Chapter-11 - Chapter 11: Signaling Games Dynamic games with...

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Unformatted text preview: Chapter 11: Signaling Games Dynamic games with incomplete information. Dynamic games with incomplete information. Signaling games Signaling games -- A player with private information (sender) moves A player with private information (sender) moves first by taking an action. first by taking an action. -- A second player (receiver) observes the sender's A second player (receiver) observes the sender's action, and then takes an action. action, and then takes an action. -- Sender's action may reveal his private information. Sender's action may reveal his private information. Job Market Signaling Why do you sit here? Receiving quality education, or Why do you sit here? Receiving quality education, or simply getting a college degree so that you can get a simply getting a college degree so that you can get a good job? good job? Good companies try to hire smart people. But whether Good companies try to hire smart people. But whether a person is smart is difficult to know. a person is smart is difficult to know. Getting a college degree is hard for dumb people. You Getting a college degree is hard for dumb people. You work hard here to get a college degree. work hard here to get a college degree. -- Sending a signal to job market that you are smart. Sending a signal to job market that you are smart. Management Trainee A management trainee tries to be hired permanently. A management trainee tries to be hired permanently. There are two types workers: lazy and industrious. There are two types workers: lazy and industrious. -- A lazy worker works normally 40 hours per week. A lazy worker works normally 40 hours per week. -- An industrious type normally work 60 hours per An industrious type normally work 60 hours per week. week. The company only wants to retain industrious workers. The company only wants to retain industrious workers. To achieve the goal, the lazy type might work 60 hours To achieve the goal, the lazy type might work 60 hours per week in the training period (mimic the other type). per week in the training period (mimic the other type). But the company will not be fooled. Working 60 hours But the company will not be fooled. Working 60 hours per week is not enough! Industrious type might need to per week is not enough! Industrious type might need to work 80 hours per week to signal his own type. work 80 hours per week to signal his own type. Perfect Bayesian-Nash Equilibrium Signaling game: Signaling game: -- Stage 1: Nature chooses the sender's type. Stage 1: Nature chooses the sender's type. -- Stage 2: The sender learns her type and chooses Stage 2: The sender learns her type and chooses an action. an action. -- Stage 3: The receiver observes the sender's action, Stage 3: The receiver observes the sender's action, modifies his beliefs about the sender's type in light modifies his beliefs about the sender's type in light of this new information, and chooses an action. of this new information, and chooses an action. Sender's strategy set: a mapping from senders' types Sender's strategy set: a mapping from senders' types to actions. to actions. Receiver's strategy set: a mapping from senders' Receiver's strategy set: a mapping from senders' actions to his own actions. actions to his own actions. Perfect Bayesian-Nash Equilibrium Perfect Bayesian-Nash equilibrium (PBE): Perfect Bayesian-Nash equilibrium (PBE): -- Sequential rationality. Sequential rationality. -- Consistent beliefs. Consistent beliefs. We make receiver's beliefs (about sender's type) We make receiver's beliefs (about sender's type) explicit. explicit. Sequential rationality: Sequential rationality: -- For sender's each type: given receiver's strategy, For sender's each type: given receiver's strategy, each type's action is optimal. each type's action is optimal. -- For receiver: given sender's strategy and receiver's For receiver: given sender's strategy and receiver's belief, receiver's strategy is optimal. belief, receiver's strategy is optimal. Consistent beliefs: receiver's beliefs are consistent Consistent beliefs: receiver's beliefs are consistent with sender's equilibrium strategy. with sender's equilibrium strategy. Perfect Bayesian-Nash Equilibrium Lazy 40 hours 60 hours 80 hours Industrious Separating strategy: two types choose different actions. Separating strategy: two types choose different actions. Consistent belief: Consistent belief: -- Observing 40 hours per week, infer that it's a lazy type. Observing 40 hours per week, infer that it's a lazy type. -- Observing 80 hours per week, infer that it's an industrious Observing 80 hours per week, infer that it's an industrious type. type. -- Observing 60 per week, belief can be any thing. (off Observing 60 per week, belief can be any thing. (off equilibrium path) equilibrium path) Perfect Bayesian-Nash Equilibrium Lazy 40 hours 60 hours 80 hours Industrious Pooling strategy: two types choose the same action. Pooling strategy: two types choose the same action. Consistent belief: Consistent belief: -- Observing 60 hours per week, nothing is learned, belief is Observing 60 hours per week, nothing is learned, belief is the same as initial belief. the same as initial belief. -- Observing 40 or 80 hours per week, belief can be any Observing 40 or 80 hours per week, belief can be any thing. (off equilibrium path) thing. (off equilibrium path) Management Trainee Two types of trainee: lazy (prob 0.75) or industrious Two types of trainee: lazy (prob 0.75) or industrious (prob 0.25). Three possible actions: 40, 60 or 80 hours (prob 0.25). Three possible actions: 40, 60 or 80 hours per week. per week. Observing trainee's action, the company decides Observing trainee's action, the company decides whether to hire or fire the trainee. whether to hire or fire the trainee. The company gets 100 ififhires an industrious worker, The company gets 100 hires an industrious worker, 25 ififhires a lazy worker, 60 ififhires no one. 25 hires a lazy worker, 60 hires no one. The trainee gets 130 ififhired and 70 ififnot hired. The trainee gets 130 hired and 70 not hired. Management Trainee Table 11.1 Personal Cost of Effort Type Lazy 40 hours 60 hours 80 hours 50 75 50 120 80 Industrious 30 Nature Lazy Prob. = .75 Industrious Prob. = .25 Trainee 40 60 80 40 60 80 Manager Hire Fire Hire Fire Trainee Manager 80 25 20 60 55 25 -5 60 Hire Fire 10 25 -50 60 Hire Fire Hire Fire 100 100 40 80 60 100 20 60 Hire Fire 50 100 -10 60 Management Trainee Trainee's strategy set: {lazy, industrious} Trainee's strategy set: {lazy, industrious} Company's strategy: {40, 60, 80} Company's strategy: {40, 60, 80} {40, 60, 80} {40, 60, 80} {hire, fire} {hire, fire} To find PBE, we start with sender's strategy: To find PBE, we start with sender's strategy: -- Separating equilibrium: sender plays separating Separating equilibrium: sender plays separating strategy strategy -- Pooling equilibrium: sender plays pooling strategy. Pooling equilibrium: sender plays pooling strategy. Management Trainee A candidate separating equilibrium: A candidate separating equilibrium: -- Lazy type chooses 40 Lazy type chooses 40 -- Industrious type chooses 60 Industrious type chooses 60 1. Pin down receiver's belief: 1. Pin down receiver's belief: -- Observing 40, lazy for sure Observing 40, lazy for sure -- Observing 60, industrious for sure Observing 60, industrious for sure -- Observing 80, belief is free (assume industrious for Observing 80, belief is free (assume industrious for sure) sure) 2. Pin down receiver's optimal strategy: 2. Pin down receiver's optimal strategy: -- Observing 40, fire Observing 40, fire -- Observing 60, hire Observing 60, hire -- Observing 80, hire Observing 80, hire Management Trainee Check whether each type of the sender has incentive to Check whether each type of the sender has incentive to deviate, given the receiver's strategy. deviate, given the receiver's strategy. -- Lazy type: choosing 40 yields payoff 20; choosing 60 Lazy type: choosing 40 yields payoff 20; choosing 60 yields payoff 55. He has incentive to deviate. yields payoff 55. He has incentive to deviate. It is not a PBE. It is not a PBE. Another candidate separating equilibrium: Another candidate separating equilibrium: -- Lazy type chooses 40 Lazy type chooses 40 -- Industrious type chooses 80 Industrious type chooses 80 Pin down receiver's belief: Pin down receiver's belief: -- Observing 40, lazy for sure Observing 40, lazy for sure -- Observing 80, industrious for sure Observing 80, industrious for sure -- Observing 60, belief is free (assume lazy with prob Observing 60, belief is free (assume lazy with prob 0.75) 0.75) Management Trainee Pin down receiver's optimal strategy: Pin down receiver's optimal strategy: -- Observing 40, fire Observing 40, fire -- Observing 60, fire Observing 60, fire -- Observing 80, hire Observing 80, hire Check whether each type of the sender has incentive to Check whether each type of the sender has incentive to deviate given the receiver's strategy. deviate given the receiver's strategy. -- Lazy type: choosing 40 yields payoff 20; choosing 60 Lazy type: choosing 40 yields payoff 20; choosing 60 yields payoff -5; choosing 80 yields payoff 10. He has yields payoff -5; choosing 80 yields payoff 10. He has no incentive to deviate. no incentive to deviate. -- Industrious type: choosing 80 yields 50, choosing 60 Industrious type: choosing 80 yields 50, choosing 60 yields 20, choosing 40 yields 40. No incentive to yields 20, choosing 40 yields 40. No incentive to deviate. deviate. It is a PBE: (40/80, fire/fire/hire; lazy/lazy with prob 0.75/ It is a PBE: (40/80, fire/fire/hire; lazy/lazy with prob 0.75/ industrious) industrious) Management Trainee A candidate pooling equilibrium: A candidate pooling equilibrium: --Both types choose 40 Both types choose 40 Pin down receiver's belief Pin down receiver's belief -- Observing 40: lazy with prob 0.75 Observing 40: lazy with prob 0.75 -- Observing 60 or 80: belief is free (to support pooling, Observing 60 or 80: belief is free (to support pooling, assume lazy with prob 0.6) assume lazy with prob 0.6) Pin down receiver's optimal strategy: Pin down receiver's optimal strategy: -- Observing 40, fire Observing 40, fire -- Observing 60 or 80, fire Observing 60 or 80, fire Management Trainee Check sender's incentive to deviate. Check sender's incentive to deviate. -- Lazy type: 20 (if 40), -5 (if 60), -50 (if 80). No incentive Lazy type: 20 (if 40), -5 (if 60), -50 (if 80). No incentive to deviate. to deviate. -- Industrious type: 40 (if 40), 20 (if 60), -10 (if 80). No Industrious type: 40 (if 40), 20 (if 60), -10 (if 80). No incentive to deviate. incentive to deviate. It is a PBE: (40/40, fire/fire/fire; lazy with prob 0.75/ lazy It is a PBE: (40/40, fire/fire/fire; lazy with prob 0.75/ lazy with prob 0.6/ lazy with prob 0.6) with prob 0.6/ lazy with prob 0.6) Separating equilibrium effectively signal the sender's type. Separating equilibrium effectively signal the sender's type. In doing that, the attractive type has to take the action that In doing that, the attractive type has to take the action that is too costly for the other type to mimic. is too costly for the other type to mimic. ...
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This note was uploaded on 07/17/2008 for the course ECON 601 taught by Professor Yang during the Spring '08 term at Ohio State.

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