Pooling Equilibrium

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.

Takeaway Points

  1. Review: Types pool when they take identical actions.
  2. 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.
  3. Finding pooling equilibria is a four step process:
    1. Identify a set of pooling strategies.
    2. Solve for the other player’s best response to those strategies.
    3. Check whether the first player can profitably deviate.
    4. Repeat Step #1 until you have exhausted all sets of pooling strategies.
  4. 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.

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