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### Transcript of 1. problem set 13 see last slide Last Slide 3 Extensive form Games with Incomplete Information 111...

• Slide 1
• 1
• Slide 2
• problem set 13 see last slide Last Slide
• 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
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• 4 Extensive form Games with Incomplete Information 111 Information set of player 1 p1p1 p2p2 p3p3 2 22 33 22 11 111 player 2 mixes What is player 1s belief at this information set ?? 1- 1
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• 5 111 p1p1 p2p2 p3p3 2 22 111 player 2 mixes What is player 1s belief at this information set ?? 33 22 11 Using Bayes Law:
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• 6 111 p1p1 p2p2 p3p3 2 22 111 33 22 11 Updating the belief is consistent with the strategy profile (whenever possible)
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• 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 players strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that players belief at the information set.
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• 8 111 p1p1 p2p2 p3p3 We require that each players strategy is optimal in that part of the game that follows an information set of this player, given the strategy profile and that players belief at the information set. (Sequential Rationality)
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• 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
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• 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(,e) where is his type. His productivity is y(,e).
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• 11 Education as a signal The payoff to the w ww worker is: w - c(,e) The payoff to the f ff firm is: y(,e) - w e Single Crossing property
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• 12 Education as a signal e
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• 13 Education as a signal e two cases:
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• 14 Education as a signal e No Envy
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• 15 Education as a signal e Envy Envy
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• 16 Pooling Equilibrium Both types choose education level: e p
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• 17 Pooling Equilibrium e
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• 18 Separating Equilibrium e Envy Envy Beliefs: H LL Payoffs for education
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• 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. )
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• 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
• Slide 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