Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result

International Journal of Game Theory - Tập 46 - Trang 167-183 - 2016
M. Ali Khan1, Yongchao Zhang2,3
1Department of Economics, The Johns Hopkins University, Baltimore, USA
2School of Economics, Shanghai University of Finance and Economics, Shanghai, China
3Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, China

Tóm tắt

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

Tài liệu tham khảo

Bogachev VI (2007) Measure theory, vol II. Springer-Verlag, Berlin Brucks KM, Bruin H (2004) Topics from one-dimensional dynamics. Cambridge University Press, Cambridge Carmona G, Podczeck K (2009) On the existence of pure-strategy equilibria in large games. J Econ Theory 144:1300–1319 Fajardo S, Keisler HJ (2002) Model theory of stochastic processes. Peters AK Ltd, Massachusetts Fremlin DH (2002) Measure theory: measure algebras, vol 3. Torres Fremlin, Colchester Fu HF (2008) Mixed-strategy equilibria and strong purification for games with private and public information. Econ Theory 37:521–432 Grant S, Meneghel I, Tourky R (2015) Savage games. Theor Econ Greinecker M, Podczeck K (2015) Purification and roulette wheels. Econ Theory 58:255–272 Hoover D, Keisler HJ (1984) Adapted probability distributions. Trans Am Math Soc 286:159–201 He W, Sun X (2014) On the diffuseness of incomplete information game. J Math Econ 54:131–137 He W, Sun X, Sun YN (2013) Modeling infinitely many agents, working paper. National University of Singapore Keisler HJ, Sun YN (2009) Why saturated probability spaces are necessary. Adv Math 221:1584–1607 Khan MA, Rath KP, Sun YN (1999) On a private information game without pure strategy equilibria. J Math Econ 31:341–359 Khan MA, Rath KP, Sun YN (2006) The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games. Int J Game Theory 34:91–104 Khan MA, Rath KP, Sun YN, Yu H (2013) Large games with a bio-social typology. J Econ Theory 148:1122–1149 Khan MA, Rath KP, Sun YN, Yu H (2014) Strategic uncertainty and the ex-post Nash property in large games. Theor Econ Khan MA, Rath KP, Yu H, Zhang Y (2013) Large distributional games with traits. Econ Lett 118:502–505 Khan MA, Rath KP, Yu H, Zhang Y (2014) Strategic representation and realization of large distributional games. Johns Hopkins University Khan MA, Sun YN (2002) Non-Cooperative games with many players. In: Aumann RJ, Hart S (eds) Handbook of game theory, vol 3. Elsevier Science, Amsterdam, pp 1761–1808 Khan MA, Zhang Y (2012) Set-Valued functions, Lebesgue extensions and saturated probability spaces. Adv Math 229:1080–1103 Khan MA, Zhang Y (2014) On the existence of pure-strategy equilibria in games with private information: a complete characterization. J Math Econ 50:197–202 Khan MA, Zhang Y (2015) On pure-strategy equilibria in games with correlated information. Johns Hopkins Unversity, Mimeo Loeb PA, Sun YN (2009) Purification and saturation. Proc Am Math Soc 137:2719–2724 Milgrom PR, Weber RJ (1985) Distributional strategies for games with incomplete information. Math Oper Res 10:619–632 Podczeck K (2009) On purification of measure-valued maps. Econ Theory 38:399–418 Qiao L, Yu H (2014) On large strategic games with traits. J Econ Theory 153:177–190 Sun X, Zhang Y (2015) Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces. Econ Theory 58:161–182 Radner R, Rosenthal RW (1982) Private information and pure strategy equilibria. Math Oper Res 7:401–409 Radner R, Ray D (2003) Robert W. Rosenthal. J Econ Theory 112:365–368 Wang J, Zhang Y (2012) Purification, saturation and the exact law of large numbers. Econ Theory 50:527–545