Wang, Jianwei and Zhang, Yongchao (2010): Purification, Saturation and the Exact Law of Large Numbers.
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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|
|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
|Depositing User:||Yongchao Zhang|
|Date Deposited:||20. Apr 2010 20:40|
|Last Modified:||31. Aug 2015 14:48|
C.D. Aliprantis and K.C. Border, Innite dimensional analysis: a hitchhiker's guide, Springer-Verlag, 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), 256--260.
A. Dvoretzky, A. Wald and J. Wolfowitz, Relations among certain ranges of vector measures, Pacic Journal of Mathematics 1 (1951a), 59--74.
A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision problems in certain statistical decision procedures and zero-sum two-person games, Annals of Mathematical Statistics 22 (1951b), 1--21.
D.A. Edwards, On a theorem of Dvoretzky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29 (1987), 205--220.
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: Set-theoretic 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), 159--201.
H.J. Keisler and Y.N. Sun, Why saturated probability spaces are necessary, Advances in Mathematics 221 (2009), 1584--1607.
M.A. Khan and K.P. Rath, On games with incomplete information and the Dvoretzky-Wald-Wolfowitz theorem with countable partitions, Journal of Mathematical Economics 45 (2009), 830--837.
M.A. Khan, K.P. Rath and Y.N. Sun, The Dvoretzky-Wald-Wolfowitz theorem and purication in atomless nite-action games, International Journal of Game Theory 34 (2006), 91--104.
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 Cournot-Nash equilibrium distributions in a nite-action, atomless game, in Equilibrium Theory in Innite Dimensional Spaces (M.A. Khan and N.C. Yannelis eds.), Springer-Verlag, Berlin, 1991, pp. 325-332.
M.A. Khan and Y.N. Sun, Non-cooperative 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. 1761--1808.
P.A. Loeb and Y.N. Sun, Purication of measure-valued maps, Illinois Journal of Mathematics 50 (2006), 747--762.
P.A. Loeb and Y.N. Sun, Purication and saturation, Proceedings of the American Mathematics Society 137 (2009), 2719--2724.
P.A. Loeb and M. Wol, Nonstandard Analysis for the Working Mathematician, Kluwer Academic Publishers, Amsterdam, 2000.
A. Mas-Colell, On a theorem of Schmeidler, Journal of Mathematical Economics 13 (1984), 200--206.
P.R. Milgrom and R.J. Weber, Distributional strategies for games with incomplete information, Mathematics of Operations Research 10 (1985), 619--632.
D. Maharam, On algebraic characterization of measure algebras, Annals of Mathematics 48 (1947), 154--167.
M. Noguchi, Existence of Nash equilibria in large games, Journal of Mathematical Economics 45 (2009), 168--184.
K. Podczeck, On purication of measure-valued maps, Economic Theory 38 (2009), 399--418.
K. Podczeck, On existence of rich Fubini extentions, Economic Theory, pubilished online, 2009.
R. Radner and R.W. Rosenthal, Private information and pure-strategy equilibria, Mathematics of Operations Research 7 (1982), 401--409.
D. Schmeidler, Equilibrium points of non-atomic games, Journal of Statistical Physics 7 (1973), 295--300.
Y.N. Sun, A theory of hypernite processes: the complete removal of individual uncertainty via exact LLN, Journal of Mathematical Economics 29 (1998), 419--503.
Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, Journal of Economic Theory 126 (2006), 31--69.
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), 861--865.
Y.N. Sun and Y.C. Zhang, Individual risk and Lebesgue extension without aggregate uncertainty, Journal of Economic Theory 144 (2009), 432--443.