1. 1

2. problem set 13 see last slide Last Slide

3. 3 Extensive form Games with Incomplete Information 111 Information set of player 1 Belief p1p1 p2p2 p3p3 p 1 + p 2 + p 3 = 1, p i ≥ 0

4. 4 Extensive form Games with Incomplete Information 111 Information set of player 1 p1p1 p2p2 p3p3 2 22 α3α3 α2α2 α1α1 111 player 2 mixes What is player 1’s belief at this information set ?? 1- α 1

5. 5 111 p1p1 p2p2 p3p3 2 22 111 player 2 mixes What is player 1’s belief at this information set ?? α3α3 α2α2 α1α1 Using Bayes’ Law:

6. 6 111 p1p1 p2p2 p3p3 2 22 111 α3α3 α2α2 α1α1 Updating the belief is consistent with the strategy profile (whenever possible)

7. 7 111 p1p1 p2p2 p3p3 Once we have a beliefs for each information set, we can define the equivalent of subgame perfect equilibrium. We require that each player’s strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that player’s belief at the information set.

8. 8 111 p1p1 p2p2 p3p3 We require that each player’s strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that player’s belief at the information set. (Sequential Rationality)

9. 9 Signalling Games The s ss sender, a player who has complete information (about the state of nature, or his own type) sends a signal to the other player, the r rr receiver. The payoffs depend on the state of nature, the signal and the action takren The receiver observes the signal and takes an action. Michael Spence Nobel Prtize, 2001

10. 10 Education as a signal The w ww worker, has skills H or L with probability q, 1-q, resp. He knows his own productivity. The f ff firm observes the signal, and pays the w ww worker a wage rate w which equals the productivity that it believes he has. He chooses a level of education e which costs him c(η,e) where η is his type. His productivity is y(η,e).

11. 11 Education as a signal The payoff to the w ww worker is: w - c(η,e) The payoff to the f ff firm is: y(η,e) - w e Single Crossing property

12. 12 Education as a signal e

13. 13 Education as a signal e two cases:

14. 14 Education as a signal e No Envy

15. 15 Education as a signal e Envy Envy

16. 16 Pooling Equilibrium Both types choose education level: e p

17. 17 Pooling Equilibrium e

18. 18 Separating Equilibrium e Envy Envy Beliefs: H LL Payoffs for education

19. 19 Separating Equilibrium e Beliefs: H LL Can the firm have these beliefs??? It is a strictly dominated (inferior) strategy for type L to send a signal in this interval Even if he is identfied as H he is better off sending e*(L). (An H is better off in this interval if he is identified as H. )

20. 20 Separating Equilibrium e Beliefs: H LL If we accept this argument then the firm ‘s belief in this interval should be H. The only separating equilibrium is when e S * is at the left of this interval The Intuitive Criterion This argument is known as The Intuitive Criterion of In-Koo Cho & David Kreps

21. 21 1.Find All sepaprating Equilibria of the Spence Model 2. Find Hybrid Equilibria, in which one type mixes, and the other plays a pure strategy Return