Perfect Bayesian equilibria not a subset of Nash equilibria?

Question

I came across a paper that contradicts what I know about PBE and NE. Since PBE is a refinement of NE, don't all PBE need to be NE?

The abstract is as follows (I'm looking for a logic error):

Perfect Bayesian equilibrium is not a subset of Nash equilibrium. Perfect Bayesian equilibrium requires players to have beliefs that are consistent with the equilibrium strategies of other players. Nash equilibrium does not explicitly specify the beliefs of the players. However, the default beliefs held by players in Nash equilibrium are that the other players are playing the equilibrium strategies (or a particular subset of the equilibrium strategies in the case of multiple equilibriums). The default beliefs and equilibrium strategies in Nash equilibrium could be more narrowly defined than that allowed under Perfect Bayesian equilibrium. Consequently, the set of equilibriums of perfect Bayesian equilibrium is not a subset of Nash equilibrium.

The paper is here.

Answer 1

It is not clear, from the paper you linked, which version of PBE the author is using.

PBE is somewhat of an elusive concept, see Fundenberg and Tirole (1991) for a discussion. A commonly used version is what Mas-Colell, Whinston, and Green (1995, aka MWG) call weak PBE, which is a strategy profile $\sigma$ and a system of beliefs $\mu$ such that :

• $\sigma$ is sequentially rational given $\mu$, and
• $\mu$ is derived from $\sigma$ using Bayes' rule whenever possible.

In the paper you linked, the author seems to have an erroneous understanding of the notion of consistent beliefs, namely, the second requirement of weak PBE. In particular, rather than using Bayes' rule to derive consistent beliefs, the author seems to think that a player indifferent between two (pure) behavioral strategies can hold any belief at an information set, regardless of how that information set is reached based on other players' strategies.

Instead of putting your faith in a little known working paper, I'd rather trust a better-established and wider cited source --- MWG, which states on p.286:

Hence, a weak perfect Bayesian equilibrium of game $\Gamma_E$ is a Nash equilibrium, but not every Nash equilibrium is a weak PBE.