Semi-Separating/Partially Pooling Equilibrium

We have learned about separating equilibria, where types choose distinct actions and therefore reveal themselves to the receiver. We have learned about pooling equilibria, where types take identical actions and therefore the receiver learns nothing from what they observe.

Semi-separating/partially pooling equilibria fall somewhere in between. One type takes a pure strategy. Another type sometimes also takes that pure strategy but sometimes takes a different pure strategy. In the former case, the receiver learns something, but the signal is noisy. In the latter case, the receiver learns the opponent’s type.

Note: Semi-separating equilibrium and partially pooling equilibrium are the same thing. The label is taste-based. I will use semi-separating equilibrium as the label for the rest of the course.

Takeaway Points

  1. The main guiding principle of semi-separating equilibrium is indifference conditions. Remembering that a type that mixes must be indifferent between the pure strategies it randomizes over is key to generating all of the equilibrium strategies.
  2. The first application of indifference conditions is the type that chooses between two different actions. The receiver’s response must make that type find such a mixture optimal.
  3. Normally, making the semi-separating type indifferent requires the receiver to also mix. Thus, we can generate the receiver’s strategy by finding what makes the semi-separating type indifferent.
  4. But the receiver mixing also requires the receiver to be indifferent. Thus, we can generate the semi-separating type’s mixing probability by finding the strategy that generates the receiver’s indifference.
  5. This last step can be the trickiest part. The receiver must be indifferent based on its posterior belief. Thus, finding the semi-separating type’s mixed strategy requires filtering through Bayes’ rule.

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