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

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).