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

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Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 21))

Abstract

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).

JEL Classification Numbers: C62, D50, D82, G13

2010 Mathematics Subject: 28C99, 28E05, 91A13, 91A44

This work draws on results in a paper titled “On Sufficiently Diffused Information and Finite-Player Games with Private Information,” and initially presented at the workshop on The Probabilistic Impulse behind Modern Economic Theory held by the Institute for Mathematical Sciences, National University of Singapore (NUS) during January 11–18, 2011. A substantial version formed the basis of a talk by Khan at The 6th Conference on Mathematical Analysis in Economic Theory held at Keio University, January 26–29, 2015. He thanks Professor Toru Maruyama for his invitation, and him and Professors Bob Anderson, Damien Eldridge, Josh Epstein, Chiaki Hara, Alexander Ioffe, Takashi Kamihigashi, Nobusumi Sagara, Takashi Suzuki and Vladimir Tikhomirov for delightful and stimulating conversation at the Conference. Both authors also thank Hülya Eraslan, John Quah and Metin Uyanik for their comments and encouragement of this project. Section 4 of the Keio version was presented at a study group on Statistical Decision Theory organized by Professors Idione Meneghel and Rabee Tourky when the author held the position of Visiting Research Fellow at the Australian National University, February 15-April 15, 2016. This final version is to be presented at a session organized by Professor Jean-Marc Bonnisseau at the 16th Conference of the Society for the Advancement of Economic Theory to be held at IMPA, July 6–9, 2016. This research program is also supported by NSFC (11201283).

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Notes

  1. 1.

    The quotations are taken from Nillsen ([37], p. 340). The authors are grateful to Ashvin Rajan for bringing Nillson’s book to their attention.

  2. 2.

    These classical papers are well-known and now collected in [36].

  3. 3.

    These papers are now classical and well-known: for Harsanyi’s papers, see [10], and for Aumann’s, [1]. In particular, we shall make extensive reference to the 1974 paper of Aumann’s and the 1983 paper of Aumann et al., and these are respectively Chaps. 31 and 30 in [1].

  4. 4.

    Note that the 1973 paper of Harsanyi’s is available in [10, Sect. B], and so we only reference the latter in our bibliography; also see Footnote 3 above. For games with incomplete information, see [10, Sect. 1]; and also Myerson [35].

  5. 5.

    The authors have revisited Aumann’s equivalence theorem in [29], and the reader should not confuse it with the core equivalence theorem. It is important for the record to note that this was the background paper at Khan’s talk at Tokyo.

  6. 6.

    See Khan et al. [21, 22] for the terms large individualized and large distributionalized games, LIG and LDG respectively, and references to the antecedent literature on the concepts they name.

  7. 7.

    There is some controversy stemming from the fact that the results in Keisler–Sun [16] were obtained in 2002; see their acknowledgement, and also the use of their results by Noguchi in 2008. It is our firm intention not to get bogged down in this controversy here.

  8. 8.

    See Hoover–Keisler [11], and the comprehensive discussion in Fajardo–Keisler [5]; also the papers of, Carmona–Podczeck [4] and Keisler–Sun [16]. In a recent important paper, Modukhovich–Sagara [32] establish the relevance of the property in stochastic models of dynamic programming.

  9. 9.

    See Khan–Rath–Sun [17], and also its footnote to the Fudenberg-Tirole text as to the possible reason why it has remained neglected. Note that this example does not invoke any order structures on the action sets.

  10. 10.

    This notation then would lead the LI to be viewed as 0-fold extended LI and the extended LI in (i) above as a 1-fold extended LI.

  11. 11.

    We shall be referring to this below as a “scrambling” operation on a particular game.

  12. 12.

    In particular, the upgraded games in (i) and (ii) above belong to the class of KRS-like games that is being singled out and studied in this paper.

  13. 13.

    The references [9, 13, 14] to the relative-saturation property are also relevant in this connection. Precise definitions of these and other properties referred to in this introduction will be offered in the sequel.

  14. 14.

    This is done on the basis of the fact that there exists a measurable mapping h from an abstract atomless probability space to the usual Lebesgue unit interval such that its induced distribution is the Lebesgue measure itself; see [16, Lemma 2.1] and the discussion in Sect. 2 below.

  15. 15.

    This is a consequence of the well-known fact that there exists a continuous onto function from any uncountable compact metric space to \([-1,1];\) see, for example, Rath–Sun–Yamashige referenced in [25] for this.

  16. 16.

    This correspondence is reproduced in Fig. 2 below, and was referred to in [26] as the Debreu correspondence simply as a mnemonic; and as indicated there, Hart-Kohlberg ascribe it to Debreu in an entirely different context and for an entirely different purpose. Our current use of the letter d for this correspondence, and for the d-property of a measure space based on it, is meant to indicate a situation where each type of agent has a dual best-response. However, if the reader wishes, he or she can capitalize d and make a non-obligatory nod in Debreu’s direction.

  17. 17.

    One of these correspondences is precisely the d-correspondence. Another derives from the celebrated example of Lyapunov; see Claims 1–3 in [26, Sect. 1]. We underscore for the general reader the intuitively-obvious fact that the Lebesgue extension is mathematically much simpler than the saturated extension of the Lebesgue interval in [45]. For Lyapunov’s theorem, see [23] and their references.

  18. 18.

    This, by itself is no longer surprising. It is now understood, at least by the cognoscenti, that one only needs a \(\sigma \)-algebra that is finer than the Lebesgue \(\sigma \)-algebra in the sense that it contains a set of measure 1/2 and which is independent of the Lebesgue \(\sigma \)-algebra; see [13] written subsequent to the first version of this paper.

  19. 19.

    There is of course a Godelian parallel here. Let \(T_1\) be a suitable theory, which is to say, complete and consistent. Then it admits an undecidable proposition, call it \(S_1.\) Let \(T_2\) be \(T_1\) extended by \(S_1,\) and denoted \(T_2 = \{T_1 + G_1\}.\) Observe that although \(G_1\) is trivially deducible in \(T_2,\) there is another undecidable in \(T_2\), say \(S_2\) etc. \(S_n\) is never decidable in \(T_{n-1}.\) In fact there is a countably-infinite series of pairs of theories and undecidables ! Extensions of this type never work to furnish a general theory. The authors are grateful to Josh Epstein for bringing the relevance of Godel’s incompleteness theorem to their attention. Josh also singled out parallels to Galois theory whose pursuit in this paper would have taken us too far afield.

  20. 20.

    We work with the simplest model; for extensions, see [7, 8, 19, 21, 39].

  21. 21.

    Since Loeb spaces are saturated, the sufficiency result generalizes previous work; see [27].

  22. 22.

    It is worthy of emphasis here that, as noted in [27], this necessary and sufficient result was already conjectured in [16]. Indeed, the necessity claim was first made by Keisler–Sun in 2002; see the relevant footnote in their paper.

  23. 23.

    Here \(\mathcal{T}^S\) is the \(\sigma \)-algebra \(\left\{ S\cap S' :S' \in \mathcal{T} \right\} \) and \(\mu ^S\) is defined on \(\mathcal{T}^S\) by \(\mu (\cdot ) = \mu (\cdot ) / \mu (S)\). The reader is referred [27] for details and references.

  24. 24.

    Even though it constitutes a rather narrow perspective from which to view this paper, one could in principle, see the results reported here as addressing themselves to the problem left open in [13].

  25. 25.

    There always exists such a function \(h_i\), see [2, Proposition 9.1.11].

  26. 26.

    A map \(h: (T, \mathcal{F}, \mu ) \rightarrow ([0,1], \mathcal{L}, \eta )\) is called measure-preserving if h is \(\mathcal{F - L}\)-measurable and the induced distribution of h is the Lebesgue measure on the unit interval.

  27. 27.

    Several excellent treatments of this standard material are available, but we hope that the following two paragraphs will not only set the stage for what is to follow but will speak to the general reader; for details, we recommend [6, 37].

  28. 28.

    But see Nillson’s singling this equality out in [37, Eq. 5.6, p. 304]. Khan would like to take this opportunity to thank Metin Uyanik for discussion concerning this “Carathéodory equation.”

  29. 29.

    See the epigraph, and the discussion in [37, Sect. 5.4].

  30. 30.

    Even though the property is intuitive, relying as it does on the notion of a length of an interval and what it means to have cover, it nevertheless requires a proof. Henceforth, by cover we shall mean a cover of half-open intervals.

  31. 31.

    See the notes and comments in Fremlin ([6]; Sect. 113); also see [3, 37].

  32. 32.

    Given a measure space \((T, \mathcal{T}, \mu )\), the associated outer measure , denoted by \(\mu ^*\), is defined as follows: for any subset \(E \subseteq T\), \(\mu ^*(E) = \inf \left\{ \Sigma _n \mu (E_n): E_n \in \mathcal{T}, E \subseteq \cup _n E_n\right\} \), it bears emphasis that the infimum is taken over all countable covers of E.

  33. 33.

    The details of each of these steps are spelt out in [26]. It is a good exercise for the interested reader to work out for herself the routine arithmetic behind each of these steps. She should note, in particular, that the proof of the claim that the outer-measure of C is unity (straightforwardly) invokes Fubini’s theorem.

  34. 34.

    As in Footnote 32, we send the reader interested in the details to [26]; and in this particular context, to Sects. 5.2.2 and 5.2.3 in that paper.

  35. 35.

    See [4, 16] for definition of these terms.

  36. 36.

    It may be worth pointing out that this is the first substantive application, and an exposition, of this extended Lebesgue interval in the economics literature.

  37. 37.

    More precisely, in the KRS-like game \(\Gamma _{h_n, h_n}\), the corresponding \(s_1, s_2\) in the proof of [27, Theorem 2] are both one.

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Authors and Affiliations

  1. Department of Economics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    M. Ali Khan

  2. School of Economics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 200433, China

    Yongchao Zhang

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  1. M. Ali Khan
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Correspondence to M. Ali Khan .

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Editors and Affiliations

  1. Professor Emeritus, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Shigeo Kusuoka

  2. Professor Emeritus, Keio University , Minato-ku, Tokyo, Japan

    Toru Maruyama

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Ali Khan, M., Zhang, Y. (2017). On Sufficiently-Diffused Information in Bayesian Games: A Dialectical Formalization. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics . Advances in Mathematical Economics, vol 21. Springer, Singapore. https://doi.org/10.1007/978-981-10-4145-7_2

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