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On Nash Equilibrium in Discontinuous Aggregative Games

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On Nash Equilibrium in Discontinuous Aggregative Games Existence of Nash equilibrium in Discontinuous Subaggregative Games Fereshteh Mohseninejad y Kevin Reffett z 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. yDepartment of Economics, Queen Mary University of London, School of Economics. Corresponding author: [email protected] zDepartment 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
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