Intro
Undergraduate and MBA students across California—including UCLA, USC, UC Berkeley, UC Irvine, UC Davis, UC Santa Cruz, UC Riverside, and the CSU system—study pooling equilibria because they appear in signaling games where different types choose the same action. A common confusion is why pooling equilibria rely on off‑path beliefs to prevent types from deviating. For additional support, visit our Game Theory Tutoring or explore related topics in the Economics Blog Hub.
Answer First
Pooling equilibria require off‑path beliefs because if all types choose the same action, the receiver cannot infer type from behavior. To prevent a type from deviating to a different signal, the receiver must hold beliefs that make deviation unprofitable. Without such beliefs, the pooling equilibrium collapses.
Problem Setup
In a signaling game:
- Sender has private type θ ∈ {θ₁, θ₂}.
- Sender chooses signal s.
- Receiver observes s and chooses action a.
A pooling equilibrium has: \[ s(θ₁) = s(θ₂). \] Off‑path beliefs μ(θ | s′) determine how the receiver interprets deviations.
Step-by-Step Explanation
1. Pooling means the receiver learns nothing from the signal
If both types choose the same signal, the receiver cannot update beliefs using Bayes’ rule. The receiver must rely on prior beliefs or off‑path beliefs.
2. A type may want to deviate to separate itself
If a high‑type sender benefits from revealing its type, it may deviate unless the receiver’s beliefs make deviation unattractive.
3. Off‑path beliefs punish deviations
To sustain pooling, the receiver must believe that any deviation comes from the type that would be harmed by the receiver’s response. Example: If the receiver believes a deviation comes from the low type, the receiver chooses an action that hurts the deviator.
4. Without off‑path beliefs, pooling is rarely stable
Most pooling equilibria unravel because:
- high types want to separate,
- low types want to mimic high types,
- the receiver cannot discipline deviations without beliefs.
5. Equilibrium refinements eliminate unreasonable pooling equilibria
Refinements like:
- Intuitive Criterion,
- D1 Criterion,
- Universal Divinity
rule out pooling equilibria supported by implausible off‑path beliefs.
Intuition
Pooling equilibria are like two students turning in identical work. If one deviates, the teacher must decide what that deviation means. The teacher’s interpretation determines whether copying continues or breaks down.
Common Exam Mistakes
- Thinking pooling equilibria do not require beliefs.
- Assuming Bayes’ rule applies to off‑path signals (it does not).
- Ignoring incentive compatibility for both types.
- Forgetting that refinements eliminate weak pooling equilibria.
Final Summary
Pooling equilibria require off‑path beliefs because the receiver cannot infer type from the observed signal. Beliefs must make deviations unprofitable for all types; otherwise, the pooling equilibrium unravels under incentive compatibility or refinement criteria.