Social conformity and equilibrium in pure strategies in games with many players
Wooders, Myrna Holtz, Cartwright, Edward and Selten, Reinhard (2002) Social conformity and equilibrium in pure strategies in games with many players. Working Paper. University of Warwick, Department of Economics, Coventry.
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We introduce a framework of noncooperative pregames, in which players are characterized by their attributes, and demonstrate that for all games with sufficiently many players, there exist approximate (e )Nash equilibria in pure strategies. In fact, every mixed strategy equilibrium can be used to construct an e-equilibrium in pure strategies, an ‘e -purification’ result. Our main result is a social conformity theorem. Interpret a set of players, all with attributes in some convex subset of attribute space and all playing the same strategy, as a society. Observe that the number of societies may be as large as the number of players. Our social conformity result dictates that, given e > 0, there is an integer L, depending on e but not on the number of players, such that any suffciently large game has an e -equilibrium in pure strategies that induces a partition of the player set into fewer than L societies.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Alternative Title:||Original version, May 15, 2001; this version, April 2002|
|Subjects:||H Social Sciences > HB Economic Theory
H Social Sciences > HM Sociology
|Divisions:||Faculty of Social Sciences > Economics|
|Library of Congress Subject Headings (LCSH):||Noncooperative games (Mathematics), Conformity, Games of strategy (Mathematics), Group theory, Game theory|
|Series Name:||Warwick economic research papers|
|Publisher:||University of Warwick, Department of Economics|
|Place of Publication:||Coventry|
|Number of Pages:||59|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|Version or Related Resource:||Revised version of: Wooders, M.H., Cartwright, E. and Selten, R. (2001). Some first results for noncooperative pregames: social conformity and equilibrium in pure strategies. [Coventry] : University of Warwick, Economics Department. (Warwick economic research papers, no.589).|
|References:|| Araujo, A., M. Pascoa and J. Orrillo (2000) “Equilibrium with default and exogenous collateral,” Mathematical Finance.  Asch, S. (1952) Social Psychology, Prentice Hall, New York (Chapter 16).  Aumann R.J. (1964) “Markets with a continuum of traders,” Econometrica 32:39-50.  Aumann, R.J. (1985) “An axiomatization of the non-transferable utility value, Econometrica 53: 599-612.  Aumann, R.J., Y. Katznelson, R. Radner, R.W. Rosenthal, and B. Weiss (1983) “Approximate purification of mixed strategies,” Mathematics of Operations Research 8, 327-341.  Binmore, K. and L. Samuelson (2001) “Evolution and Mixed Strategies,” Games and Economic Behavior 34, 200-226.  Blume, L.E. (1993) “The statistical mechanics of strategic interaction,” Games and Economic Behaviour 5:387-424.  Bjornerstedt, J. and J.Weibull (1995), “Nash equilibrium and evolution by imitation,” in The Rational Foundations of Economic Behavior, ed by K.Arrow et al, Macmillan.  Cartwright, E. and M. Wooders (2001) “Social conformity in games with incomplete information,” Notes.  Deutsch, M. and H.B. Gerard, (1955) “A study of normative and informational social influences upon individual judgement,” Journal of Abnormal and Social Psychology 51:629-636.  Ellison, G. (1993) “Learning, local interaction and coordination,” Econometrica 61:1047-1071.  Ellison, G. and D. Fudenberg (1993) “Rules of thumb for social learning,” Journal of Political Economy 101:612-643.  Ellison, G. and D. Fudenberg (1995) “Word-of-mouth communication and social learning,” Quarterly Journal of Economics 110: 93-125.  Follmer, H. (1974) “Random economies with many interacting agents,” Journal of Mathematical Economics 1, 51-62.  Fudenberg, D. and D.K. Levine (1998) The Theory of Learning in Games, Cambridge: MIT press.  Gale, D. and R. Rosenthal, 1999, “Experimentation, Imitation, and Strategic Stability,” Journal of Economic Theory 84, 1-40.  Green, E.J. (1984) “Continuum and finite-player noncooperative models of competition,” Econometrica vol.52, no. 4: 975-993.  Green, J. and W.P. Heller (1991) “Mathematical analysis and convexity with applications to economics,” in Handbook of Mathematical Economics, Vol.1, K.J. Arrow and W. Intriligator, eds. North Holland: Amsterdam, New York, Oxford.  Greenberg, J. and S. Weber (1986) “Strong Tiebout equilibrium with restricted preference domain,” Journal of Economic Theory 38:101-117.  Gross, R. (1996) Psychology. The science of Mind and Behaviour, Hodder and Stoughton.  Hildenbrand, W. (1971) “Random preferences and equilibrium analysis,” Journal of Economic Theory 3, 414-429.  Kandori, M., G.J. Mailath and R. Rob (1993) “Learning, mutation and long run equilibria in games,” Econometrica 61:29-56.  Kalai, E. (2000) “Private information in large games,” Northwestern University Discussion Paper 1312.  Khan, A. (1989) “On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi continuous payoffs," Transactions of the American Mathematical society 293: 737-749.  Khan, A. and Y. Sun (1999) “Non-cooperative games on hyperfinite Loeb spaces,” Journal of Mathematical Economics 31, 455-492.  Khan, A., K.P. Rath and Y.N. Sun (1997) “On the existence of pure strategy equilibria with a continuum of players,” Journal of Economic Theory 76:13-46.  Kirman, A.P. (1981) “Measure Theory,” Handbook of Mathematical Economics, K. Arrow and M. Intrilligator (eds.), North Holland Amsterdam/ New York/Oxford.  Kirman, A.P. (1993) “Ants, rationality, and recruitment,” Quarterly Journal of Economics, 137-156.  Kovalenkov, A. and M. Wooders (1997) “Epsilon cores of games with limited side payments; Nonemptiness and equal treatment,” Autonoma University of Barcelona Working Paper 392.97 revised, Games and Economic Behavior August 2001, vol. 36, no. 2, pp. 193-218 (26). .  Kovalenkov, A. and M. Wooders (1997) “An exact bound on epsilon for non-emptiness of the epsilon-core of an arbitrary game with side payments,” Autonoma University of Barcelona Working Paper 393.97 revised, Mathematics of Operations Research Vol 26, No 4, Nov 2001, pp 654-678.  Mas-Colell, A. (1984) “On a theorem of Schmeidler,” Journal of Mathematical Economics 13: 206-210.  O’Neill, B. (1987) “Nonparametric test of the Minimax Theory of twoperson, zero sum games,” Proceedings of the National Academy of Sciences 84, 2106-2109.  Pascoa, M. (1998) “Nash equilibrium and the law of large numbers,” International Journal of Game Theory 27: 83-92.  Pascoa, M. (1993a) “Approximate equlibrium in pure strategies for nonatomic games,” Journal of Mathematical Economics 22: 223-241.  Pascoa, M. (1993b) “Noncooperative equilibrium and Chamberlinian monopolistic competition,” Journal of Economic Theory, 69: 335-353.  Radner, R and R.W. Rosenthal (1982) “Private information and purestrategy equilibria,” Mathematics of Operations Research 7: 401-409  Rashid, S. (1983) “Equilibrium points of nonatomic games; Asymptotic results,” Economics Letters 12: 7-10.  Rath, K.P., Y. Sun, S. Yamashige (1995) “The nonexistence of symmetric equilibria in anonymous games with compact action spaces,” Journal of Mathematical Economics 24: 331-346.  Reny, P.J. (1999) “On the existence of pure and mixed strategy Nash equilibria in discontinuous games,” Econometrica 67: 1029-1056.  Scharfstein, D.S. and J.C. Stein (1990) “Herd behaviour and investment,” The American Economic Review, vol 80, no. 3, 465-479.  Selten, R. (1980) “A note on evolutionarily stable strategies in asymmetric animal contesets,” Journal of Theoretical Biology 84, 93-101.  Schleifer, A., (2000) Inefficient Markets, an introduction to behavioural finance, Oxford University Press.  Schmeidler, D. (1973) “Equilibrium points of nonatomic games,” Journal of Statistical Physics 7: 295-300.  Walker, M. and J. Wooders (2001) “Minimax play at Wimbledon,” The American Economic Review; Dec 200, Volume: 91, Issue: 5, 1521-1538.  Wooders, M. (1983) “The epsilon core of a large replica game,” Journal of Mathematical Economics 11, 277-300.  Wooders, M.H. (1993) “On Auman’s markets with a continuum of traders; The continuum, small group effectiveness, and social homogeneity,” University of Toronto Department of Economics Working Paper No. 9401.  Wooders, M. (1994) “Equivalence of games and markets,” Econometrica 62, 1141-1160.  Young, H.P. (2001) Individual Strategy and Social Structure, Princeton University Press.|
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