DOCSLIB.ORG

Large Population Aggregative Potential Games∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Large Population Aggregative Potential Games∗ Large Population Aggregative Potential Games∗ Ratul Lahkary April 27, 2016 Abstract We consider population games in which payoff depends upon the aggregate strategy level and which admit a potential function. Examples of such aggregative potential games include the tragedy of the commons and the Cournot competition model. These games are technically simple as they can be analyzed using a one{dimensional variant of the potential function. We use such games to model the presence of externalities, both positive and negative. We characterize Nash equilibria in such games as socially inefficient. Evolutionary dynamics in such games converge to socially inefficient Nash equilibria. Keywords: Externalities, Aggregative Games, Potential Games, Evolutionary Dynamics. JEL classification: C72; C73; D62. ∗I thank an anonymous associate editor and two anonymous referees for various comments and suggestions. Any remaining error or omission is my responsibility. ySchool of Economics, Ashoka University, Rajiv Gandhi Education City, Kundli, Haryana, 131 028, India. email: [email protected]. 1 1 Introduction Potential games are games in which information about players' payoffs can be summarized using a real valued function. Monderer and Shapley (1996) define and analyze the fundamental properties of a normal form potential game. Sandholm (2001, 2009) extends the notion of a potential game to population games. In the context of a population game, a potential game is one in which payoffs are equal to the gradient of a real valued function called the potential function. Potential games are of interest in evolutionary game theory because a variety of evolutionary dynamics converge to Nash equilibria in such games.1 In this paper, we consider a particularly simple class of potential games which we call aggregative potential games. These are potential games which belong to the class of aggregative population games, i.e. population games in which payoffs depend upon the strategy used, and the aggregate strategy level at a population state. This notion of an aggregative population game is an extension of the original concept introduced by Corch´on(1994) and elaborated further by, for example, Acemoglu and Jensen (2013), in the context of finite player games. We describe aggregative potential games as \simple" because, as we show here, they can be analyzed using a one{dimensional analogue of the potential function, which we call the quasi{ potential function. In general, Nash equilibria of a potential game are related to the maximizers of its potential function (Sandholm, 2001). For aggregative potential games, the maximizers of the potential function are well approximated by the maximizers of the quasi{potential function, partic- ularly if the strategy set of the underlying game is sufficiently dense. Since finding the maximizers of the quasi{potential function is a trivial task in comparison to computing the maximizers of the potential function, identifying Nash equilibria in aggregative potential games becomes particularly easy. In addition to their technical simplicity, aggregative potential games are of interest because they include important economic models like the tragedy of the commons, the model of Cournot competition and models of search. Further, aggregative potential games provide a parsimonious framework to model both negative and positive externalities in the context of population games. This allows us to use existing results on evolution in potential games to explain how, in the presence of externalities, societies may converge to inefficient aggregate economic outcomes. We consider aggregative potential games with a finite set of positive strategies. We analyze two types of such games|one with negative externalities and the other with positive externalities. Our results on games with negative externalities are particularly interesting. In this case, under reasonable assumptions, the equilibrium aggregate strategy level is uniquely defined. This equilib- rium aggregate strategy level is either exactly characterized by or well{approximated by the unique maximizer of the quasi{potential function. Games with positive externalities do not have unique equilibrium aggregate strategy levels. Nor is their characterization as elegant as in the case of neg- ative externalities. Nevertheless, with some relatively strong assumptions on the quasi{potential 1See Sandholm (2010) for a review of such results. 1 function, we are able to relate the equilibrium aggregate strategy levels under positive externalities to the maximizers of the quasi{potential function. Externalities imply a difference between Nash equilibria and the socially efficient state, i.e. the state that maximizes aggregate payoff in the population. For aggregative potential games, we estab- lish this distinction by using a technique similar to characterizing Nash equilibria in these games. We construct a one-dimensional analogue to the aggregate payoff and show that the maximizer of this function corresponds to the socially efficient state, either exactly or with a high degree of approximation. We then show that under negative externalities, the unique equilibrium aggregate social state is higher than the socially efficient aggregate state. Under positive externalities, aggre- gate states at equilibria are lower than the socially efficient state. Such conclusions are, of course, consistent with standard microeconomic theory. The novelty lies in the technique through which we arrive at these conclusions. Use of the one{dimensional analogues of the potential function and the aggregate payoff function makes it particularly easy to relate Nash equilibria and efficient states in aggregative potential games. We can then apply standard results on evolution in potential games (Sandholm, 2001) to con- clude that the social state in aggregative potential games converge to a state that is different from the socially efficient state. Deterministic evolutionary dynamics like the replicator dynamic, the logit dynamic, the BNN dynamic and the Smith dynamic converge either to Nash equilibria or to a perturbed version of Nash equilibria in potential games from all or almost all initial states. In aggregative potential games, therefore, all such dynamics converge to an aggregate social state that differs from the socially efficient aggregate state. From a broader perspective, the evolutionary analysis of aggregative potential games allows us to appreciate how inefficient aggregate behavior becomes prevalent in societies. If we regard the revision protocols that generate the dynamics commonly studied in evolutionary game theory as reasonable descriptions of human behavior, then our analysis helps explain why we should expect social inefficiency under a wide variety of behavioral norms and from a wide range of initial conditions. There are precedents to the evolutionary analysis of the models considered here. For example, Vega{Redondo (1997) and Al´os{Ferrer and Ania (2005) consider an imitative model of Cournot competition with a finite population of players. Their main result is that the evolutionarily stable strategy (ESS) in the finite population model is not the Nash equilibrium of the model, but the Walrasian or competitive equilibrium. The key reason why Nash equilibria are not finite population ESS is that a strategy change by single player affects a finite population state. Hence, when a player playing, say, strategy i observes the payoff of strategy j he is seeking to imitate, he is not observing the payoff that he would actually obtain when he changes the strategy to j. This is because when the i−player shifts to j, the number of j−players increases by one, thereby changing the payoff to strategy j. The failure to appreciate this distinction may make it advantageous to imitate a mutant and change strategy at a Nash equilibrium in a finite population Cournot model. Al´os{Ferrer and Ania (2005) also extend this result on finite population ESS to other models like the tragedy of the 2 commons and the search model that we have considered here. Schaffer (1988) notes an important implication of the finite population ESS. It is as if players seek to maximize not absolute payoffs but relative payoffs. Thus, deviation from a Nash equilibrium becomes worthwhile because even though it may reduce one's absolute payoff, it can reduce oppo- nents' payoff even more, thereby increasing relative payoff. Another significant model that explores the distinction between Nash equilibria and finite population ESS under imitative learning due to the concern with relative payoff is Bergin and Bernhardt (2004). In their model, this distinction does not arise if agents' learn from one's own experience instead of imitating the experience of others. Unlike the models discussed in the previous two paragraphs, in our paper, only Nash equilib- ria can be evolutionarily stable. This is because in an infinite population model such as ours, a strategy change by a single agent has no effect on the population state and on payoffs. Hence, the difference between the payoff observed before a strategy change and the payoff obtained after the strategy change cannot arise in this context. In terms of Schaffer’s (1988) interpretation, concern with relative payoffs is irrelevant and so, deviation from a Nash equilibrium is never worthwhile. We should note, however, a stable Nash equilibrium in our model coincides with the finite popu- lation ESS identified in these other models. Thus, in our Cournot model, Nash equilibria are also Walrasian equilibria. Intuitively, this is because with an infinite
Load more

Recommended publications
  • Aggregate Comparative Statics
    This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights Author's personal copy Games and Economic Behavior 81 (2013) 27–49 Contents lists available at SciVerse ScienceDirect Games and Economic Behavior www.elsevier.com/locate/geb Aggregate comparative statics ✩ ∗ Daron Acemoglu a, , Martin Kaae Jensen b a Department of Economics, Massachusetts Institute of Technology, United States b Department of Economics, University of Birmingham, United Kingdom article info abstract Article history: In aggregative games, each player’s payoff depends on her own actions and an aggregate of Received 31 August 2011 the actions of all the players. Many common games in industrial organization, political Available online 23 April 2013 economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in JEL classification: aggregative games. We focus on two classes of games: (1) aggregative games with strategic C60 C72 substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies.
    [Show full text]
  • Bounded Rationality in Keynesian Beauty Contests: a Lesson for Central Bankers?
    Discussion Paper No. 2019-53 | November 06, 2019 | http://www.economics-ejournal.org/economics/discussionpapers/2019-53 Please cite the corresponding Journal Article at http://www.economics-ejournal.org/economics/journalarticles/2020-16 Bounded rationality in Keynesian beauty contests: a lesson for central bankers? Felix Mauersberger, Rosemarie Nagel, and Christoph Bühren Abstract The goal of this paper is to show how adding behavioral components to micro-foundated models of macroeconomics may contribute to a better understanding of real world phenomena. The authors introduce the reader to variations of the Keynesian Beauty Contest (Keynes, The General Theory of Employment, Interest, and Money, 1936), theoretically and experimentally with a descriptive model of behavior. They bridge the discrepancies of (benchmark) solution concepts and bounded rationality through step-level reasoning, the so-called level-k or cognitive hierarchy models. These models have been recently used as building blocks for new behavioral macro theories to understand puzzles like the lacking rise of inflation after the financial crisis, the effectiveness of quantitative easing, the forward guidance puzzle and the effectiveness of temporary fiscal expansion. (Published in Special Issue Bio-psycho-social foundations of macroeconomics) JEL E12 E13 E7 D80 D9 C91 Keywords Beauty contest game; expectation formation; equilibration; level-k reasoning; macroeconomics; game theory; experimental economics Authors Felix Mauersberger, University of Bonn, Germany Rosemarie Nagel, ICREA-UPF Barcelona GSE, Spain, [email protected] Christoph Bühren, TU Clausthal, Germany Citation Felix Mauersberger, Rosemarie Nagel, and Christoph Bühren (2019). Bounded rationality in Keynesian beauty contests: a lesson for central bankers? Economics Discussion Papers, No 2019-53, Kiel Institute for the World Economy.
    [Show full text]
  • Aggregative Oligopoly Games with Entry1
    Aggregative Oligopoly Games with Entry1 Simon P. Anderson2, Nisvan Erkal3 and Daniel Piccinin4 First version: November 2009 This version: December 2016 1We thank David Byrne, Chris Edmond, Maxim Engers, Daniel Halbheer, Joe Harring- ton, Simon Loertscher, Phil McCalman, Claudio Mezzetti, Volker Nocke, Martin Peitz, Frank Stähler, Jidong Zhou, and especially Richard Cornes for comments and discussion. We also thank seminar participants at the Federal Trade Commission, Johns Hopkins Uni- versity, New York University, National University of Singapore, University of Mannheim, and conference participants at the Australian National University (Workshop in Honor of Richard Cornes), North American Winter Meeting of the Econometric Society (2013), Australasian Economic Theory Workshop (2011), EARIE (2010), Université Catholique de Louvain (Conference in Honor of Jacques Thisse) for their comments. Imogen Halstead, Jingong Huang, Boon Han Koh, Charles Murry, and Yiyi Zhou have provided excellent research assistance. The …rst author thanks National Science Foundation (0752923) for …nancial support and the Department of Economics at the University of Melbourne for its hospitality. The second author gratefully acknowledges funding from the Australian Research Council (DP0987070). 2Department of Economics, University of Virginia P.O. Box 400182, Charlottesville, VA 22904-4182, USA. [email protected]. 3Department of Economics, University of Melbourne, VIC 3010, Australia. [email protected]. 4Brick Court Chambers, 7-8 Essex Street, London, WC2R 3LD, United Kingdom. [email protected]. Abstract We compile an IO toolkit for aggregative games and use inclusive best reply functions to show strong neutrality properties for long-run equilibria across market structures. The IIA property of demand functions (CES and logit) implies that consumer surplus depends on the aggregate alone, and the Bertrand pricing game is aggregative.
    [Show full text]
  • Aggregative Oligopoly Games with Entry1
    Aggregative Oligopoly Games with Entry1 Simon P. Anderson2, Nisvan Erkal3 and Daniel Piccinin4 First version: November 2009 This version: February 2016 1We thank David Byrne, Chris Edmond, Maxim Engers, Daniel Halbheer, Joe Harring- ton, Simon Loertscher, Phil McCalman, Claudio Mezzetti, Volker Nocke, Martin Peitz, Frank Stähler, Jidong Zhou, and especially Richard Cornes for comments and discussion. We also thank seminar participants at the Federal Trade Commission, Johns Hopkins Uni- versity, New York University, National University of Singapore, University of Mannheim, and conference participants at Université Catholique de Louvain (Conference in Honor of Jacques Thisse), EARIE (2010), Australasian Economic Theory Workshop (2011), North American Winter Meeting of the Econometric Society (2013) for their comments. Imogen Halstead, Boon Han Koh, Charles Murry, and Yiyi Zhou have provided excellent research assistance. The …rst author thanks National Science Foundation (0752923) for …nancial support and the Department of Economics at the University of Melbourne for its hospi- tality. The second author gratefully acknowledges funding from the Australian Research Council (DP0987070). 2Department of Economics, University of Virginia P.O. Box 400182, Charlottesville, VA 22904-4182, USA. [email protected]. 3Department of Economics, University of Melbourne, VIC 3010, Australia. [email protected]. 4Brick Court Chambers, 7-8 Essex Street, London, WC2R 3LD, United Kingdom. [email protected]. Abstract We compile an IO toolkit for aggregative games and use inclusive best reply func- tions to compare long-run market structures. We show strong neutrality properties across market structures. The IIA property of demands (CES and logit) implies that consumer surplus depends on the aggregate alone, and the Bertrand pricing game is aggregative.
    [Show full text]
  • Aggregative Oligopoly Games with Entry1
    Aggregative Oligopoly Games with Entry1 Simon P. Anderson2, Nisvan Erkal3 and Daniel Piccinin4 First version: November 2009 This version: January 2014 1We thank Maxim Engers, Daniel Halbheer, Joe Harrington, Simon Loertscher, Clau- dio Mezzetti, Volker Nocke, Frank Stähler, Jidong Zhou, and especially Richard Cornes for comments and discussion. We also thank seminar participants at the Federal Trade Commis- sion, Johns Hopkins University, University of Mannheim, National University of Singapore, and conference participants at Université Catholique de Louvain (Conference in Honor of Jacques Thisse), EARIE (2010), Australasian Economic Theory Workshop (2011), North American Winter Meeting of the Econometric Society (2013) for their comments. Imogen Halstead, Charles Murry, and Yiyi Zhou have provided excellent research assistance. The …rst author thanks National Science Foundation (0752923) for …nancial support and the Department of Economics at the University of Melbourne for its hospitality. The second author gratefully acknowledges funding from the Australian Research Council (DP0987070). 2Department of Economics, University of Virginia P.O. Box 400182, Charlottesville, VA 22904-4182, USA. [email protected]. 3Department of Economics, University of Melbourne, VIC 3010, Australia. [email protected]. 4Brick Court Chambers, 7-8 Essex Street, London, WC2R 3LD, United Kingdom. [email protected]. Abstract We compile an IO toolkit for aggregative games and use inclusive reaction functions to compare long-run market structures in aggregative oligopoly games. We show strong neutrality properties across market structures. The IIA property of demands (CES and logit) implies that consumer surplus depends on the aggregate alone, and that the Bertrand pricing game is aggregative. We link together the following results: merging parties’pro…ts fall but consumer surplus is unchanged, Stackelberg leadership raises welfare, monopolistic competition is the market structure with the highest surplus.
    [Show full text]
  • Aggregative Games with Entry1
    Aggregative Games with Entry1 Simon P. Anderson2, Nisvan Erkal3 and Daniel Piccinin4 This version: May 2010 Extremely preliminary Please do not circulate without the authors’consent 1 We thank Daniel Halbheer, and seminar participants at the Federal Trade Commission and Johns Hopkins University for their comments. 2 Department of Economics, University of Virginia P.O. Box 400182, Charlottesville, VA 22904- 4182, USA. [email protected]. 3 Department of Economics, University of Melbourne, VIC 3010, Australia. [email protected]. 4 Linklaters LLP, One Silk Street, London, EC2Y 8HQ, United Kingdom. [email protected]. Abstract Aggregative games are used to model strategic interaction in many …elds of economics, including industrial organization, political economy, international trade, and public …nance. In such games, each player’s payo¤ depends on his/her own actions and an aggregate of all player’s actions. Examples in industrial organization are the Cournot oligopoly model, logit and CES di¤erentiated products Bertrand models (and linear demand models in the short run), and R&D games. We suppose a change a¤ects some of the players, such as cost shocks (subsidies, tari¤s), privatization, and a merger or RJV. In a unifying framework, we determine the impact of the change on the aggregate variable, producer surplus and consumer surplus under free entry. We also show that the IIA property of demands implies that consumer surplus depends on the aggregate alone, and the corresponding Bertrand pricing game is aggregative. JEL Classi…cations: D43, L13 Keywords: Aggregative games; entry; strategic substitutes and complements; IIA prop- erty. 1 Introduction Consider a mixed oligopoly industry of a public …rm and private …rms producing di¤eren- tiated products and demand characterized by the CES model and a free entry zero pro…t condition.
    [Show full text]
  • Game Theory and Management
    GRADUATE SCHOOL OF MANAGEMENT FACULTY OF APPLIED MATHEMATICS & CONTROL PROCESSES St. Petersburg University THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES (Russian Chapter) GAME THEORY AND MANAGEMENT The Fourth International Conference Game Theory and Management GTM2010 June 28-30, 2010, St. Petersburg, Russia ABSTRACTS Edited by Leon A. Petrosyan and Nikolay A. Zenkevich Graduate School of Management St. Petersburg University St. Petersburg 2010 УДК 518.9, 517.9, 681.3.07 GAME THEORY AND MANAGEMENT. Collected abstracts of papers presented on the Fourth International Conference Game Theory and Management / Editors Leon A. Petrosyan, Nikolay A. Zenkevich. – SPb.: Graduate School of Management SPbU, 2010. – 264 p. The collection contains abstracts of papers accepted for the International Conference Game Theory and Management (June 28–30, 2010, St. Petersburg University, Russia). The presented abstracts belong to the field of game theory and its applications to management. The abstract volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments. Computer design: Anna V. Iljina, Andrew V. Zyatchin © Copyright of the authors only, 2010 ТЕОРИЯ ИГР И МЕНЕДЖМЕНТ. Сб. тезисов 4-й международной конференции по теории игр и менеджменту / Под ред. Л. А Петросяна, Н. А. Зенкевича. – СПб.: Высшая школа менеджмента СПбГУ, 2010. – 264 с. Сборник содержит тезисы докладов участников 4-й международной конференции по теории игр и менеджменту (28–30 июня 2010 года, Высшая школа менеджмента, Санкт-Петербургский государственный университет, Россия). Представленные тезисы относятся к теории игр и её приложениям в менеджменте. Тезисы представляют интерес для научных работников, аспирантов и студентов старших курсов университетов, специализирующихся по менеджменту, экономике и прикладной математике.
    [Show full text]
  • Game Theory and Management
    ST. PETERSBURG UNIVERSITY Graduate School of Management Faculty of Applied Mathematics & Control Processes THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES (Russian Chapter) GAME THEORY AND MANAGEMENT The Fifth International Conference Game Theory and Management GTM2011 June 27-29, 2011, St. Petersburg, Russia ABSTRACTS Edited by Leon A. Petrosyan and Nikolay A. Zenkevich Graduate School of Management St. Petersburg University St. Petersburg 2011 УДК 518.9, 517.9, 681.3.07 GAME THEORY AND MANAGEMENT. Collected abstracts of papers presented on the Fifth International Conference Game Theory and Management / Editors Leon A. Petrosyan and Nikolay A. Zenkevich. – SPb.: Graduate School of Management SPbU, 2011. – 268 p. The collection contains abstracts of papers accepted for the Fifth International Conference Game Theory and Management (June 27–29, 2011, St. Petersburg University, St. Petersburg, Russia). The presented abstracts belong to the field of game theory and its applications to management. The abstract volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments. Computer design: Tatyana Grigorova and Andrew Zyatchin © Copyright of the authors, 2011 © Graduate School of Management, SPbU, 2011 ТЕОРИЯ ИГР И МЕНЕДЖМЕНТ. Сб. тезисов 5-ой международной конференции по теории игр и менеджменту / Под ред. Л.А. Петросяна и Н.А. Зенкевича. – СПб.: Высшая школа менеджмента СПбГУ, 2011. – 268 с. Сборник содержит тезисы докладов участников 5-ой международной конференции по теории игр и менеджменту (27–29 июня 2011 года, Высшая школа менеджмента, Санкт-Петербургский государственный университет, Санкт- Петербург, Россия). Представленные тезисы относятся к теории игр и её приложениям в менеджменте. Тезисы представляют интерес для научных работников, аспирантов и студентов старших курсов университетов, специализирующихся по менеджменту, экономике и прикладной математике.
    [Show full text]
  • On the Gamma-Core of Asymmetric Aggregative Games
    Munich Personal RePEc Archive On the gamma-core of asymmetric aggregative games Stamatopoulos, Giorgos 11 August 2018 Online at https://mpra.ub.uni-muenchen.de/88722/ MPRA Paper No. 88722, posted 02 Sep 2018 22:49 UTC On the γ-core of asymmetric aggregative games Giorgos Stamatopoulos∗ Department of Economics, University of Crete, Greece August 11, 2018 Abstract This paper analyzes the core of cooperative games generated by asymmetric aggrega- tive normal-form games, i.e., games where the payoff of each player depends on his strategy and the sum of the strategies of all players. We assume that each coalition calculates its worth presuming that the outside players stand alone and select individ- ually optimal strategies (Chander & Tulkens 1997). We show that under some mild monotonicity assumptions on payoffs, the resulting cooperative game is balanced, i.e. it has a non-empty γ-core. Our paper thus offers an existence result for a core notion that is considered quite often in the theory and applications of cooperative games with externalities. Keywords: cooperative game; aggregative game; balancedness; core JEL Classification: C71 1 Introduction The core is one of the most widely used solution concepts in cooperative games. This notion describes all the ways of splitting the benefits created by the entire set of players that are immune to coalitional rejections. To define the core, one needs to first define the characteristic function of a coalition, i.e., its worth. For a cooperative game with externalities, namely a game where the worth of a coalition depends on the actions of the players outside the coalition, the specification of the characteristic function requires a forecast or a belief about the behavior of the outsiders, and in particular about their coalition structure.
    [Show full text]
  • Aggregative Games with Entry1
    Aggregative Games with Entry1 Simon P. Anderson2,NisvanErkal3 and Daniel Piccinin4 This version: March 2011 Preliminary and Incomplete 1 We thank Daniel Halbheer, Frank Stähler, seminar participants at the Federal Trade Commis- sion, Johns Hopkins University, University of Mannheim, and conference participants at Université Catholique de Louvain (Conference in Honor of Jacques Thisse), EARIE (2010), Australasian Eco- nomic Theory Workshop (12011) for their comments. 2 Department of Economics, University of Virginia P.O. Box 400182, Charlottesville, VA 22904- 4182, USA. [email protected]. 3 Department of Economics, University of Melbourne, VIC 3010, Australia. [email protected]. 4 Brick Court Chambers, 7-8 Essex Street, London, WC2R 3LD, United Kingdom. [email protected]. Abstract For aggregative games we use a cumulative reaction function approach to determine the long-run effects of an exogenous change (e.g., cost changes, cooperation, leadership) affect- ing some players. Even though the affected players’ equilibrium actions and payoffs, and the number of active players change, the aggregator stays the same: free entry completely un- does the short-run impact. Unaffected players’ equilibrium payoffs are unchanged, whether or not actions are strategic complements or strategic substitutes. When the payoffs of agents outside the game are solely a function of the aggregator, the welfare (total payoffs) change from an exogenous change are measured simply as the change in payoffs to the directly affected player(s). An important case covers Logit and CES models in Bertrand differenti- ated products oligopoly: we show that the IIA property of demands implies that consumer surplus depends on the aggregator alone, and conversely, and the Bertrand pricing game is aggregative.
    [Show full text]
  • Aggregative Games and Oligopoly Theory: Short-Run and Long-Run Analysis1
    Aggregative Games and Oligopoly Theory: Short-run and Long-run Analysis1 Simon P. Anderson2, Nisvan Erkal3 and Daniel Piccinin4 July 2019 1An earlier version of this paper was circulated as CEPR Discussion Paper No. 9511, with the title "Aggregative Oligopoly Games with Entry." We thank David Myatt (the Editor) and three anonymous referees for their constructive comments. We are also thankful to Suren Basov, David Byrne, Chris Edmond, Maxim Engers, Daniel Halbheer, Joe Harrington, Simon Loertscher, Phil McCalman, Claudio Mezzetti, Volker Nocke, Martin Peitz, Frank Stähler, Jun Xiao, Jidong Zhou, and especially the late Richard Cornes for comments and discussion. We also thank seminar participants at the Federal Trade Commission, Johns Hopkins Uni- versity, New York University, National University of Singapore, University of Mannheim, and conference participants at the Monash IO Workshop (2018), Australian National University Workshop in Honor of Richard Cornes (2016), North American Winter Meeting of the Econo- metric Society (2013), Australasian Economic Theory Workshop (2011), EARIE (2010), CORE Conference in Honor of Jacques Thisse (2010) for their comments. Imogen Halstead, Jingong Huang, Boon Han Koh, Charles Murry, and Yiyi Zhou have provided excellent research as- sistance. The first author thanks National Science Foundation for financial support and the Department of Economics at the University of Melbourne for its hospitality. The second author gratefully acknowledges funding from the Australian Research Council (DP0987070). 2Department of Economics, University of Virginia; [email protected]. 3Department of Economics, University of Melbourne; [email protected]. 4Brick Court Chambers; [email protected]. Abstract We compile an IO toolkit for aggregative games with positive and normative comparative statics results for asymmetric oligopoly in the short and long run.
    [Show full text]
  • 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.
    [Show full text]