Wang, Jianwei and Zhang, Yongchao (2010): Purification, Saturation and the Exact Law of Large Numbers.

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Abstract
Purification results are important in game theory and statistical decision theory. The purpose of this paper is to prove a general purification theorem that generalizes many authors' results. The key idea of our proof is to make use of the exact law of large numbers. As an application, we show that every mixed strategy in games with finite players, general action spaces, and diffused, conditionally independent incomplete information has many strong purifications.
Item Type:  MPRA Paper 

Original Title:  Purification, Saturation and the Exact Law of Large Numbers 
Language:  English 
Keywords:  Exact law of large numbers, Fubini extension, Incomplete information, Purification, Saturated probability space 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C70  General C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60  General 
Item ID:  22119 
Depositing User:  Yongchao Zhang 
Date Deposited:  20 Apr 2010 20:40 
Last Modified:  29 Sep 2019 12:56 
References:  C.D. Aliprantis and K.C. Border, Innite dimensional analysis: a hitchhiker's guide, SpringerVerlag, Berlin, 1994. J.L. Doob, Stochastic Processes, Wiley, New York, 1953. A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain problems of statistics and of the theory of games, Proceedings of National Academy of Science, USA 36 (1950), 256260. A. Dvoretzky, A. Wald and J. Wolfowitz, Relations among certain ranges of vector measures, Pacic Journal of Mathematics 1 (1951a), 5974. A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision problems in certain statistical decision procedures and zerosum twoperson games, Annals of Mathematical Statistics 22 (1951b), 121. D.A. Edwards, On a theorem of Dvoretzky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29 (1987), 205220. S. Fajardo and H.J. Keisler, Model Theory of Stochastic Processes, Lecture Notes in Logic, vol. 14, Assoc. Symbolic Logic, Urbana, IL, 2002. D.H. Fremlin, Measure Theory, Volume 5: Settheoretic Measure Theory (version 8.5.03/29.6.06), 2005. Available at http://www.essex.ac.uk/maths/sta/fremlin/mt.htm D.N. Hoover and H.J. Keisler, Adapted probability distributions, Transactions of the American Mathematical Society 286 (1984), 159201. H.J. Keisler and Y.N. Sun, Why saturated probability spaces are necessary, Advances in Mathematics 221 (2009), 15841607. M.A. Khan and K.P. Rath, On games with incomplete information and the DvoretzkyWaldWolfowitz theorem with countable partitions, Journal of Mathematical Economics 45 (2009), 830837. M.A. Khan, K.P. Rath and Y.N. Sun, The DvoretzkyWaldWolfowitz theorem and purication in atomless niteaction games, International Journal of Game Theory 34 (2006), 91104. M.A. Khan, K.P. Rath, Y.N. Sun and H.M. Yu, On large games with a multiplicities of types of players, The Johns Hopkins University, mimeo., 2010. M.A. Khan, Y.N. Sun, On symmetric CournotNash equilibrium distributions in a niteaction, atomless game, in Equilibrium Theory in Innite Dimensional Spaces (M.A. Khan and N.C. Yannelis eds.), SpringerVerlag, Berlin, 1991, pp. 325332. M.A. Khan and Y.N. Sun, Noncooperative games with many players, in Handbook of Game Theory with Economic Applications, Volume III (R.J. Aumann and S. Hart eds), Elsevier Science, Amsterdam, 2002, pp. 17611808. P.A. Loeb and Y.N. Sun, Purication of measurevalued maps, Illinois Journal of Mathematics 50 (2006), 747762. P.A. Loeb and Y.N. Sun, Purication and saturation, Proceedings of the American Mathematics Society 137 (2009), 27192724. P.A. Loeb and M. Wol, Nonstandard Analysis for the Working Mathematician, Kluwer Academic Publishers, Amsterdam, 2000. A. MasColell, On a theorem of Schmeidler, Journal of Mathematical Economics 13 (1984), 200206. P.R. Milgrom and R.J. Weber, Distributional strategies for games with incomplete information, Mathematics of Operations Research 10 (1985), 619632. D. Maharam, On algebraic characterization of measure algebras, Annals of Mathematics 48 (1947), 154167. M. Noguchi, Existence of Nash equilibria in large games, Journal of Mathematical Economics 45 (2009), 168184. K. Podczeck, On purication of measurevalued maps, Economic Theory 38 (2009), 399418. K. Podczeck, On existence of rich Fubini extentions, Economic Theory, pubilished online, 2009. R. Radner and R.W. Rosenthal, Private information and purestrategy equilibria, Mathematics of Operations Research 7 (1982), 401409. D. Schmeidler, Equilibrium points of nonatomic games, Journal of Statistical Physics 7 (1973), 295300. Y.N. Sun, A theory of hypernite processes: the complete removal of individual uncertainty via exact LLN, Journal of Mathematical Economics 29 (1998), 419503. Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, Journal of Economic Theory 126 (2006), 3169. Y.N. Sun, Large Bayesian games, National University of Singapore, mimeo., 2007a. Y.N. Sun, On the Characterization of Individual Risks and Fubini Extension, National University of Singapore, mimeo., 2007b. Y.N. Sun, N.C. Yannelis, Saturation and the integration of Banach valued correspondences, Journal of Mathematical Economics 44 (2008), 861865. Y.N. Sun and Y.C. Zhang, Individual risk and Lebesgue extension without aggregate uncertainty, Journal of Economic Theory 144 (2009), 432443. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/22119 