Equilibria existence in Bayesian games: climbing the countable Borel equivalence relation hierarchy

Hellman, Z. and Levy, Y. J. (2022) Equilibria existence in Bayesian games: climbing the countable Borel equivalence relation hierarchy. Mathematics of Operations Research, 47(1), pp. 367-383. (doi: 10.1287/moor.2021.1135)

Abstract

The solution concept of a Bayesian equilibrium of a Bayesian game is inherently an interim concept. The corresponding ex ante solution concept has been termed a Harsányi equilibrium; examples have appeared in the literature showing that there are Bayesian games with uncountable state spaces that have no Bayesian approximate equilibria but do admit a Harsányi approximate equilibrium, thus exhibiting divergent behaviour in the ex ante and interim stages. Smoothness, a concept from descriptive set theory, has been shown in previous works to guarantee the existence of Bayesian equilibria. We show here that higher rungs in the countable Borel equivalence relation hierarchy can also shed light on equilibrium existence. In particular, hyperfiniteness, the next step above smoothness, is a sufficient condition for the existence of Harsányi approximate equilibria in purely atomic Bayesian games.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Levy, Dr John
• Authors: Hellman, Z., and Levy, Y. J.
• College/School: College of Social Sciences > Adam Smith Business School > Economics
• Journal Name: Mathematics of Operations Research
• Publisher: INFORMS
• ISSN: 0364-765X
• ISSN (Online): 1526-5471
• Published Online: 31 August 2021
• Copyright Holders: Copyright © 2021 INFORMS
• First Published: First published in Mathematics of Operations Research 47(1): 367-383
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher