Off-the-Path Beliefs

When a player needs to update his beliefs on the equilibrium path, Bayes’ rule gives a well-defined way to do this. However, there is no obvious way to update beliefs off the equilibrium path. Any such information sets are supposed to occur with probability 0. Attempting to apply Bayes’ rule therefore produces an undefined expression.

As a result, perfect Bayesian equilibrium allows for players to adopt any belief off the path. In turn, to fully exhaust all pooling equilibrium candidates, we have to consider every belief.

Takeaway Points

  1. When seeing what off-the-path beliefs can sustain a set of pooling strategies in equilibrium, begin by considering one type’s deviations. Then note which off-the-path beliefs sustain that player’s preference to maintain the pooling strategy.
  2. Repeat this process with all other types.
  3. Any belief that sustains all types’ preferences.
  4. If none do, then the pooling strategies are not a part of an equilibrium.

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