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Extensive Game with Imperfect Information III. Topic One: Costly Signaling Game. Spence’s education game. Players: worker (1) and firm (2) 1 has two types: high ability H with probability p H and low ability L with probability p L .
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Spence’s education game • Players: worker (1) and firm (2) • 1 has two types: high ability H with probability p Hand low ability L with probability p L . • The two types of worker choose education level e H and e L (messages). • The firm also choose a wage w equal to the expectation of the ability • The worker’s payoff is w – e/
Pooling equilibrium • e H = e L = e* L pH (H - L) • w* = pHH + pLL • Belief: he who chooses a different e is thought with probability one as a low type • Then no type will find it beneficial to deviate. • Hence, a continuum of perfect Bayesian equilibria
Separating equilibrium • e L = 0 • H (H - L) ≥ e H ≥L (H - L) • w H = H and w L = L • Belief: he who chooses a different e is thought with probability one as a low type • Again, a continuum of perfect Bayesian equilibria • Remark: all these (pooling and separating) perfect Bayesian equilibria are sequential equilibria as well.
L type equilibrium payoff w H type equilibrium payoff Increase in payoff H type payoff by choosing e=0 wH wL e eL=0 eH The most efficient separating equilibrium
The signal is costly Single crossing condition holds (i.e., signal is more costly for the low-type than for the high-type) When does signaling work?
Refinement of sequential equilibrium • There are too many sequential equilibria in the education game. Are some more appealing than others? • Cho-Kreps intuitive criterion • A refinement of sequential equilibrium—not every sequential equilibrium satisfies this criterion
Two sequential equilibria with outcomes: (R,R) and (L,L), respectively (L,L) is supported by belief that, in case 2’s information set is reached, with high probability 1 chose M. If 2’s information set is reached, 2 may think “since M is strictly dominated by L, it is not rational for 1 to choose M and hence 1 must have chosen R.” 1 R L M 2 2 2,2 R L R L 5,1 0,0 0,0 1,3 An example where a sequential equilibrium is unreasonable (slided deleted)
1,1 1,0 3,0 0,1 F N F N 2 Q Q 0.9 0.1 c 1 1 strong weak B B 2 N F N F 1,1 1,0 0,0 3,1 Beer or Quiche (Slide deleted)
If player 1 is weak she should realize that the choice for B is worse for her than following the equilibrium, whatever the response of player 2. If player 1 is strong and if player 2 correctly concludes from player 1 choosing B that she is strong and hence chooses N, then player 1 is indeed better than she is in the equilibrium. Hence player 2’s belief is unreasonable and the equilibrium is not appealing under scrutiny. 1,1 1,0 3,0 0,1 F N F N 2 Q Q 0.9 0.1 c 1 1 strong weak B B 2 N F N F 1,1 1,0 0,0 3,1 Why the second equilibrium is not reasonable? (slide deleted)
Cho-Kreps Intuitive Criterion • Consider a signaling game. Consider a sequential equilibrium (β,μ). We call an action that will not reach in equilibrium as an out-of-equilibrium action (denoted by a). • (β,μ) is said to violate the Cho-Kreps Intuitive Criterion if: • there exists some out-of-equilibrium action a so that one type, say θ*, can gain by deviating to this action when the receiver interprets her type correctly, while every other type cannot gain by deviating to this action even if the receiver interprets her as type θ*. • (β,μ) is said to satisfy the Cho-Kreps Intuitive Criterion if it does not violate it.
Spence’s education game • Only one separating equilibrium survives the Cho-Kreps Intuitive criterion, namely: e L = 0 and e H =L (H - L) • Any separating equilibrium where e L = 0 and e H >L (H - L) does not satisfy Cho-Kreps intuitive criterion. • A high type worker after choosing an e slightly smaller will benefit from it if she is correctly construed as a high type. • A low type worker cannot benefit from it however. • Hence, this separating equilibrium does not survive Cho-Kreps intuitive criterion.
L type equilibrium payoff w H type equilibrium payoff wH wL e eL=0 eH The most efficient separating equilibrium
Inefficient separating equilibrium L type worse off by deviating to e# if believed to be High type w H type better off by deviating to e# if believed to be High type L type equilibrium payoff H type equilibrium payoff wH’ wL e eH e# eL=0 eH’
Spence’s education game • All the pooling equilibria are eliminated by the Cho-Kreps intuitive criterion. • Let e satisfy w* – e*/ L > H – e/ L and w* – e*/ H > H – e/ L (such a value of e clearly exists.) • If a high type work deviates and chooses e and is correctly viewed as a good type, then she is better off than under the pooling equilibrium • If a low type work deviates and successfully convinces the firm that she is a high type, still she is worse off than under the pooling equilibrium. • Hence, according to the intuitive criterion, the firm’s belief upon such a deviation should be such that the deviator is a high type rather than a low type. • The pooling equilibrium break down!
Perfect Information Transmission? • An equilibrium in which each type will report honestly does not exist unless b=0.
No information transmission • There always exists an equilibrium in which no useful information is transmitted. • The receiver regards every message from the sender as useless, uninformative. • The sender simply utters uninformative messages.
Final Remark: • Relationship among different equilibrium concepts: • Sequential equilibrium satisfying Cho-kreps => sequential equilibrium => Perfect Bayesian equilibrium => subgame perfect equilibrium => Nash equilibrium
