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I'm struggling to understand why the notion of "belief revision" is an important concept. In particular, why does the belief over information sets with probability zero matter?

When comparing to the notion of "weak sequential equilibriums" (i.e. an assessment that satisfies sequential rationality and Bayesian updating at reached information sets), since both equilibria satisfy sequential rationality, does this mean that for any profile $\sigma_W$ of a weak sequential equilibrium, there exists a profile $\sigma_P$ belonging to a perfect Bayesian equilibrium such that $\sigma_W$ and $\sigma_P$ agree on information sets with positive probability?

Finally, suppose that all information sets have non-zero probability. In this case, is every weak sequential equilibrium also a perfect Bayesian equilibrium?

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The key is that "since both equilibria satisfy sequential rationality" is no longer true when you consider weak sequential equilibria.

Both concepts satisfy sequential rationality on-path, but the whole point of weak equilibria is that off-path, we allow for any beliefs that can disregard sequential rationality, while a PBE would force you to have sequential rationality even on paths of play that are never reached.

No, having $\sigma_W$ as a weak equilibria might require empty threats or crazy beliefs off-path, so there may not be a PBE supporting the same path of play (the reverse is obviously true).

If all information sets are reached with non-zero probability, then you must always use Bayes rule, so you are correct.

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