in a a b b zi, such X ∗(zi; t) and X ∗(zi; t) (resp., X ∗(zi; t) and X ∗(zi; t) ) are decreasing (resp., increasing) ∨ i ∧ i ∨ i ∧ i selections.

Define the subaggregate correspondence

a b a b Q∗(zi; t) = (Q ∗,Q ∗)(zi; t) = q∗(zi; t) q∗(zi; t) = gi(x ∗(zi; t), x ∗(zi; t)), x∗(zi; t) X∗(zi; t) i i i { i | i i i i ∈ i }

a b As the subaggregate function gi(xi) is continuous, increasing in xi , and decreasing in xi ,and the greatest a a b b and least selections X ∗(zi; t) and X ∗(zi; t) are decreasing in zi (resp., X ∗(zi; t) and X ∗(zi; t) are ∨ i ∧ i ∨ i ∧ i increasing in zi), the greatest and least selections Q∗(zi; t) and Q∗(zi; t) are decreasing selections in zi, ∨ i ∧ i each t. Finally, again, as the subaggregate best reply only depends on zi = j=i qj, the Sa-GSS is a linear 6 aggregative game in subaggregates. That completes the proof of the lemmaP under conditions (a) and (b.ii). Finally, for conditions (a) and (b.iii), the result of the lemma follows for the "mixed heterogeneity case" where N1 players satisfy (a) and (b.i) and N2 players satisfy (a) and (b.ii) follows from the fact that for each zi = j=i qj Y. 6 } ⊂ P

15 player i, their subaggregate correspondence:

a b a b Q∗(zi; t) = (Q ∗,Q ∗)(zi; t) = q∗(zi; t) q∗(zi; t) = gi(x ∗(zi; t), x ∗(zi; t)), x∗(zi; t) X∗(zi; t) i i i { i | i i i i ∈ i }

has Q∗(zi; t) and Q∗(zi; t) decreasing in zi, each t T (as the N1 players have the normal-form best ∨ i ∧ i ∈ reply correspondence of the game under (a) and (b.i), and the N2 players have the normal-form best reply

correspondences for the game under (a) and (b.ii), each leading to Q∗(zi; t) and Q∗(zi; t) decreasing in ∨ i ∧ i

zi = j=i qj, each t T. 6 ∈ WeP make an number of important remarks per lemma 6. relative to its importance to the next section of the paper concerning the existence of Nash equilibrium. Although in the normal-form structure of the

original aggregative game we do require each player to be supermodular in their own normal-form actions, the

key point is we do not require the normal-form game to have require decreasing or increasing differences with

the normal-form strategies of other players. Indeed, at a normal-form level, the game is neither necessarily a

GSS or GSC. Therefore, per the Nash equilibrium existence question for linear Sa-GSS and linear Sa-GSC in

i the next section of the paper, the assumptions on the normal-form aggregative game (U ,Xi)i N , φ, t T { ∈ ∈ } that have subaggregative representations via Lemma 3. via Lemma 6. differs from the approach to the

existence question in discontinuous games in Prokopovych and Yannelis ([57]). That is, the subaggregative

i representation of the aggregative game (U ,Xi)i N , φ, t T has a standard single-crossing conditions; yet, { ∈ ∈ } the the normal-form game need to be a GSS or GSC (or one with a "directional" single crossing condition as in their work, or the work of Heikkila and Reffett ([33]).

Also, when the subaggregate functions gi(xi) are decreasing functions, then it "order reverses" the complementarity structure of the subaggregate game. That is, for games with primitives given by Lemma

i 6. imply the aggregative game (U ,Xi)i N , φ, t T admits a linear Sa-GSS that is a linear aggregative { ∈ ∈ } game in subaggregates, if the subaggregate functions are changed to have gi(xi) decreasing, the resulting subaggregative representation of the game will be linear Sa-GSC that is a linear aggregative game in sub-

aggregates. Actually, in that case, the normal-form they will be GSC also. Dually, if the primitives of the

game in Lemma 6. implies the subaggregative representation is a linear Sa-GSC, if gi(xi) is now decreasing, the new game will be a linear Sa-GSS.

Also, we should mention, although an aggregative game that admits a generalized aggregator φ(x) that

is additionally a normal-form GSS (resp., GSC) is an example of linear Sa-GSC (resp., linear Sa-GSS) (i.e.,

if the subaggregates are each increasing functions), the converse need not be the case. This is an example of

a linear Sa-GSS (resp., linear Sa-GSC) that is aggregative under condition (b.ii) of Lemma 6.:

16 Example 7. Linear Sa-GSS (resp, linear Sa-GSC) are not GSS (resp, GSC) : Consider the

i following subaggregative game: y = i gi(xi), qi = gi(xi),and each player has a real-valued payoffs U (xi) + a 2 Ui (gi(xi) + zi) for i N, where zPi =i j=i qj, xi Xi E+, Xi is a nonempty, convex, compact, ∈ 6 ∈ ⊂ i a sublattice. Assume U (xi) is continuous andP supermodular, U (y) strictly increasing and strictly concave

a b (resp., strictly increasing and strictly convex), and gi(xi) = x x for all i N, so player i’s payoff is i − i ∈ i a a b q U (xi)+U ((xi xi )+ j=i qj). This payoff function satisfies the conditions of Lemma 6.. That is, if U (y) − 6 is strictly concave (resp,P strictly convex), it is supermodular in xi, and also has decreasing differences (resp,

a b increasing differences) in (xi , zi) and increasing (resp., decreasing) differences in (xi ; zi). Hence, by Lemma 3. this aggregate game admits a subaggregate representation, and by Lemma 6. it is a linear Sa-GSS (resp.,

linear Sa-GSC) that is aggregative in subaggregates. This game is never a GSS or GSC.

III. Comparing Nash Equilibrium in Subaggregative Games with

Complementarities/Substitutes

We now prove two new results on the existence of pure-strategy NE equilibrium in Sa-GSS and Sa-GSC

that are aggregative in subaggregates for games with complete information. Our results also imply results

on aggregate comparative statics in both cases. The proof of existence of Sa-GSC can easily be extended

to linear Sa-GSC of incomplete information by adapting the results of Van Zandt ([74]) to the interim

Bayesian aggregative game induced by a linear Sa-GSC after each player choose her normal-form action

(xi)i N (Xi)i N . ∈ ∈ ∈ We first prove the following existence result for pure-strategy Nash equilibrium in linear Sa-GSS.

i Theorem 8. Let (Π ,Xi)i N , φ, t T be an aggregative game with generalized aggregator φ(x) = { ∈ ∈ } H( qi)), qi = gi(xi) for all i N satisfies conditions (a) and (b.i), (b.ii), or (b.iii) of Lemma 6. for ∈ i theP case of a linear Sa-GSS. Then, the aggregative game (Π ,Xi)i N , φ, t T has pure-strategy Nash { ∈ ∈ } equilibrium.

i Proof. As (Π ,Xi)i N , φ, t T is an aggregative game with generalized aggregator φ(x) = H( qi)), { ∈ ∈ } qi = gi(xi) for all i N that satisfies the assumptions of Lemma 6.. Then, by Lemma 6., player i PN has ∈ ∈ a subaggregate best reply correspondence given by

Q∗(zi; t) = q∗(zi; t) q∗(zi; t) = gi(x∗(zi; t)), x∗(zi; t) X∗(zi; t) i { i | i i i ∈ i }

17 where her normal-form best reply map is

i Xi∗(zi, t) = arg max Πq(xi, gi(xi) + zi, t) xi Xi ∈

Define

Q∗(q, t) = (Q1∗(z1, t), ..., Qn∗ (zn, t))

which for a linear Sa-GSS has Q∗(z; t) and Q∗(q(z; t) decreasing selections of Q∗(q, t) in in z = (z1, ..., zn), ∨ ∧

zi = j=i qj. Then, by either Kukushkin ([39] Propositions 1-3) or Jensen ([35], Corollary 1), the game has 6 a pureP strategy Nash equilibrium q∗(t) = (q∗(t), q∗(t), .., q∗ (t)) Ψq(t) in subaggregates. By Lemma 3., the 1 2 n ∈ set of pure strategy normal-form Nash equilibrium x∗(t) Ψx(t) is non-empty. ∈ Notice, this result is not contained in either KuKushkin’s or Jensen’s result as we allow (a) the action

spaces in the normal-form game to be compact Hausdorff topological lattices Xi for each player i N, and ∈ i (b) the normal-form game itself (Π ,Xi, gi)i N , H, t T is not necessarily a aggregative GSS. It is also not { ∈ ∈ } contained in Prokopovych and Yannelis ([57]) as action spaces are not totally ordered, and in normal-form, we the game does not necessarily have directional single crossing conditions.

Remark 9. The proof of the existence result in Lemma 8. is built on the subaggregative representation of

i aggregate game (Π ,Xi, gi)i N , H, t T . That subaggregative representation of the aggregative game is { ∈ ∈ } also aggregative. Therefore, we immediately arrive at the conclusions per the existence of least and greatest aggregate comparative statics results reported in Acemoglu and Jensen ([2], Theorem 2-4) concerning "shocks to the aggregator" for our linear SaGSS in Theorem 8.. In particular, they hold on this aggregative games studied in Lemma 8.. Notice, as our games are not necessary normal-form GSS, that result is not contains in Acemoglu and Jensen’s results (as the application of their arguments must be made to the subaggregate

representation of the game, and the underlying game in normal-form does not have to be a GSS.

Next, we turn to aggregative games which are linear Sa-GSC.16

i Theorem 10. Let (Π ,Xi)i N , φ, t T be an aggregative game with generalized aggregator φ(x) = { ∈ ∈ } H( qi)), qi = gi(xi) for all i N which satisfies assumptions (a) and (b.i), (b.ii), or (b.iii) of Lemma 6. for ∈ i linearP Sa-GSC. Then, (a) the game (Π ,Xi)i N , φ, t T has a pure-strategy Nash equilibrium. Further, { ∈ ∈ } 16 If each subaggregate function gi(xi) is additionally order preserving, and gi(xi) is assumed to be order continuous, Q is a n complete lattice in E . This very relevant in the proof/result in Theorem 10.. In particular, as Q is a complete lattice, Q∗(q, t) will be subcomplete latticed-valued, and therefore set of subaggregate Nash equilibria is a nonempty complete lattice for each parameter t T, by Veinott’s version of Tarski (e.g., see Veinott ([75]), chapter 4, Theorem 10). The fixed point comparative statics result∈ also follows from the same theorem in Veinott.

18 i if T is a chain, and U (xi, y; t) have increasing differences in (xi, y; t), the least and greatest subaggregates

(resp, aggregates) q∗(t) and q∗(t), (resp., y∗(t) and y∗(t)) are increasing in t. ∧ i ∨ i ∧ ∨

i Proof. (a) fix t T. As (Π ,Xi)i N , φ, t T is an aggregative game with generalized aggregator ∈ { ∈ ∈ } φ(x) = H( qi)), qi = gi(xi) for all i N that satisfies the assumptions of Lemma 6. for both conditions (a) ∈ and (b.i), orP (b.ii), or (b.iii) for the lin Sa-GSC, player i N has a subaggregate best reply correspondence ∈ given by

Q∗(zi; t) = q∗(zi; t) q∗(zi; t) = gi(x∗(zi; t)), x∗(zi; t) X∗(zi; t) i { i | i i i ∈ i } where her normal-form best reply map is

i Xi∗(zi, t) = arg max Πq(xi, gi(xi) + zi, t) xi Xi ∈ such that Q∗(zi; t) and Q∗(zi; t) are increasing selections in zi, each t T. Define the mapping: ∨ i ∧ i ∈

Q∗ (q, t) = Q∗(q, t), Q∗(q, t) f {∧ ∨ } which for a Sa-GSC is strong set order ascending in q, and subchained valued for each t T. As Q is ∈ chain complete, By Heikkila and Reffett’s version of Markowsky’s theorem (e.g., Heikkila and Reffett ([33,

Theorem 2.1 and Theorem 2.2), the set of fixed points of Q∗(q; t), denoted by Ψq∗(t), has least and greatest 17 selections Ψ∗(t) and Ψ∗(t) both existing. By Lemma 3., the set of normal-from pure-strategy Nash ∧ q ∨ q equilibria are then given by the following: x∗(q∗(t), t) X∗(q∗(t), t) Ψ∗ (t), q∗(t) Ψ∗(t). ∈ ∈ x ∈ q (b) Under our assumptions of Lemma 6. for conditions (a) and (b.i), we add the assumption that

i Πq(xi, gi(xi) + zi, t) also has increasing differences in (xi; t), each zi. For the case of condition (a) and (b.ii), i a b a b a we add the assumption that Πq(xi , xi , gi(xi , xi ) + zi, t) has increasing differences in (xi ; t) and decreasing b differences in (xi ; t). For conditions (a) and (b.iii), we add the obvious corresponding mixture of conditions above for (a) and (b.i) (for players N1) and (a) and (b.ii) (for players N2).

Then, for each player i N = N1 N2, given qj Qj for j = i, her best reply correspondence is ∈ ∪ ∈ 6

a b i Xi∗(zi, t) = (Xi ∗,Xi ∗)(zi; t) = arg max Πq(xi, gi(xi) + zi, t) xi Xi ∈

a b which in addition to the properties of Xi∗(zi, t) = (Xi ∗,XI∗)(zi, t) in zi proved in Lemma 6. for the linear 17 Actually, by a simple extension of Markowsky’stheorem ([45, Theorem 9), you can show the set of fixed points is also chain complete.

19 a b Sa-GSC case, by Topkis’theorem, Xi ∗(zi, t) (resp., Xi ∗(zi.t)) is now additionally strong set order ascending a a b b (resp., descending) in t, such X ∗(zi, t) and X ∗(zi, t) (resp., X ∗(zi, t) and X ∗(zi, t) ) are increasing ∨ i ∧ i ∨ i ∧ i (resp., decreasing) selections. Recall, the subaggregate correspondence in this situation is given by

a b a b Q∗(zi, t) = (Q ∗,Q ∗)(zi, t) = q∗(zi, t) q∗(zi, t) = gi(x ∗(zi, t), x ∗(zi, t)), x∗(zi, t) X∗(zi; t) i i i { i | i i i i ∈ i }

with Q∗(zi, t) and Q∗(zi, t) now increasing selections of (zi, t) jointly. ∨ i ∧ i Define the mapping

Q∗ (q, t) = Q∗(q, t), Q∗(q, t) f {∧ ∨ }

where by construction, the correspondence Q∗ : Q T Q is now subcomplete valued, and strong set order f × ⇒ ascending in (q, t) Q T. Then, by the fixed point comparative statics result in Heikkila and Reffett’s ∈ × extension of Veinott’sfixed point comparative statics theorem (again, Heikkila and Reffett ([33], Theorems

2.1 and 2.2), we now have Ψ∗(t) and Ψ∗(t) increasing equilibrium subaggregative selections. ∧ q ∨ q Finally, as y∗(t) = H( q∗(t)) (resp., y∗(t) = H( q∗(t)), the equilibrium aggregates are also ∧ ∧ i ∨ ∨ i increasing in t. P P This last result is very important. For example, we will be able to apply our existence result guarantee our applications of aggregate and subaggregate lattice programming techniques to interim Bayesian subag- gregate GSCs are not vacuous. Further, as obtaining existence results for pure strategy Nash equilibrium in discontinuous Bayesian games without "monotone in type" equilibrium is a well-known question, such an existence result provides an important extension of the results of Van Zandt ([74]) to interim Bayesian games that are not necessarily an interim Bayesian supermodular games.

IV. Appendix: Lattices, Posets, and Mappings

We begin by discussing some terminology we use in the paper and their relevance to the constructions in the paper. In this appendix, we define the class of objective functions with complementarity structure over lattices in their most general form (i.e., in their extended-real valued form). We do this to make clear that as long as our payoffs in the paper are of the superextremal class mapping to a posemigroup (P, , ) ∗ ≥ with suffi ciently properly increasing, all the results of the paper go through. ∗

Sets and Relations. Let X and Y be sets. A binary relation is a mapping on a set X defined as

: X X X. A semigroup is a pair (X, ) where is an associative binary operation. A partially ordered ∗ × → ∗ ∗

20 set (or Poset) P is a set equipped with an relation that is transitive, reflexive, and antisymmetric. A ≥ posemigroup is a triple (P, , ) where (P, ) is a semigroup and (P, ) is a poset. If (P, ) is a poset, ≥ ∗ ∗ ≥ ≥ subset C P is a chain if it is linearly ordered (i.e., all elements x, y C are ordered). ⊂ ∈ Let (L, ) be a poset. A upper (resp. lower) bound for a set X L is any element xu(resp. xl) L ≥ ⊂ ∈ such that xu majorizes (minorizes) each point in X. If there is a point xu (resp, xl) such that xu is the least element in a subset of upper bounds of X L (resp, the greatest element in a subset of lower bounds of ⊂ X L), we say xu (resp, x;) is a supremum (resp, infimum) of A. We say a poset (L, ) is a lattice X if ⊂ ≥ for any two elements x and x0 in L, L is closed under the operation of infimum in L , denoted x x0, and ∧ supremum in L, denoted x x0. A subset of a lattice is complete (resp, meet-complete, join-complete) if any ∨ subset X of L, X and X are in L (resp., X is in , X is in L). A subset X of L is a sublattice (resp, ∨ ∧ ∧ ∨ meet-sublattice, join-sublattice) of L contains the sup and the inf (resp, meet, sup) with respect to L of any pair of points in L. If X is only complete for any sequence S L (i.e, we only have S and S in L), then ⊂ ∨ ∧ L is σ-complete. If a sublattice S in X is complete, S is subcomplete in X.

Mappings in Posets. Let (P1, ) and (P2, 0) be two posets. A function f : P1 P2 is isotone on ≥ ≥ → X , f(x0) 0 f(x), when x0 x, for x, x0 P. If f(x0) 0 f(x), f(x0) = f(x) when x0 x for x = x0, for ≥ ≥ ∈ ≥ 6 ≥ 6 x0, x P we say the function f is increasing. If f(x0) >0 f(x), when x0 > x, for x0, x P, we say the function ∈ ∈ f is strictly increasing. Dually, a function f(x) is antitone ( resp, decreasing, and strictly decreasing). A function that is either isotone or antitone is monotone.

In some cases, correspondences can be viewed as isotone mappings. Let (X, ) be a lattice, 2X the ≥ powerset of X,L(X) to be the set of all the nonempty sublattices of X. We define the following order relation on compatible with (X, ), namely, the so-called strong set order. Let X1 and X2 be elements of ≥ L(X). We say X1 is lower than X2 in Veinott’s strong set order if for all x1 X1, x2 X2, the (a) meet ∈ ∈ x1 x2 X1 and the (b) join x1 x2 X2. In this case, we denote this by X2 v X1 , where v is Veinott’s ∧ ∈ ∨ ∈   strong set order. It is well-known that the set (L(X), ) is the largest poset is 2X. (e.g., Veinott ([75]), v 18 19 Theorem 1, Chapter 4). We shall refer to Lv(X) = (L(X), ) as Veinott’s Poset Powerdomain. v We can define various notations of "ascending correspondences". Let X and T be posets, 2X be a powerset, F : T X, (2X, ) the powerset equipped with an order relation . We say F (t) is ascending ⇒    − (resp, descending) if F (t0) F (t) (resp., F (t) F (t0)) when t0 t. Researchers studying constrained  −   ≥ monotone comparatives have proposed alternative methods of ordering subclasses of 2X (where X in this

18 The strong set order is the union of two other set orders, both inducing poset order on subclasses of 2X . In particular, if the base set X is a meet lattice (resp., join) lattice, then one can compare meet (resp., join) sublattices by (a) (resp, (b)). The resulting pair of relations define the "lower meet" (resp., "lower join" ) set relation discussed in Veinott ([75]). 19 In general, Lv(X) is only a poset; if X is a completely distributive complete lattice, then an order relation on Lv(X) can be introduced such that Lv(X) is a lattice (actually, a completely distributive complete lattice). See Veinott ([75], chapter 2).

21 case represents a choice space, T a set or parameters). In all cases, these order relations induce their order

structure for at least the poset structure of X; often, they also use the lattice structure of X). As needed, we shall define and to these various orders later in the paper as needed.

In our work we shall always be working with lattice-induced partial orders (e.g., the strong set order).

When the order relation induces a partial order on a subclass P (X) 2X, we say F (t) is ascending in such  ⊂ an order relation, we shall say F (t) is an isotone (resp, antitone) correspondence. For our work, this is case

when F (t) is ascending in vThat is, let Lv(X) be a Veinott Poset Powerdomain for the lattice X, (T, ) a  ≥ poset, and define Z = X T with (X, ) given its product order. Then F : T Lv(X) is strong set order × ≥ → ascending (resp., descending), it is precisely an isotone (resp., antitone) mapping.

When studying the complementarity properties of value functions in parameterized optimization prob-

lems, its important to consider the sublatticed-graph structure of a correspondence. This is a stronger

property than strong order increasing. Let Z be a lattice, X T a sublattice. Then F : T L(X) has a × → sublatticed-graph if for t, t0 T, any x F (t), x0 F (t0), x x0 F (t t0) and x x0 F (t t0). ∈ ∈ ∈ ∧ ∈ ∧ ∨ ∈ ∨

Extended-real Valued Supermodular Functions on a Lattice. Let L is a lattice, f : L P, P → a chain. Let (P, , ) be a posemigroup for = +, , , . A function f (x) is super (resp, sub ) from a ∗ ≥ ∗ { · ∨ ∧} ∗ ∗ lattice L to a chain P if for all x, y X ∈

f(x y) f(x y) f(x) f(y) (resp, ) ∧ ∗ ∨ ≥ ∗ ≤

Let (P, , ) be a semigroup, a, b, c, d X. We say is increasing (resp, properly increasing; strictly increas- ∗ ≥ ∈ ∗ ing) if: 20

(a, b) (c, d)(resp, a < c, b < d;(a, b) < (c, d)) ≤

⇒ a b c d (resp, a b < c d; a b < c d) ∗ ≤ ∗ ∗ ∗ ∗ ∗

In this note, we shall primarily focus on super functions where =+, P R∗ (the extended-real line), ∗ ∗ ⊂ with the super function associated with a posemigroup where at least properly increasing. In the case of ∗ ∗ a properly increasing operation, we shall refer to the function as super function. So, for example, in this ∗ ∗ paper our concern is about extended real valued super functions with = + (i.e., extended-real valued ∗ ∗ supermodular functions).

20 The notation (a, b) < (c, d) means (a, b) (c, d), (a, b) = (c, d) • ≤ 6

22 A real-valued function f : X R is supermodular (resp., strictly supermodular) in x if (x, y) X X, → ∀ ∈ × we have

(6) f(x y) + f(x y) (resp., >) f(x) + f(y) ∨ ∧ ≥

A function f is submodular (resp., strictly submodular) the inequalities in (6) are reversed. In this sense, one can view a supermodular function (resp, strictly supermodular function) as a super function from a ∗ lattice X to a posemigroup (P, , ) where P=R is a chain, and with the binary operator = +.21 ∗ ≥ ∗ We could consider "ordinal versions of these super (or superextremal) functions for our aggregative ∗ games. That is, for example, we could study the cases of payoffs , aggregates, and subaggregates that are super and/or superextremal (where = or instead of the supermodular case. The point is ∗ ∗ {∨ ∧} in this note, as we are working extensively with additively separable aggregators, so ordinal cases do not seem to be of special interest. But we could restate all the results for games of complete information using ordinal notions of complementarities (e.g., see Reny’sremarks on the need for "ordinal payoff" conditions in discontinuous games).

Dually, we can define an extended real-valued submodular (resp., strictly submodular) function from a lattice X to a posemigroup (R∗ , +, ). + ≥ Next, consider a partially ordered set Ψ = X1 P (with order ), and B X1 P . The function × ≥ ⊂ × f : B R has increasing differences (resp, strict increasing differences) in (x1, p) if for all p1, p2 P , −→ ∈ p1 p2 = f(x, p2) f(x, p1) is non-decreasing (strictly increasing) in x Bp , where Bp is the p section ≤ ⇒ − ∈ 1 of B. If these differences are non-increasing (resp, strictly decreasing), we say f has dual (resp., strict dual) increasing differences.22

Key Theorems on Monotone Controls and Preserving Complementarities to Value func- tions. We state two key theorems we use repeatedly throughout this note.

Proposition 11. Topkis’Theorem. Let T be a poset, X a lattice, Γ be a nonempty correspondence,

Γ: T Lv(X) strong set order ascending, and f(x, t) an extended real-valued supermodular function of x → with increasing differences in (x; t). Then (i) X∗(t) is strong set order increasing in t (hence, sublatticed- valued) with X∗(t) and X∗(t) isotone selections. ∨ ∧

Proposition 12. Topkis Value Function Projection Theorem: Let T be a lattice, X a lattice, X T × given its product order, Γ be a nonempty correspondence, Γ: T Lv(X) that has a sublatticed-graph, and → 21 See LiCalzi and Veinott ([44]) and Veinott ([75, chapter 3] for extensive discussion of super * and superextremal functions. 22 Sometimes, less formally, we shall say f has decreasing (resp., strict decreasing) differences.

23 f(x, t) an (extended real-valued) quasisupermodular function on X T . Then V (t) is supermodular. ×

Both of these results are proven for the case of extended-real valued objective functions in LiCalzi and

Veinott ([44], Theorem 3 and corollary 11, respectively) for the case = +. When the posemigroup for ∗ payoffs is (R, +, ), Proposition 11. is Topkis (([73, Theorem 2.8.2), and Proposition 12. is in Topkis ([73], ≥ theorem 2.7.6).

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