Abstract
Purification results are important in game theory and statistical decision theory. We prove a new purification theorem that generalizes several earlier 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.
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The authors are very grateful to Yi-Chun Chen, Sengkee Chua, Haifeng Fu, M. Ali Khan, Konrad Podczeck, Xiao Luo, Yeneng Sun, Nicholas C. Yannelis for helpful comments. The first draft of this paper was prepared when Yongchao Zhang participated in the Trimester Program on Mechanism Design held at the Hausdorff Research Institute for Mathematics (HIM), Bonn University, May–August 2009, and financial support from HIM is specially acknowledged. This paper was presented at the tenth SAET (Society for the Advancement of Economic Theory) conference held at Singapore, August 13–15, 2010 and the first CGTEEA (Chinese Game Theory and Experimental Economics Association) conference held at Beijing, August 24–26, 2010.
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Wang, J., Zhang, Y. Purification, saturation and the exact law of large numbers. Econ Theory 50, 527–545 (2012). https://doi.org/10.1007/s00199-010-0593-3
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DOI: https://doi.org/10.1007/s00199-010-0593-3
Keywords
- Exact law of large numbers
- Fubini extension
- Incomplete information
- Purification
- Saturated probability space