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