Non-Cooperative Games with Many Players

In this survey article, we report results on the existence of pure-strategy Nash equilibria in games with an atomless continuum of players, each with an action set that is not necessarily finite. We also discuss purification and symmetrization of mixed-strategy Nash equilibria, and settings in which private information, anonymity and idiosyncratic shocks are given particular prominence.

[1]  E. Chamberlin.  The Theory of Monopolistic Competition , 1933 .

[2]  Joan V. Robinson.  Economics of imperfect competition , 1969 .

[3]  J. L. Doob,et al.  Stochastic processes depending on a continuous parameter , 1937 .

[4]  E. Chamberlin,et al.  Monopolistic or Imperfect Competition , 1937 .

[5]  E. Chamberlin.  The Theory of Monopolistic Competition , 1933 .

[6]  J Wolfowitz,et al.  Elimination of Randomization in Certain Problems of Statistics and of the Theory of Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[7]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[8]  H. E. Vaughan,et al.  The Marriage Problem , 1950 .

[9]  Abraham Wald,et al.  Two Methods of Randomization in Statistics and the Theory of Games , 1951 .

[10]  J. Wolfowitz,et al.  Relations among certain ranges of vector measures , 1951 .

[11]  I. Glicksberg.  A FURTHER GENERALIZATION OF THE KAKUTANI FIXED POINT THEOREM, WITH APPLICATION TO NASH EQUILIBRIUM POINTS , 1952 .

[12]  K. Fan.  Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[14]  J. Doob.  Stochastic processes , 1953 .

[15]  R. Aumann.  Markets with a continuum of traders , 1964 .

[16]  K. Fan.  Applications of a theorem concerning sets with convex sections , 1966 .

[17]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[18]  G. Debreu.  Integration of correspondences , 1967 .

[19]  B. Peleg.  EQUILIBRIUM POINTS FOR GAMES WITH INFINITELY MANY PLAYERS. , 1969 .

[20]  On sets with convex sections , 1969 .

[21]  J. Chipman.  External Economies of Scale and Competitive Equilibrium , 1970 .

[22]  J. Mirrlees.  An Exploration in the Theory of Optimum Income Taxation an Exploration in the Theory of Optimum Income Taxation L Y 2 , 2022 .

[23]  Yakar Kannai,et al.  Continuity Properties of the Core of a Market: A Correction , 1972 .

[24]  J. Vial,et al.  Oligopoly "à la Cournot" in a general equilibrium analysis , 1972 .

[25]  A Quick Proof for a One-Dimensional Version of Liapounoff's Theorem , 1974 .

[26]  E. Prescott,et al.  Equilibrium search and unemployment , 1974 .

[27]  W. Hildenbrand.  Core and Equilibria of a Large Economy. , 1974 .

[28]  Sergiu Hart,et al.  On equilibrium allocations as distributions on the commodity space , 1974 .

[29]  Sergiu Hart,et al.  Equally distributed correspondences , 1974 .

[30]  B. Bollobás,et al.  Representation of systems of measurable sets , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[31]  Hugo Sonnenschein,et al.  Equilibrium in abstract economies without ordered preferences , 1975 .

[32]  Distributions of agents' characteristics☆ , 1975 .

[33]  D. J. Roberts,et al.  THE INCENTIVES FOR PRICE-TAKING BEHAVIOR IN LARGE EXCHANGE ECONOMIES , 1976 .

[34]  Robert M. Anderson,et al.  A non-standard representation for Brownian Motion and Itô integration , 1976 .

[35]  Noncooperative exchange with a continuum of traders , 1977 .

[36]  F. Hiai,et al.  Integrals, conditional expectations, and martingales of multivalued functions , 1977 .

[37]  L. Hurwicz.  On informationally decentralized systems , 1977 .

[38]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[39]  L. S. Shapley,et al.  Values of Large Games II: Oceanic Games , 1978, Math. Oper. Res..

[40]  David Schmeidler,et al.  Efficiency in an Atomless Economy with Fiat Money , 1978 .

[41]  Oliver Hart,et al.  Monopolistic competition in a large economy with differentiated commodities , 1979 .

[42]  Jean-Jacques Laffont,et al.  A General Equilibrium Entrepreneurial Theory of Firm Formation Based on Risk Aversion , 1979, Journal of Political Economy.

[43]  P. Hammond.  Straightforward Individual Incentive Compatibility in Large Economies , 1979 .

[44]  Peter J. Hammond,et al.  Symposium on Incentive Compatibility: Introduction , 1979 .

[45]  Douglas Gale.  Large Economies with Trading Uncertainty , 1979 .

[46]  R. Radner,et al.  Allocation of Resources in Large Teams , 1979 .

[47]  Kevin Roberts,et al.  The limit points of monopolistic competition , 1980 .

[48]  Andreu Mas-Colell,et al.  Noncooperative Approaches to the Theory of Perfect Competition , 1980 .

[49]  Oliver Hart,et al.  Perfect competition and optimal product differentiation , 1980 .

[50]  P. Dasgupta,et al.  Fully progressive taxation , 1980 .

[51]  M. Shubik,et al.  Efficiency properties of strategies market games: An axiomatic approach , 1980 .

[52]  Robert J. Weber,et al.  Topologies on Information and Strategies in Games with Incomplete Information , 1981 .

[53]  Zvi Artstein,et al.  Law of Large Numbers for Random Sets and Allocation Processes , 1981, Math. Oper. Res..

[54]  P. Champsaur,et al.  A Note on Incentives in Large Economies , 1982 .

[55]  Andreu Mas-Colell,et al.  Perfect Competition and the Core , 1982 .

[56]  A. Mas-Colell,et al.  Efficiency Properties of Strategic Market Games: An Axiomatic Approach* , 1982 .

[57]  Small Efficient Scale as a Foundation for Walrasian Equilibrium , 1982 .

[58]  Oliver Hart,et al.  A Model of Imperfect Competition with Keynesian Features , 1982 .

[59]  O. Hart.  Imperfect Competition in General Equilibrium: An Overview of Recent Work (Now published in Frontiers of Economics, edited by K. Arrow and S. Honkapohja, (Basil Blackwell, Oxford, 1985).) , 1983 .

[60]  Nigel J. Cutland.  Nonstandard Measure Theory and its Applications , 1983 .

[61]  Andreu Mas-Colell,et al.  Walrasian Equilibria as Limits of Noncooperative Equilibria. Part I: Mixed Strategies* , 1983 .

[62]  E. Green.  Continuum and Finite-Player Noncooperative Models of Competition , 1984 .

[63]  Kevin Roberts,et al.  The Theoretical Limits to Redistribution , 1984 .

[64]  A. Mas-Colell.  On a theorem of Schmeidler , 1984 .

[65]  M. A. Khan.  An alternative proof of Diestel's theorem , 1984, Glasgow Mathematical Journal.

[66]  The Profit Motive in the Theory of Monopolistic Competition , 1984 .

[67]  G. Weinrich.  On the Theory of Effective Demand Under Stochastic-Rationing , 1984 .

[68]  J. Diestel.  Sequences and series in Banach spaces , 1984 .

[69]  Robert J. Weber,et al.  Distributional Strategies for Games with Incomplete Information , 1985, Math. Oper. Res..

[70]  S. Rashid.  The approximate purification of mixed strategies with finite observation sets , 1985 .

[71]  O. Hart.  Monopolistic Competition in the Spirit of Chamberlin: A General Model , 1985 .

[72]  K. Judd.  The law of large numbers with a continuum of IID random variables , 1985 .

[73]  O. Hart.  MONOPOLISTIC COMPETITION IN THE SPIRIT OF CHAMBERLIN: SPECIAL RESULTS , 1985 .

[74]  Mark Feldman,et al.  An expository note on individual risk without aggregate uncertainty , 1985 .

[75]  A counterexample and an exact version of Fatou's lemma in infinite dimensional spaces , 1986 .

[76]  Beth E Allen,et al.  Price-Setting Firms and the Oligopolistic Foundations of Perfect Competition , 1986 .

[77]  Mamoru Kaneko,et al.  The core of a game with a continuum of players and finite coalitions: The model and some results , 1986 .

[78]  On Cournot-Nash equilibria in generalized qualitative games with a continuum of players , 1987 .

[79]  Entry, Fixed Costs and the Aggregation of Private Information , 1987 .

[80]  N. Yannelis.  Equilibria in noncooperative models of competition , 1987 .

[81]  P. Loeb,et al.  Lyapunov’s Theorem , 1987 .

[82]  Leo K. Simon,et al.  Games with Discontinuous Payoffs , 1987 .

[83]  P. Hammond.  Markets as Constraints: Multilateral Incentive Compatibility in Continuum Economies , 1987 .

[84]  Joseph M. Ostroy,et al.  Vickrey-Clarke-Groves Mechanisms in Continuum Economies: Characterization and Existence , 1988 .

[85]  David Housman.  Infinite Player Noncooperative Games and the Continuity of the Nash Equilibrium Correspondence , 1988, Math. Oper. Res..

[86]  R. Rosenthal,et al.  Anonymous sequential games , 1988 .

[87]  H. Jerome Keisler,et al.  Nonstandard Analysis and its Applications: INFINITESIMALS IN PROBABILITY THEORY , 1988 .

[88]  Economies with many agents : an approach using nonstandard analysis , 1988 .

[89]  H. Haller.  Competition in a stock market with small firms , 1988 .

[90]  Ton Lindstrøm.  Nonstandard Analysis and its Applications: AN INVITATION TO NONSTANDARD ANALYSIS , 1988 .

[91]  Steven R. Williams,et al.  The Rate of Convergence to Efficiency in the Buyer's Bid Double Auction as the Market Becomes Large , 1989 .

[92]  Thomas A. Gresik,et al.  The rate at which a simple market converges to efficiency as the number of traders increases: An asymptotic result for optimal trading mechanisms , 1989 .

[93]  Sergiu Hart,et al.  Existence of Correlated Equilibria , 1989, Math. Oper. Res..

[94]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[95]  The approximation of competitive equilibria by Bertrand-Edgeworth equilibria in large markets , 1989 .

[96]  M. Ali Khan,et al.  On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs , 1989 .

[97]  The core of a continuum economy with widespread externalities and finite coalitions: From finite to continuum economies , 1989 .

[98]  R. Rosenthal,et al.  More on the "anti-folk theorem" , 1989 .

[99]  H. Haller,et al.  Tax systems and direct mechanisms in large finite economies , 1990 .

[100]  Andrew Postlewaite,et al.  Asymmetric Information Bargaining Problems with Many Agents , 1990 .

[101]  Equilibrium in abstract economies with a non-compact infinite dimensional strategy space, an infinite number of agents and without ordered preferences , 1990 .

[102]  On a reformulation of Cournot-nash equilibria , 1990 .

[103]  Edi Karni,et al.  FIXED PREFERENCES AND CHANGING TASTES , 1990 .

[104]  Nicholas C. Yannelis,et al.  Set-Valued Functions of Two Variables in Economic Theory , 1991 .

[105]  M. Ali Khan,et al.  On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atomless Game , 1991 .

[106]  Subir K. Chakrabarti,et al.  Equilibria of Large Games with Imperfect Observability , 1991 .

[107]  Robert M. Anderson,et al.  Non-standard analysis with applications to economics , 1991 .

[108]  N. Yannelis.  Integration of Banach-Valued Correspondence , 1991 .

[109]  R. Anderson.  The core in perfectly competitive economies , 1992 .

[110]  Kali P. Rath.  A direct proof of the existence of pure strategy equilibria in games with a continuum of players , 1992 .

[111]  Faruk Gul,et al.  Asymptotic efficiency in large exchange economies with asymmetric information , 1992 .

[112]  Guoqiang Tian.  On the existence of equilibria in generalized games , 1992 .

[113]  A. Mas-Colell,et al.  Implementation in Economies with a Continuum of Agents , 1993 .

[114]  Aldo Rustichini,et al.  Cournot-Nash equilibrium distributions of games with uncertainty and imperfect information , 1993 .

[115]  Jordi Massó.  Undiscounted equilibrium payoffs of repeated games with a continuum of players , 1993 .

[116]  P. Hammond.  Markets, information, and uncertainty: History as a widespread externality in some Arrow–Debreu market games , 1993 .

[117]  WIDESPREAD EXTERNALITIES AND PERFECTLY COMPETITIVE MARKETS: EXAMPLES , 1994 .

[118]  Gideon Schwarz,et al.  Game theory and statistics , 1994 .

[119]  Edi Karni,et al.  Social Attributes and Strategic Equilibrium: A Restaurant Pricing Game , 1994, Journal of Political Economy.

[120]  L. Shapley,et al.  Noncooperative general exchange with a continuum of traders: Two models , 1994 .

[121]  Randomization and the limit points of monopolistic competition , 1994 .

[122]  Roger B. Myerson,et al.  Communication, correlated equilibria and incentive compatibility , 1994 .

[123]  An equilibrium result for games with private information and infinitely many players , 1994 .

[124]  R. Selten,et al.  Game theory and evolutionary biology , 1994 .

[125]  Peter J. Hammond,et al.  FOUR CHARACTERIZATIONS OF CONSTRAINED PARETO EFFICIENCY IN CONTINUUM ECONOMIES WITH WIDESPREAD EXTERNALITIES , 1995 .

[126]  Roger Guesnerie,et al.  A contribution to the pure theory of taxation , 1996 .

[127]  Kali P. Rath.  Representation of finite action large games , 1995 .

[128]  M. A. Khan,et al.  On Large Games with Finite Actions: A Synthetic Treatment , 1995 .

[129]  Kali P. Rath,et al.  The nonexistence of symmetric equilibria in anonymous games with compact action spaces , 1995 .

[130]  On the Representation of Sets in Finite Measure Spaces , 1996 .

[131]  Kali P. Rath.  Existence and upper hemicontinuity of equilibrium distributions of anonymous games with discontinuous payoffs , 1996 .

[132]  D. Fudenberg,et al.  When Are Nonanonymous Players Negligible , 1996 .

[133]  Yeneng Sun,et al.  Hyperfinite Law of Large Numbers , 1996, Bulletin of Symbolic Logic.

[134]  M. A. Khan,et al.  Nonatomic games on Loeb spaces. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[135]  Existence of non-cooperative equilibria in social systems , 1996 .

[136]  Comments on the existence of equilibrium distributions , 1996 .

[137]  Continuous Nash equilibria , 1997 .

[138]  Yeneng Sun.  Integration of Correspondences on Loeb Spaces , 1997 .

[139]  M. Ali Khan,et al.  Pure-Strategy Nash Equilibrium Points in Large Non-Anonymous Games , 1997 .

[140]  Yeneng Sun,et al.  A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN1 , 1998 .

[141]  Mário R. Páscoa,et al.  Nash equilibrium and the law of large numbers , 1998, Int. J. Game Theory.

[142]  Frontiers of Research in Economic Theory: On the Theory of Perfect Competition (1984) , 1998 .

[143]  A Result on Large Anonymous Games: An Elementary And Self-Contained Exposition , 1999 .

[144]  M. Páscoa,et al.  Manipulation-proof equilibrium in atomless economies with commodity differentiation , 1999 .

[145]  P. Hammond.  Rights, Free Exchange, and Widespread Externalities , 2000 .

[146]  Heterogeneous beliefs, price dispersion, and welfare-improving price controls , 2001 .

[147]  R. Dudley,et al.  Real Analysis and Probability: Complex Numbers, Vector Spaces, and Taylor's Theorem with Remainder , 2002 .

[148]  John C. Harsanyi,et al.  Games with Incomplete Information Played by "Bayesian" Players, I-III: Part I. The Basic Model& , 2004, Manag. Sci..