Advances in Mathematical Economics pp 47-73 | Cite as

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

## 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).

## Keywords

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

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