Market games and clubs

Wooders, Myrna (2008): Market games and clubs. Published in: Springer Encyclopeida of Complexity and Systems Science (2009): pp. 5359-5377.

Abstract

The equivalence of markets and games concerns the relationship between two sorts of structures that appear fundamentally different -- markets and games. Shapley and Shubik (1969) demonstrates that: (1) games derived from markets with concave utility functions generate totally balanced games where the players in the game are the participants in the economy and (2) every totally balanced game generates a market with concave utility functions. A particular form of such a market is one where the commodities are the participants themselves, a labor market for example. But markets are very special structures, more so when it is required that utility functions be concave. Participants may also get utility from belonging to groups, such as marriages, or clubs, or productive coalitions. It may be that participants in an economy even derive utility (or disutility) from engaging in processes that lead to the eventual exchange of commodities. The question is when are such economic structures equivalent to markets with concave utility functions. This paper summarizes research showing that a broad class of large economies generate balanced market games. The economies include, for example, economies with clubs where individuals may have memberships in multiple clubs, with indivisibile commodities, with nonconvexities and with non-monotonicities. The main assumption are: (1) that an option open to any group of players is to break into smaller groups and realize the sum of the worths of these groups, that is, essential superadditivity is satisfied and :(2) relatively small groups of participants can realize almost all gains to coalition formation. The equivalence of games with many players and markets with many participants indicates that relationships obtained for markets with concave utility functions and many participants will also hold for diverse social and economic situations with many players. These relationships include: (a) equivalence of the core and the set of competitive outcomes; (b) the Shapley value is contained in the core or approximate cores; (c) the equal treatment property holds -- that is, both market equilibrium and the core treat similar players similarly. These results can be applied to diverse economic models to obtain the equivalence of cooperative outcomes and competitive, price taking outcomes in economies with many participants and indicate that such results hold in yet more generality.

• Item Type: MPRA Paper
• Language: English
• Keywords: Markets; games; market games; clubs; core; market-game equivalence; Shapley value; price taking equilibrium; small group effectiveness; inessentiality of large groups; per capita boundedness; competitive equilibrium; games with side payments; balanced games; totally balanced games; local public goods, core convergence; equal treatment property; equal treatment core; approximate core; strong epsilon core; weak epsilon core; cooperative game; asymptotic negligibility
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games; H - Public Economics > H4 - Publicly Provided Goods > H41 - Public Goods; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General

References

Allouch, N. and M. Wooders (2008) "Price taking equilibrium in economies with multiple memberships in clubs and unbounded club sizes," Journal of Economic Theory, doi:10,1016/j.jet.2007.07.06.

Allouch, N., J.P. Conley and M. Wooders (2008) "Anonymous price taking equilibrium in Tiebout economies with a continuum of agents: Existence and characterization," Journal of Mathematical Economics, doi:10.1016/j.jmateco.2008.06.003.

Azrieli, Y. and E. Lehrer (2007) "Market games in large economies with a finite number of types," Economic Theory 31:327-342.

R.J. (1964) "Markets with a continuum of traders," Econometrica 32, 39-50.

Aumann, R.J. (1987) "Game theory," The New Palgrave: A Dictionary of Economics, J. Eatwell, M. Milgate, and P. Newman, eds. The Macmillan Press.

Aumann, R. J. (1964) "Markets with a continuum of traders," Econometrica 32, 39-50.

Aumann, R.J. and Dreze, J. (1974) "Cooperative games with coalition structures," International Journal of Game Theory 3, 217-37.

Aumann, R.J. and S. Shapley (1974) Values of Non-Atomic Games, Princeton University Press, Princeton, New Jersey.

Bennett, E. and Wooders, M. (1979) `Income distribution and firm formation', Journal of Comparative Economics 3, 304-317, on-line at http://www.myrnawooders.com/

Bergstrom, T. and Varian, H. R. (1985). "When do market games have transferable utility?," Journal of Economic Theory, Elsevier, vol. 35(2), 222-233.

Billera, L.J. (1974) "On games without side payments arising from a general class of markets," Journal of Mathematical Economics 1(2), 129-139.

Billera, L.J. and R.E. Bixby (1974) "Market representations of n-person games," Bulletin of the American Mathematical Society 80(3), 522-526.

Böhm, V. (1974) "The core of an economy with production," Review of Economic Studies 41, 429--436.

Bondareva, O. (1963) "Some applications of linear programming to the theory of cooperative games," Problemy kibernetiki 10 (in Russian, see English translation in Selected Russian papers in game theory 1959-1965, Princteon University Press, Princeton New Jersey.

Buchanan, J. (1965) "An economic theory of clubs," Economica 33, 1-14.

Casella, A. and Feinstein, J. S. (2002) "Public goods in trade on the formation of markets and jurisdictions," ,International Economic Review 43, 437-462.

Cartwright, E., J. Conley and M. Wooders (2006) "The Law of Demand in Tiebout Economies," in The Tiebout Model at 50: Essays in Public Economics in honor of Wallace Oates, W. A. Fischel (ed), Lincoln Institute of Land Policy, Cambridge, Mass.

Champsaur, P. (1975) "Competition vs. cooperation," Journal of Economic Theory 11, 394-417.

Cheng, H.C. (1981) "On dual regularity and value convergence theorems," Journal of Mathematical Economics 8, 37-57.

Conley, J. and S. Smith (2005) "Coalitions and clubs; Tiebout equilibrium in large economies," in Group Formation in Economics; Networks, Clubs and Coalitions, G. Demange and M. Wooders (eds), Cambridge University Press, Cambridge, UK.

Conley, J.P. and M. Wooders (1995) "Hedonic independence and taste-homogeneity of optimal jurisdictions in a Tiebout economy with crowding types," Annales D'Economie et de Statistique 75/76, 198-219.

Crawford, V.P. and A. S. Kelso (1982) "Job matching, coalition formation, and gross substitutes," Econornetrica 50, 1483-1504.

Debreu, G. and H. Scarf (1963) "A limit theorem on the core of an economy," International Economic Review 4, 235-246.

Demange G. (1994) "Intermediate preferences and stable coalition structures," Journal of Mathematical Economics, 1994, 45-48.

Ellickson, B., B. Grodal, S. Scotchmer, and W. Zame (1999) "Clubs and the market," Econometrica 67, 1185-1218.

Gale, D. and Shapley, L.S. (1962) "College admissions and the stability of marriage," American Mathematical Monthly 69, 9-15.

Garratt, R. and C-Z Qin (1997) "On a market for coalitions with indivisible agents and lotteries, Journal of Economic Theory 77, #1, 81-101.

Gillies, D.B. (1953) "Some theorems on n-person games' Ph.D. Dissertation, Department of Mathematics, Princeton University, Princeton.

Haimanko, O., M. Le Breton and S. Weber (2004) "Voluntary formation of communities for the provision of public projects," Journal of Economic Theory 115, 1-34.

Hildenbrand, W. (1974) Core and Equilibria of a Large Economy, Princeton University Press, Princeton, New Jersey.

Hurwicz, L. and H. Uzawa (1977) "Convexity of asymptotic average production possibility sets," in Studies in Resource Allocation Processes, K.J. Arrow and L. Hurwicz eds. Cambridge University Press.

Kalai, E. and E. Zemel (1982a) "Totally balanced games and games of flow," Mathematics of Operations Research 7, 476-478.}

Kalai, E. and E. Zemel (1982b) "Generalized network problems yielding totally balanced games," Operations Research 30, 998-1008.

Kaneko, M. and Wooders, M. (1982) "Cores of partitioning games," Mathematical Social Sciences 3, 313-327.

Kaneko, M. and M. Wooders (2004) "Utility theories in cooperative games," Handbook of Utility Theory vol. 2, Chapter 19, 1065-1098. Kluwer Academic Press.

Kaneko, M. and M. Wooders (1986) "The core of a game with a continuum of players and finite coalitions; the model and some results," Mathematical Social Sciences 12, 105-137, on line at http://www.myrnawooders.com/

Kannai, Y. (1972) "Continuity properties of the core of a market," Econometrica 38, 791-815.

Konishi, H., M. Le Breton, and S. Weber (1998) "Equilibrium in a finite local public goods economy," Journal of Economic Theory 79, 224-244.

Kovalenkov, A. and M. Wooders (2001). "Epsilon cores of games with limited side payments: nonemptiness and equal treatment," Games and Economic Behavior. 36 (2), 193-218.

Kovalenkov, A. and M. Wooders (2003) "Approximate cores of games and economies with clubs," Journal of Economic Theory 110, 87-120.

Kovalenkov, A. and M. Wooders (2005) "A law of scarcity for games," Economic Theory 26, 383-396.

Kovalenkov, A. and M. Wooders (2006) "Comparative statics and laws of scarcity for games," in (C.D. Aliprantis, R.L. Matzkin, D.L. McFadden, J.C. Moore and N.C. Yannelis, eds.), Rationality and Equilibrium: A Symposium in Honour of Marcel K. Richter, Studies in Economic Theory Series 26, Springer: Springer Verlag, 141-169.

Mas-Colell, A. (1975) "A further result on the representation of games by markets," Journal of Economic Theory 10 (1), 117-122.

Mas-Colell, A. (1977) "Indivisible commodities and general equilibrium theory," Journal of Economic Theory 16, 2, 443-4

Mas-Colell, A. (1979) "Competitive and value allocations of large exchange economies," Journal of Economic Theory 14, 307-310.

Mas-Colell, A. (1980) "Efficiency and decentralization in the pure theory of public goods," Quarterly Journal of Economics 94, 625-641.

Mas-Colell, A. (1985) The Theory of General Economic Equilibrium, Economic Society Publication No. 9, Cambridge University Press: Cambridge.

Moulin, M. (1988) Axioms of Cooperative Decision Making, Econometric Society Monograph No.15, Cambridge Press, Cambridge.

Moulin, H. (1992) "Axiomatic cost and surplus sharing," Handbook of Social Choice and Welfare, K. Arrow, A. K. Sen, K. Suzumura, eds. Edition 1, volume 1, chapter 6, pages 289-357, Elsevier.

von Neumann. J. and Morgenstern, O. (1953) Theory of Games and Economic Behavior, Princeton University Press, Princeton New Jersey.

Owen, G. (1975) "On the core of linear production games," Mathematical Programming 9, 358-370.6.

Page, F. H. and M. Wooders (2007) Networks and clubs," Journal of Economic Behavior & Organization, 64, 406-425.

Page, F.H. and M. Wooders (2010) "Club networks with multiple memberships and noncooperative stability," Games and Economic Behavior, Volume 70, Issue 1, September 2010, Pages 12-20.

Pauly, M. (1970) "Cores and clubs," Public Choice 9, 53-65.

Qin, C-Z, L.S. Shapley, and K-I Shimomura (2006) "The Walras core of an economy and its limit theorem," Journal of Mathematical Economics 42, Issue 2, 180-197.

Roth, A. and Sotomayer, M. (1990) Two-sided Matching; A Study in Game-theoretic Modeling and Analysis, Cambridge University Press, Cambridge.

Scotchmer, S. and Wooders, M. (1988) `Monotonicity in games that exhaust gains to scale', IMSSS Technical Report No. 525, Stanford University.

L.S. (1964) "Values of large games -VII: A general exchange economy with money," Rand Memorandum RM-4248-PR.

Shapley, L.S. (1967) "On balanced sets and cores," Navel Research Logistics Quarterly 9, 45-48.

Shapley, L.S. (1952) "Notes on the N-Person game III: Some variants of the von-Neumann-Morgenstern definition of solution," Rand Corporation research memorandum, RM- 817, 1952.

Shapley, L. S. and M. Shubik (1960) "On the core of an economic system with externalities," The American Economic Review 59, 678-684.

Shapley, L.S. and Shubik, M. (1966) "Quasi-cores in a monetary economy with nonconvex preferences," Econometrica 34, 805-827.

Shapley, L.S. and Shubik, M. (1969) "On market games", Journal of Economic Theory 1, 9-25.

Shapley, L.S. and Shubik, M. (1972) "The Assignment Game 1; The core', International Journal of Game Theory 1, 11-30.

Shapley, L.S. and Shubik, M. (1975) "Competitive outcomes in the cores of market games," International Journal of Game Theory 4, 229-237.

Shapley, L.S. and Shubik, M. (1977) "Trade using one commodity as a means of payment," Journal of Political Economy 85, 937-68.

Shubik, M. (1959) `Edgeworth market games', in Luce, F.R. and Tucker, A.W. (eds.), Contributions to the Theory of Games IV, Annals of Mathematical Studies 40, Princeton: Princeton University Press, 267-278.

Shubik, M., Wooders, M., 1982. Clubs, markets, and near-market games. In: Wooders, M. (Ed.), Topics in Game Theory and Mathematical Economics: Essays in Honor of Robert J. Aumann. Field Institute Communication Volume, American Mathematical Society, originally Near Markets and Market Games, Cowles Foundation, Discussion Paper No. 657.

M. and Wooders, M. (1983) `Approximate cores of replica games and economies: Part II. Set-up costs and firm formation in coalition production economies', Mathematical Social Sciences 6, 285-306.

Shubik, M. (1959) "Edgeworth market games," in Luce, F.R. and Tucker, A.W. (eds.) Contributions to the Theory of Games IV, Annals of Mathematical Studies 40, Princeton: Princeton University Press, 267-278.

Shubik, M. and M. Wooders (1982) "Near markets and market games," Cowles Foundation Discussion Paper No. 657, on line at http://www.myrnawooders.com/

Shubik, M., Wooders, M., 1982. Clubs, markets, and near-market games. In: Wooders, M. (Ed.), Topics in Game Theory and Mathematical Economics: Essays in Honor of Robert J. Aumann. Field Institute Communication Volume, American Mathematical Society, originally Near Markets and Market Games, Cowles Foundation, Discussion Paper No. 657.

Shubik, M. and Wooders, M. (1983) "Approximate cores of replica games and economies: Part I. Replica games, externalities, and approximate cores," Mathematical Social Sciences 6, 27-48.

Shubik, M. and Wooders, M. (1983) "Approximate cores of replica games and economies: Part II. Set-up costs and firm formation in coalition production economies." Mathematical Social Sciences 6, 285-306.

Shubik, M. and Wooders, M. (1986) "Near-markets and market-games," Economic Studies Quarterly 37, 289-299.

Sondermann, D. (1974) "Economics of scale and equilibria in coalition production economies," Journal of Economic Theory 8, 259--291.

Sun, N., W. Trockel, and Z. Yang (2008) "Competitive outcomes and endogenous coalition formation in an n-person game," Journal of Mathematical Economics 44, 853-860.

Tauman, Y. (1987) "The Aumann-Shapley prices: A survey", in The Shapley Value: Essays in Honor of Lloyd S. Shapley, ed. A. Roth, Cambridge University, Cambridge, MA.

Tauman, Y., A. Urbano and J. Watanabe (1997) "A model of multiproduct price competition," Journal of Economic Theory 77, 377-401.

Tiebout, C. (1956) "A pure theory of local expenditures," Journal of Political Economy 64, 416-424.

Weber, S. (1979) On ε-cores of balanced games, International Journal of Game Theory 8, 241-250.

Weber, S. (1981) Some results on the weak core of a non-sidepayment game with infinitely many players, Journal of Mathematical Economics 8:101-111.

Winter, E. and M. Wooders (1990) "On large games with bounded essential coalition sizes," University of Bonn Sondeforschungsbereich 303 Discussion Paper B-149, on-line at http://www.myrnawooders.com/ International Journal of Economic Theory (2008) 4, 191 - 206.

Wooders, M. (1977) "Properties of quasi-cores and quasi-equilibria in coalition economies," SUNY-Stony Brook Department of Economics Working Paper No. 184, revised (1979) as "A characterization of approximate equilibria and cores in a class of coalition economies", State University of New York Stony Brook Economics Department, on-line at http://www.myrnawooders.com/ (on-line at www.myrnawooders.com)

Wooders M. (1978) "Equilibria, the core, and jurisdiction structures in economies with a local public good," Journal of Economic Theory 18, 328-348.

Wooders, M. (1983) "The epsilon core of a large replica game," Journal of Mathematical Economics 11, 277-300, on-line at http://www.myrnawooders.com/

Wooders, M. (1988) "Large games are market games 1. Large finite games," C.O.R.E. Discussion Paper No. 8842 http://www.myrnawooders.com/

Wooders, M.H. (1989) `A Tiebout Theorem', Mathematical Social Sciences 18, 33-55.

Wooders, M. (1991a) "On large games and competitive markets 1: Theory," University of Bonn Sonderforschungsbereich 303 Discussion Paper No. (B-195, Revised August 1992), on line at http://www.myrnawooders.com/

Wooders, M. (1991b) "The efficaciousness of small groups and the approximate core property in games without side payments," University of Bonn Sonderforschingsbereich 303 Discussion Paper No. B-179on-line at http://www.myrnawooders.com/

Wooders, M. (1992a) "Inessentiality of large groups and the approximate core property; An equivalence theorem," Economic Theory 2, 129-147.

Wooders, M. (1992b) "Large games and economies with effective small groups," University of Bonn Sonderforschingsbereich 303 Discussion Paper No. B-215.(revised) in Game-Theoretic Methods in General Equilibrum Analysis, eds. J-F. Mertens and S. Sorin, Kluwer Academic Publishers Dordrecht/Boston/London, on-line at http://www.myrnawooders.com/

Wooders, M. (1993) "The attribute core, core convergence, and small group effectiveness; The effects of property rights assignments on the attribute core," University of Toronto Working Paper No. 9304

Wooders, M.(1994) "Equivalence of games and markets," Econometrica 62, 1141-1160, on-line at www.myrnawooders.com/ Wooders, M. (1997) "Equivalence of Lindahl equilibria with participation prices and the core," Economic Theory 9,113-127. Wooders, M. (2007) " Cores of many-player games; Nonemptiness and equal treatment," Review of Economic Design (to appear). Wooders, M. (2008a) "Small group effectiveness, per capita boundedness and nonemptiness of approximate cores," Journal of Mathematical Economics 44, 888-906.

Wooders, M. (2009) "Market Games and Clubs," Springer Encyclopedia of Complexity and Systems Science 2009: 5359-5377. Wooders, M. and Zame, W.R. (1987) "Large games; Fair and stable outcomes', Journal of Economic Theory 42, 59-93. Zajac, E. (1972) "Some preliminary thoughts on subsidization," presented at the Conference on Telecommunications Research, Washington