Values of games with infinitely many players

This chapter studies the theory of value of games with infinitely many players.Games with infinitely many players are models of interactions with many players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. The interactions are modeled as cooperative games with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or oceanic player-sets, or indeed any mixture of these.The value is defined as a map from a space of cooperative games to payoffs that satisfies the classical value axioms: additivity (linearity), efficiency, symmetry and positivity. The chapter introduces many spaces for which there exists a unique value, as well as other spaces on which there is a value.A game with infinitely many players can be considered as a limit of finite games with a large number of players. The chapter studies limiting values which are defined by means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players.Various formulas for the value which express the value as an average of marginal contribution are studied. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0,1]. In the case of smooth games the value formula is a diagonal formula: an integral of marginal contributions which are expressed as partial derivatives and where the integral is over all perfect samples of the set of players. The domain of the formula is further extended by changing the order of integration and derivation and the introduction of a well-crafted infinitesimal perturbation of the perfect samples of the set of players provides a value formula that is applicable to many additional games with essential nondifferentiabilities.

[1]  M. Osborne.  Why do some goods bear higher taxes than others , 1984 .

[2]  Andreu Mas-Colell,et al.  Egalitarian Solutions of Large Games: I. A Continuum of Players , 1995, Math. Oper. Res..

[3]  Igal Milchtaich.  Vector Measure Games Based on Measures with Values in an Infinite Dimensional Vector Space , 1998 .

[4]  Rudolf Henn,et al.  Operations Research-Verfahren , 1963 .

[5]  Yair Tauman,et al.  The Existence of Nondiagonal Axiomatic Values , 1976, Math. Oper. Res..

[6]  R. Gardner.  Wealth and Power in a Collegial Polity , 1981 .

[7]  A. Mas-Colell.  Competitive and value allocations of large exchange economies , 1977 .

[8]  Y. Tauman.  Value on the space of all scalar integrable games , 1979 .

[9]  Louis J. Billera,et al.  Cores of Nonatomic Linear Production Games , 1981, Math. Oper. Res..

[10]  Robert J. Aumann,et al.  Power and Taxes in a multi-commodity economy , 1977 .

[11]  Robert J. Weber,et al.  Probabilistic Values for Games , 1977 .

[12]  G. Owen.  Multilinear Extensions of Games , 1972 .

[13]  Sylvain Sorin,et al.  Game-theoretic methods in general equilibrium analysis , 1994 .

[14]  Haruo Imai,et al.  On Harsanyi's solution , 1983 .

[15]  Abraham Neyman,et al.  Semivalues of Political Economic Games , 1985, Math. Oper. Res..

[16]  L. Shapley,et al.  The core and value , 1971 .

[17]  Abraham Neyman,et al.  Renewal Theory for Sampling Without Replacement , 1982 .

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

[19]  Dov Samet,et al.  The Determination of Marginal-Cost Prices Under a Set of Axioms , 1982 .

[20]  H. Imai.  Voting, bargaining, and factor income distribution , 1979 .

[21]  Jean-François Mertens.  Non differentiable T.U. markets. The value , 1987 .

[22]  Y. Tauman.  Value on a class of non-differentiable market games , 1981 .

[23]  D. Monderer.  A probabilistic problem arising in economics , 1988 .

[24]  A. Schotter,et al.  Applied Game Theory , 1979, Physica-Verlag HD.

[25]  G. Owen,et al.  Game Theory (2nd Ed.). , 1983 .

[26]  Ori Haimanko,et al.  Cost sharing: the nondifferentiable case , 2001 .

[27]  Yair Tauman,et al.  Demand Compatible Equitable Cost Sharing Prices , 1982, Math. Oper. Res..

[28]  L. Shapley.  A Value for n-person Games , 1988 .

[29]  Olli Lehto.  Proceedings of the International Congress of Mathematicians : Helsinki, 1978 , 1980 .

[30]  R. Aumann,et al.  Power and taxes in a multi-commodity economy (updated) , 1978 .

[31]  Some results on values of finite and infinite games , 1975 .

[32]  Ori Haimanko,et al.  Payoffs in Nondifferentiable Perfectly Competitive TU Economies , 2001, J. Econ. Theory.

[33]  Abraham Neyman,et al.  Continuous Values are Diagonal , 1977, Math. Oper. Res..

[34]  Dov Samet,et al.  An Axiomatic Approach to the Allocation of a Fixed Cost Through Prices , 1983 .

[35]  Ori Haimanko,et al.  Partially Symmetric Values , 2000, Math. Oper. Res..

[36]  Jean-Pierre Aubin,et al.  Cooperative Fuzzy Games , 1981, Math. Oper. Res..

[37]  Sergiu Hart,et al.  Values of non-atomic vector measure games: Are they linear combinations of the measures? , 1988 .

[38]  Sergiu Hart,et al.  The number of commodities required to represent a market game , 1982 .

[39]  Lloyd S. Shapley,et al.  Additive and non-additive set functions , 1953 .

[41]  Sergiu Hart,et al.  Values of mixed games , 1973 .

[42]  Ori Haimanko.  Value theory without symmetry , 2000, Int. J. Game Theory.

[43]  R. Aumann,et al.  Power and public goods , 1987 .

[44]  D. Butnariu.  Decompositions and range for additive fuzzy measures , 1983 .

[45]  Leonard J. Mirman.  A Sufficient Condition on f for f.u to be in pNAD , 1980 .

[46]  The value of certain pNA games through infinite-dimensional Banach spaces , 1993 .

[47]  D. Pallaschke,et al.  Game Theory and Related Topics , 1980 .

[48]  The linearizing projection, global theories , 1988 .

[49]  On orderable set functions and continuity I , 1973 .

[50]  Myrna Holtz Wooders.  Topics in mathematical economics and game theory : essays in honor of Robert J. Aumann , 1999 .

[51]  R. Aumann,et al.  Voting for Public Goods , 1983 .

[52]  S. Hart,et al.  Potential, value, and consistency , 1989 .

[53]  L. Shapley,et al.  Values of Non-Atomic Games , 1974 .

[54]  J. R. Isbell.  Review: R. J. Aumann and L. S. Shapley, Values of non-atomic games , 1975 .

[55]  Rann Smorodinsky,et al.  Asymptotic Values of Vector Measure Games , 2004, Math. Oper. Res..

[56]  Robert E. Verrecchia,et al.  A Unique Procedure For Allocating Common Costs From A Production Process , 1981 .

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

[58]  Louis J. Billera,et al.  Internal Telephone Billing Rates - A Novel Application of Non-Atomic Game Theory , 1978, Oper. Res..

[59]  Abraham Neyman,et al.  Singular Games have Asymptotic Values , 1981, Math. Oper. Res..

[60]  Françoise Fogelman,et al.  Asymptotic Value of Mixed Games , 1980, Math. Oper. Res..

[61]  Jean-François Mertens,et al.  A value on ′AN , 2003, International Journal of Game Theory.

[62]  Yair Tauman,et al.  The Partition Value , 1979, Math. Oper. Res..

[63]  A. Neyman,et al.  PAYOFFS IN NONATOMIC ECONOMIES: AN AXIOMATIC APPROACH , 1984 .

[64]  R. Aumann,et al.  Orderable Set Functions and Continuity. III: Orderability and Absolute Continuity , 1977 .

[65]  H. Berbee.  On Covering Single Points by Randomly Ordered Intervals , 1981 .

[66]  Andreu Mas-Colell,et al.  Egalitarian Solutions of Large Games: II. The Asymptotic Approach , 1995, Math. Oper. Res..

[67]  Dov Monderer.  A Milnor Condition for Nonatomic Lipschitz Games and Its Applications , 1990, Math. Oper. Res..

[68]  The values of nonstandard exchange economies , 1976 .

[69]  Sergiu Hart,et al.  Asymptotic value of games with a continuum of players , 1977 .

[70]  Abraham Neyman,et al.  Diagonality of Cost Allocation Prices , 1984, Math. Oper. Res..

[71]  Yair Tauman,et al.  A sufficient condition on ƒ for ƒ∘μ to be in pNAD , 1982 .

[72]  Jean-François Mertens.  Values and Derivatives , 1980, Math. Oper. Res..

[73]  J. Mertens,et al.  The Shapley value in the non differentiate case , 1988 .

[74]  Robert J. Aumann,et al.  Power and Taxes , 1977 .

[75]  R. Rosenthal.  Lindahl's Solution and Values for A Public-Goods Example , 1976 .

[76]  Sergiu Hart,et al.  Measure-Based Values of Market Games , 1980, Math. Oper. Res..

[77]  William W. Sharkey,et al.  Weighted Aumann-Shapley pricing , 1998, Int. J. Game Theory.

[78]  A. Neyman.  Decomposition of ranges of vector measures , 1981 .

[79]  R. Aumann,et al.  Core and value for a public-goods economy: An example , 1977 .

[80]  Sergiu Hart,et al.  Harsanyi Values of Large Economies: Nonequivalence to Competitive Equilibria , 1996 .

[81]  P. Dubey.  On the uniqueness of the Shapley value , 1975 .

[82]  Pradeep Dubey,et al.  Value Theory Without Efficiency , 1981, Math. Oper. Res..

[83]  Yair Tauman,et al.  The Space of Polynomials in Measures is Internal , 1985, Math. Oper. Res..

[84]  An extension of the Aumann-shapley value concept to functions on arbitrary banach spaces , 1982 .

[85]  D. Butnaiu,et al.  Values and cores of fuzzy games with infinitely many players , 1987 .

[86]  Dov Monderer,et al.  The Shapley value: Values of smooth nonatomic games: the method of multilinear approximation , 1988 .

[87]  N. Z. Shapiro,et al.  Values of Large Games, I: A Limit Theorem , 1978, Math. Oper. Res..

[88]  A. Mas-Colell,et al.  Remarks on the game-theoretic analysis of a simple distribution of surplus problem , 1980 .

[89]  On dual regularity and value convergence theorems , 1981 .

[90]  Robert J. Weber,et al.  Subjectivity in the Valuation of Games , 1979 .

[91]  Martin L. Leibowitz,et al.  Cooperation versus Competition , 2000 .

[92]  Roger Guesnerie,et al.  Monopoly, syndicate, and shapley value: About some conjectures , 1977 .

[93]  Abraham Neyman,et al.  Values of non-atomic vector measure games , 2001 .

[94]  Dov Samet.  Vector Measures Are Open Maps , 1984, Math. Oper. Res..

[95]  Recent Developments in the Theory of the Shapley Value , 1978 .

[96]  Itzhak Gilboa,et al.  Additivizations of Nonadditive Measures , 1989, Math. Oper. Res..

[97]  L. Shapley,et al.  PURE COMPETITION, COALITIONAL POWER, AND FAIR DIVISION* , 1969 .

[98]  Mirman,et al.  Valeur de Shapley et répartition équitable des couts de production , 1981 .

[99]  José Manuel Zarzuelo,et al.  Weighted values for non-atomic games: an axiomatic approach , 1999, Math. Methods Oper. Res..

[100]  Dov Monderer,et al.  Potentials and Weighted Values of Nonatomic Games , 1997, Math. Oper. Res..

[101]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[102]  Pradeep Dubey.  Asymptotic Semivalues and a Short Proof of Kannai's Theorem , 1980, Math. Oper. Res..

[103]  Ori Haimanko.  Nonsymmetric Values of Nonatomic and Mixed Games , 2000, Math. Oper. Res..

[104]  Yair Tauman.  A Nondiagonal Value on a Reproducing Space , 1977, Math. Oper. Res..

[105]  L. Raut.  Construction of a Haar measure on the projective limit group and random order values of non-atomic games , 1997 .

[106]  Yair Tauman,et al.  The continuity of the Aumann-Shapley price mechanism , 1982 .

[107]  Dov Monderer,et al.  Measure-Based Values of Nonatomic Games , 1986, Math. Oper. Res..

[108]  Dov Samet.  Continuous Selections for Vector Measures , 1987, Math. Oper. Res..

[109]  Values of Large Market Games , 1979 .

[110]  José Manuel Zarzuelo,et al.  Mixing weighted values of non-atomic games , 1998, Int. J. Game Theory.

[111]  On non-atomic games: Conditions forf ο μ εp N A , 1973 .

[112]  Values of games with denumerably many players , 1971 .

[113]  L. Shapley,et al.  The Shapley Value , 1994 .

[114]  William H. Ruckle.  Projection in Certain Spaces of Set Functions , 1982, Math. Oper. Res..

[115]  L. Shapley.  Values of Large Games - VII: A General Exchange Economy with Money , 1964 .

[116]  Dov Samet,et al.  An Application of the Aumann-Shapley Prices for Cost Allocation in Transportation Problems , 1984, Math. Oper. Res..

[117]  Sergiu Hart,et al.  Values of non-differentiable markets with a continuum of traders , 1977 .

[118]  H. Young.  Producer Incentives in Cost Allocation , 1985 .

[119]  Yakar Kannai,et al.  Values of games with a continuum of players , 1966 .

[120]  H. Young.  Monotonic solutions of cooperative games , 1985 .

[121]  On the symmetry axiom for values of nonatomic games , 1990 .

[122]  Shapley value and disadvantageous monopolies , 1977 .

[123]  Yair Tauman,et al.  A characterization of vector measure games inpNA , 1982 .

[124]  Joseph M. Ostroy,et al.  A REFORMULATION OF THE MARGINAL PRODUCTIVITY THEORY OF DISTRIBUTION , 1984 .

[125]  An Equivalence Principle for Perfectly Competitive Economies , 1997 .

[126]  R. Aumann,et al.  A variational problem arising in economics , 1965 .

[127]  R. Aumann.  Values of Markets with a Continuum of Traders , 1975 .

[128]  Louis J. Billera,et al.  Allocation of Shared Costs: A Set of Axioms Yielding A Unique Procedure , 1982, Math. Oper. Res..

[130]  Addendum: The Shapley value of a perfectly competitive market may not exist , 1994 .

[131]  Abraham Neyman.  Value of Games with a Continuum of Players , 1994 .

[132]  Abraham Neyman.  Weighted Majority Games Have Asymptotic Value , 1988, Math. Oper. Res..

[133]  Dov Monderer.  Asymptotic Measure-Based Values of Nonatomic Games , 1989, Math. Oper. Res..

[134]  The Shapley value for countably many players , 1997 .