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

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.

1. In alphabetical order, see Carmona and Podczeck (2009), Keisler and Sun (2009), Loeb and Sun (2009), Podczeck (2009), and for more recent work in chronological order Wang and Zhang (2012), He and Sun (2014), He et al. (2013), Qiao and Yu (2014), Khan et al. (2013a, b, 2014a, b), Khan and Zhang (2014), Greinecker and Podczeck (2015), Sun and Zhang (2015). All of this work revolves around so-called saturated or super-atomless probability spaces, the measure-theoretic property that constitutes the subtext of this work.

2. Keisler and Sun (2009) take their cue from the saturation property formalized in Hoover and Keisler (1984) in the form of a saturated filtration, a motivation that cames from the need for a systematic study of the existence of strong solutions for stochastic integral equations; see Fajardo and Keisler (2002) for a comprehensive discussion which connects the property to Maharam’s classification of measure algebras. For an approach to the subject that begins with Maharam, and thereby refers to the intrinsic super-atomless property of a probability space, see Carmona and Podczeck (2009), a paper that also has a synthetic thrust.

3. See Khan et al. (1999), and note that a general existence theorem with countably many moves is available (see Khan and Sun 2002 and their references), and so it an action set with uncountably many moves that poses the difficulties.

4. See the recent work of the authors Khan and Zhang (2014) and of He and Sun (2014), and prior to them, to Khan and Sun (2002), and its full set of references, for the earlier work involving Loeb spaces. Also see He et al. (2013) and Greinecker and Podczeck (2015) and their references. We comment on this work below.

5. Also see the description in the last but one paragraph in Radner and Ray (2003). Relative to the earlier 1974 work of Aumann’s, the RR work in a setting where the players’ payoffs depend on individual types or private information, and also consider information part of which is public, but we abstain from this latter aspect here.

6. Since Loeb spaces are saturated, the sufficiency result generalizes previous work; see Khan and Zhang (2014).

7. It is worthy of emphasis here that, as noted in Khan and Zhang (2014), this closing of the circle for finite-player Bayesian games was already conjectured in Keisler and Sun (2009); see their Footnote 2 for a sketch of the sufficiency result.

8. We are grateful to an anonymous referee for emphasizing Fremlin’s treatment, and specifically his Definition 331A. This concept is presented in He and Sun (2014) under the phraseology that “\(\mathscr {F}_i\) is setwise coarser than \(\mathscr {T}_i\)”.

9. Here \(\mathscr {T}^S\) is the \(\sigma \)-algebra \(\left\{ S\cap S' :S' \in \mathscr {T} \right\} \) and \(\mu ^S\) is defined on \(\mathscr {T}^S\) by \(\mu (\cdot ) = \mu (\cdot ) / \mu (S)\). The reader is referred Khan and Zhang (2014) for details and references.

10. Fact 1(i) is Proposition 2 of Khan et al. (1999), and Fact 1(ii) is a combination of Claims 1 and 2 therein.

11. There always exists such a function \(h_i\), see (Bogachev 2007, Proposition 9.1.11) .

12. For the latter, think of a function h manufactured on 2n intervals of the interval, and with a tent-map on each of the n pairs of intervals. For the dynamics of such maps, see Brucks and Bruin (2004).

13. We suppress the fact that the game \(\varGamma _{h_1, h_2}\) also depends on \((T_i, \mathscr {T}_i, \mu _i),\; i=1,2.\)

14. We are grateful to an anonymous referee for this observation.

15. This RR example, just like the KRS example, does not satisfy the main assumption in Grant et al. (2015), for example.

16. See e.g. Lemma 4 of Khan and Zhang (2012).

17. The construction of h is rather standard in the literature: see, for example, Lemma 5 in Podczeck (2009), and its mention in the proof of Theorem 3.7 in Keisler and Sun (2009).

The authors are grateful to an Associate Editor, and two anonymous referees, of this journal for their comments. This work reports results extracted from a larger and more diffused essay entitled “On sufficiently diffused information and finite-player games with private information”, originally circulated in 2012. For encouragement and stimulating discussion concerning the authors’ research program, they thank Josh Epstein, Hülya Eraslan, Micheal Greneicker, Wei He, Paulo Klinger Monteiro, Konrad Podczeck, John Quah, Kali Rath, Nobusumi Sagara, Xiang Sun, Yeneng Sun, Metin Uyanik and Haomiao Yu.

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

3. Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, 200433, China

   Yongchao Zhang

Corresponding author

Correspondence to Yongchao Zhang.

Y. Zhang acknowledges financial support from National Natural Science Foundation of China (No. 11201283).

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