Lecture 9. Signalling (signalling)
Recommended reading : 【Microeconomics】 Microeconomics Table of Contents
1. Overview
2. Separating equilibrium
3. Pooling equilibrium
4. Hybrid equilibrium
1. Overview
**⑴ Signalling theory:** Mainly applied to the job market
① A worker’s action becomes a signal, allowing the employer to infer the worker’s type.
② A sequential version of incomplete information.
○ (Reference) Contract theory is the simultaneous version of incomplete information.
③ Example : Education level
○ Education is used as one means to distinguish capable and less capable people.
○ In other words, the fact that someone received education conveys information about ability.
○ Since education is unrelated to real productivity gains, the theory argues that higher education should not lead to higher wages.
⑵ Stage 1. The worker’s type is determined by a chance node.
① If t = H, the worker is a high-skilled worker.
② If t = L, the worker is a low-skilled worker.
⑶ Stage 2. After observing their type, the worker chooses E or NE (E : education; NE : no education).
① If the worker chooses E, a type-H worker and a type-L worker incur costs cH and cL, respectively.
② In general, the cost of education for a high-skilled worker is not large.
③ If the worker is hired, they receive a wage w.
⑷ Stage 3. After observing the worker’s choice, the employer decides whether to hire (J) or not hire (NJ).
① (Reference) Joint probability
② (Reference) Conditional probability
③ (Reference) Bayes’ rule (Bayes’ theorem)
④ If the employer hires a type-L worker, the payoff is -w; if the employer hires a type-H worker, the payoff is π - w.
⑤ Depending on the specific design of the problem, the information available to the employer can differ.
⑸ Perfect Bayesian equilibrium
① Nash equilibrium condition : At each information set, players choose actions that maximize their payoffs.
② Bayes’ rule : Beliefs about types are updated according to Bayes’ rule.
2. Separating equilibrium
**⑴ Overview:** The case where H and L choose different actions.
⑵ Step 1. Worker’s strategy
① Assume Pr(E H) = 1 and Pr(NE L) = 1.
② That is, assume the worker’s strategy is (E H, NE L).
③ Also assume cH ≤ w ≤ cL.
⑶ Step 2. Compute the employer’s beliefs.
⑷ Step 3. Employer’s best response
① Employer’s payoff if the worker chooses E and the employer chooses J
② Employer’s payoff if the worker chooses E and the employer chooses NJ
③ Employer’s payoff if the worker chooses NE and the employer chooses J
④ Employer’s payoff if the worker chooses NE and the employer chooses NJ
⑤ Conclusion : The employer’s best response to (E H, NE L) is (J E, NJ NE).
⑸ Step 4. Existence of a perfect Bayesian equilibrium
① The worker’s best response to (J E, NJ NE) is (E H, NE L).
○ (Reference) cH ≤ w ≤ cL
○ If a type-H worker deviates to (NE H, NE L), their payoff decreases from w - cH to 0.
○ If a type-L worker deviates to (E H, E L), their payoff decreases from 0 to w - cL.
② Conclusion : [(J E, NJ NE), (E H, NE L)] is a perfect Bayesian equilibrium.
⑹ Step 5. Uniqueness of the perfect Bayesian equilibrium
① Example : [(NJ E, J NE), (E L, NE H)]
○ Given (E L, NE H), the employer’s best response is indeed (NJ E, J NE).
○ But if a type-L worker deviates to (NE L, NE H), their payoff increases from w - cL to 0.
○ Therefore, (E L, NE H) is not the worker’s best response to (NJ E, J NE).
② By checking all cases, one can show that there is no perfect Bayesian equilibrium other than [(J E, NJ NE), (E H, NE L)].
3. Pooling equilibrium
**⑴ Definition:** H and L take the same action, so the employer obtains no information.
⑵ Step 1. Worker’s strategy
① Assume Pr(E H) = 1 and Pr(E L) = 1.
② That is, assume the worker’s strategy is (E H, E L).
③ Also assume cH ≤ cL ≤ w.
⑶ Step 2. Compute the employer’s beliefs.
⑷ Step 3. Employer’s best response
① Employer’s payoff if the worker chooses E and the employer chooses J
② Employer’s payoff if the worker chooses E and the employer chooses NJ
③ Employer’s payoff if the worker chooses NE and the employer chooses J
④ Employer’s payoff if the worker chooses NE and the employer chooses NJ
⑤ Conclusion : For the employer’s best response to (E H, E L) to be (J E, NJ NE), the following condition is needed:
○ (Note) The formula above is a general expression; in this example it is difficult to define Pr(H NE) immediately.
○ Conclusion : A Nash equilibrium exists only if there are sufficiently many type-H workers and the fraction of type-H among the non-educated is sufficiently small.
⑸ Step 4. Existence of a perfect Bayesian equilibrium
① The worker’s best response to (J E, NJ NE) is (E H, E L).
○ (Reference) cH ≤ cL ≤ w
○ If a type-H worker deviates to (NE H, E L), their payoff decreases from w - cH to 0.
○ If a type-L worker deviates to (E H, NE L), their payoff decreases from w - cL to 0.
② Conclusion : [(J E, NJ NE), (E H, E L)] is a perfect Bayesian equilibrium.
| ⑹ Step 5. Uniqueness of the perfect Bayesian equilibrium : In fact, [(J | E, NJ | NE), (E | H, E | L)] is not the only solution. |
4. Hybrid equilibrium
**⑴ Definition:** A mixed strategy is used, so even “noisy” information is conveyed to the employer.
⑵ Step 1. Worker’s strategy
① A type-H worker always chooses E, while a type-L worker chooses E with probability 0 < e < 1.
② That is, assume the worker’s strategy is (E H, e L).
③ Also assume cH ≤ cL ≤ w.
⑶ Step 2. Employer’s strategy
① The employer chooses J with probability 0 < j < 1, and chooses NJ with probability 1 - j.
② That is, assume the employer’s strategy is (j E, NJ NE).
⑷ Step 3. Compute the employer’s beliefs.
⑸ Step 4. Employer’s best response : If π(E) > π(NE), then the employer would choose e = 1, and if π(E) < π(NE), then the employer would choose e = 0—so a contradiction.
⑹ Step 5. Worker’s best response : If π(J) > π(NJ), then the worker would choose j = 1, and if π(J) < π(NJ), then the worker would choose j = 0—so a contradiction.
⑺ Step 6. Perfect Bayesian equilibrium
① [(j* E, NJ NE), (e* L, E H)] is a perfect Bayesian equilibrium.
② (Reference) It is easy to show that the employer’s best response to NE is NJ.
③ (Reference) It is easy to show that a type-H worker’s best response to (j* E, NJ NE) is E.
Posted: 2020.06.13 16:26