On Sufficiently-Diffused Information in Bayesian Games: A Dialectical Formalization

  • M. Ali KhanEmail author
  • Yongchao Zhang
Part of the Advances in Mathematical Economics book series (MATHECON, volume 21)


There have been substantive recent advances in the existence theory of pure-strategy Nash equilibria (PSNE) of finite-player Bayesian games with diffused and dispersed information. This work has revolved around the identification of a saturation property of the space of information in the formalization of such games. In this paper, we provide a novel perspective on the theory through the extended Lebesgue interval presented in Khan and Zhang (Adv Math 229:1080–1103, 2012) [26] in that (i) it resolves the existing counterexample of Khan–Rath–Sun (J Math Econ 31:341–359, 1999) [17], and yet (ii) allows the manufacture of new examples. Through the formulation of a d-property of an abstract probability space, we exhibit a process under which a game without a PSNE in a specific class of games can be upgraded to one with: a (counter)example on any n-fold extension of the Lebesgue interval resolved by its \((n+1)\)-fold counterpart. The resulting dialectic that we identify gives insight into both the saturation property and its recent generalization proposed by He–Sun–Sun (Modeling infinitely many agents, working paper, National University of Singapore, 2013) [14] and referred to as nowhere equivalence. The primary motivation of this self-contained essay is to facilitate the diffusion and use of these ideas in mainstream non-cooperative game theory.                                     (190 words).


Bayesian games d-property Saturation property KRS-like games Lebesgue extension Nowhere equivalent \(\sigma \)-algebras 


  1. 1.
    Aumann RJ (2000) Collected papers, vol 1 and 2. MIT Press, CambridgeGoogle Scholar
  2. 2.
    Bogachev VI (2007) Measure theory, vol II. Springer, BerlinCrossRefzbMATHGoogle Scholar
  3. 3.
    Brucks KM, Bruin H (2004) Topics from one-dimensional dynamics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. 4.
    Carmona G, Podczeck K (2009) On the existence of pure-strategy equilibria in large games. J Econ Theory 144:1300–1319MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fajardo S, Keisler HJ (2002) Model theory of stochastic processes. A. K. Peters Ltd, MassachusettsGoogle Scholar
  6. 6.
    Fremlin DH (2011) Measure theory: the irreducible minimum, vol 1. Torres Fremlin, ColchesterzbMATHGoogle Scholar
  7. 7.
    Fu HF (2008) Mixed-strategy equilibria and strong purification for games with private and public information. Econ Theor 37:521–432MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grant S, Meneghel I, Tourky R (2015) Savage games, Theoretical Economics, forthcomingGoogle Scholar
  9. 9.
    Greinecker M, Podczeck K (2015) Purification and roulette wheels. Econ Theor 58:255–272MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Harsanyi JC (1983) Papers in game theory. D. Reidel Pub. Co., DordrechtzbMATHGoogle Scholar
  11. 11.
    Hoover D, Keisler HJ (1984) Adapted probability distributions. Trans Am Math Soc 286:159–201MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    He W, Sun YN (2013) The necessity of nowhere equivalence, working paper, National University of SingaporeGoogle Scholar
  13. 13.
    He W, Sun X (2014) On the diffuseness of incomplete information game. J Math Econ 54:131–137MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    He W, Sun X, Sun YN (2013) Modeling infinitely many agents, working paper, National University of Singapore. (Revised version, June 28, 2015. Forthcoming Theoretical Economics)Google Scholar
  15. 15.
    Kakutani S (1944) Construction of a non-separable extension of the Lebesque measure space. Proc Imp Acad 20:115–119CrossRefzbMATHGoogle Scholar
  16. 16.
    Keisler HJ, Sun YN (2009) Why saturated probability spaces are necessary. Adv Math 221:1584–1607MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khan M. Ali, Rath KP, Sun YN (1999) On a private information game without pure strategy equilibria. J Math Econ 31:341–359Google Scholar
  18. 18.
    Khan M. Ali, Rath KP, Sun YN (2006) The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games. Int J Game Theory 34:91–104Google Scholar
  19. 19.
    Khan M. Ali, Rath KP, Sun YN, Yu H (2013) Large games with a bio-social typology. J Econ Theory 148:1122–1149Google Scholar
  20. 20.
    Khan M. Ali, Rath KP, Sun YN, Yu H (2015) Strategic uncertainty and the ex-post Nash property in large games. Theor Econ 10:103–129Google Scholar
  21. 21.
    Khan M. Ali, Rath KP, Yu H, Zhang Y (2013) Large distributional games with traits. Econ Lett 118:502–505Google Scholar
  22. 22.
    Khan M. Ali, Rath KP, Yu H, Zhang Y (2014) Strategic representation and realization of large distributional games, Johns Hopkins University, mimeo. An earlier version presented at the Midwest Economic Theory Group Meetings held in Lawrence, Kansas, 14–16 October 2005Google Scholar
  23. 23.
    Khan M. Ali, Sagara N (2013) Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Ill J Math 57:145–169Google Scholar
  24. 24.
    Khan M. Ali, Sun YN (1996) Nonatomic games on Loeb spaces. Proc Nat Acad Sci USA 93, 15518–15521Google Scholar
  25. 25.
    Khan M. Ali, 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 Chapter 46Google Scholar
  26. 26.
    Khan M. Ali, Zhang YC (2012) Set-valued functions, Lebesgue extensions and saturated probability spaces. Adv Math 229:1080–1103Google Scholar
  27. 27.
    Khan M. Ali, Zhang YC (2014) On the existence of pure-strategy equilibria in games with private information: a complete characterization. J Math Econ 50:197–202Google Scholar
  28. 28.
    Khan M. Ali, Zhang YC (2017) Existence of pure-strategy equilibria in Bayesian games: A sharpened necessity result. Int J Game Theory 46:167–183Google Scholar
  29. 29.
    Khan M. Ali, Zhang YC (2014) On pure-strategy equilibria in games with correlated information. Johns Hopkins University, mimeoGoogle Scholar
  30. 30.
    Loeb PA (1975) Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans Amer Math Soc 211:113–122MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Loeb PA, Sun YN (2009) Purification and saturation. Proc Am Math Soc 137:2719–2724MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mordukhovich BS, Sagara N, Subdifferentials of nonconvex integral functionals in Banach spaces with applications to stochastic dynamic programming. J Convex Anal, forthcoming.
  33. 33.
    Maharam D (1942) On homogeneous measure algebras. Proc Nat Acad Sci USA 28:108–111MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Milgrom PR, Weber RJ (1985) Distributional strategies for games with incomplete information. Math Oper Res 10:619–632MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Myerson R (2004) Harsanyi’s games with incomplete information. Manag Sci 50:1818–1824CrossRefGoogle Scholar
  36. 36.
    Nash J (2007) The essential John Nash. In: Kuhn HW, Nassar S (eds). Princeton University Press, PrincetonGoogle Scholar
  37. 37.
    Nillsen R (2010) Randomness and recurrence in dynamical systems, Carus Mathematical Monographs no 31. MAA Service Center, WashingtonGoogle Scholar
  38. 38.
    Podczeck K (2009) On purification of measure-valued maps. Econ Theor 38:399–418MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Qiao L, Yu H (2014) On large strategic games with traits. J Econ Theory 153:177–190CrossRefGoogle Scholar
  40. 40.
    Radner R, Rosenthal RW (1982) Private information and pure strategy equilibria. Math Oper Res 7:401–409MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Radner R, Ray D (2003) Robert W. Rosenthal. J Econ Theory 112:365–368Google Scholar
  42. 42.
    Rashid S (1985) The approximate purification of mixed strategies with finite observation sets. Econ Lett 19:133–135MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Reny P (2011) On the existence of monotone pure strategy equilibria in Bayesian games. Econometrica 79:499–553MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schmeidler D (1973) Equilibrium points of non-atomic games. J Stat Phys 7:295–300CrossRefzbMATHGoogle Scholar
  45. 45.
    Sun YN, Zhang YC (2009) Individual risk and Lebesgue extension without aggregate uncertainty. J Econ Theory 144:432–443MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wang J, Zhang YC (2012) Purification, saturation and the exact law of large numbers. Econ Theory 50:527–545MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of EconomicsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.School of EconomicsShanghai University of Finance and EconomicsShanghaiChina

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