Having seen a simple example of perfect Bayesian equilibrium in a signaling model, we now turn to pooling equilibrium. This is more complicated than a separating equilibrium because the uninformed actor must develop off-the-path beliefs, and PBE provides no guidance on what to do here.
- Review: Types pool when they take identical actions.
- On the equilibrium path, the uninformed actor’s belief must be identical to its prior. It cannot meaningfully update its information because there is no separation occurring between the types.
- Finding pooling equilibria is a four step process:
- Identify a set of pooling strategies.
- Solve for the other player’s best response to those strategies.
- Check whether the first player can profitably deviate.
- Repeat Step #1 until you have exhausted all sets of pooling strategies.
- Checking for profitable deviations is the most difficult part of finding pooling equilibria because of off-the-path beliefs. This is not a concern for the strategies considered in this lecture because all of the off-the-path information sets can only arise from a single type. When that is not the case, the pooling equilibria become more difficult to find. The next lecture describes what to do then.