روش‌های جبری در نظریه بازی‌ها

نوع مقاله : مقاله مروری

نویسندگان

1 دانشگاه تهران - پردیس علوم - دانشکده ریاضی ، آمار و علوم کامپیوتر

2 دانشجوی دانشکده ریاضی، آمار و علوم کامپیوتر دانشگاه تهران

چکیده

در این مقاله، ضمن مروری بر روش های به کار رفته در اثبات وجود تعادل نش طی ٧٠ سال اخیر، نشان می دهیم که محور این روش ها، قضیۀ نقطۀ ثابت براوئر و تعمیم های آن بوده است و سپس به تبیین روشی جدید می پردازیم که مبتنی بر استفاده از روش های جبری در اثبات وجود تعادل است. گرچه این روش هنوز دوران طفولیت خود را می گذراند، پیشینۀ استفاده از روش های جبری در حل مسائل ریاضی نشان می دهد که این روش، نویدبخش یک رویکرد پژوهشی گسترده در آینده است.

کلیدواژه‌ها

موضوعات


Aliprantis, C. D., Border, K. C., Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer-Verlag, Heidelberg, 2003.
Aumann, R. J., Maschler, M.,  The bargaining set for cooperative games, Annals of Mathematics Studies, 52 (1964), 443--476.
Baer, R.,   Finiteness properties of groups, Duke Math. J., 15 (1948),1021--1032.
Balder, E. J.,   An equilibrium closure result for discontinuous games, Economic Theory, 48 (2010), 47--65.
Barelli, P., Govindan, S., Wilson, R. B.,   Competition for a Majority, Technical Report 2104, Stanford School of Business, 2012.
 
Barelli, P., Meneghel, I.,   A note on the equilibrium existence problem in discontinuous games, Econometrica, 81 (2013), 813--824.
 
Başar, T., Olsder, G. J., Dynamic Noncooperative Game Theory,  SIAM, 1998.
Bhaskara Rao, K. P. S., Bhaskara Rao, M.,  Theory of Charges, Academic Press, New York, 1983.
Bich, P., Laraki, R.,  A unified approach to equilibrium existence in discontinuous strategic games, Technical Report 12040, Université Panthéon Sorbonne (Paris1), Centre d’Economie de la Sorbonne, 2012.
Brouwer, L. E. J.,   Über abbildung von mannigfaltigkeiten,  Math. Ann., 71 (1911), 97--115.
 
Capraro, V., Morrison, K. E.,  Optimal strategies for a game on amenable semigroups, Int. J. Game Theory, 42  (2012), 917--929.
Capraro, V., Scarsini, M., Existence of equilibriain countable games: An algebraic approach, Games and Eonomic Behavior, 79 (2013), 163--180.
 
Carmona, G., On the existence of equilibriain discontinuous games: Three counterexamples, Int. J. GameTheory, 33 (2005), 181--187.
 
Carmona, G., Symposium on: existence of Nash equilibria in discontinuous games, Economic Theory, 48 (2011), 1--4.
Carmona, G.,  Understanding some recent existence results for discontinuous games, Economic Theory, 48 (2010), 31--45.
Cotter, K. D., Correlated equilibrium in games with type-dependent strategies, J. Econ. Theory, 54 (1991), 48--68.
 
Dasgupta, P., Maskin, E., The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud., 53 (1986), 1--26.
Davis, M., Maschler, M., The kernel of a cooperative game, Naval Research Logistics Quarterly, 12 (1965), 223--259.
 
De Castro, L. I., Equilibrium existence and approximation of regular discontinuous games, Economic Theory, 48 (2010), 67--85.
 
De Finetti, B., Probability, Induction and Statistics:The Art of Guessing, John Wiley & Sons, London, NewYork, Sidney, 1972.
Debreu, G., A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA, 38 (1952), 886--893.
 
Debreu, G., Theory of Value: An Axiomatic Analysis of Economic Equilibrium, New Haven and London, Yale University Press, 1959.
 
Dunford, N., Schwartz, J. T.,  Linear Operators: Part I., Wiley Classics Library, John Wiley & Sons, New York, 1988.
 
Eilenberg, S., Montgomery, D., Fixed point theorems for multi-valued transformations, Amer. J. Math., 68 (1946), 214--222.
 
Fan, K., Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Natl. Acad. Sci. USA,  38  (1952), 121–126.
 
Fan, K., Minimax theorems,  Proc. Natl. Acad. Sci. USA, 39 (1953), 42--47.
 
Gillies, D. B., Solutions to general non-zero-sum games, Annals of Mathematics Studies, 40 (1959), 47-85.
 
Glicksberg, I. L., A further generalization of the Kakutani fixed point  theorem with application to Nash equilibrium points   Proc. Amer. Math. Soc., 3 (1952), 170--174.
 
Hildebrandt, T. H., On bounded linear functional operations, Trans. Amer. Math. Soc., 36 (1934), 868--875.
 
Kakutani, S., A generalization of Brouwer's fixed point theorem, Duke Math. J., 8 (1941), 457--459.
 
Kindler, J., A general solution concept for two-person zero-sum games, J. Optim. Theory Appl., 40 (1983), 105--119.
Maitra, A., Sudderth, W., Finitely additive and measurable stochastic games, Int. J. Game Theory, 22 (1993), 201--223.
 
Maitra, A., Sudderth, W., Finitely additive stochastic games with Borel measurable payoffs, Int. J. Game Theory, 27  (1998), 257--267.
 
Marinacci, M., Finitely additive and epsilon Nash equilibria, Int. J. Game Theory, 26 (1997), 315--333.
 
Myerson, R., Reny, P. J., Sequential equilibria of multi-stage games with infinite sets of types and actions, Unpublished, 2012.
 
Nash, J., Equilibrium points in n-person games, Proc. Natl. Acad. Sci. USA, 36 (1950) , 48--49.
 
Nash, J., Non-cooperative games, Annals of Mathematics Studies, 54 (1951), 286-295.
 
Nash, J., The bargaining problem  Econometrica, 18 (1950), 155--162.
 
Neumann, B. H., Groups with finite classes of conjugate elements , Proc. Lond. Math. Soc.,  1 (1951),178--187.
 
Neumann, J. V., Morgenstern, O.,   Theory of Games and Economic Behavior, Princeton university press, 1944.
 
Reny, P. J., On the existence of pure and mixed strategy Nash equilibriain discontinuous games, Econometrica, 67 (1999), 1029--1056.
 
Schervish, M. J., Seidenfeld, T.,  A fair minimax theorem for two-person (zero-sum) games involving finitely additive strategies. In: Berry, D. A., Chaloner, K. M., Geweke, J. K. (eds.), Bayesian Analysis in Statistics and Econometrics. John Wiley & Sons, New York, 557–568, 1996.
 
Schmeidler, D., The nucleolus of a characteristic function game, SIAM Journal on Applied Mathematics, 17 (1969), 1163--1170.
 
Shapley, L. S., A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307--317.
 
Simon, L. K., Games with discontinuous payoffs,  Rev. Econ. Stud., 54 (1987), 569--597.
 
Simon, L. K., Zame, W. R., Discontinuous games and endogenous sharing rules, Econometrica, 58 (1990), 861--872.
 
Wald, A., Generalization of a theorem by von Neumann concerning zero-sum two person games, Annals  of Math.,  46 (1945), 281--286.
 
Yanovskaya, E. B., The solution of infinite zero-sum two-person games with finitely additive strategies, Theory Probab. Appl., 15 (1970), 153--158.