EconS 424 - Signalling Games I - Felix Munoz-Garcia

EconS 424 - Signalling Games IFélix Muñoz-GarcíaWashington State Universityfmunoz@wsu.eduApril 28, 2014Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20141 / 33

Watson, Ch. 29 Exercise 3There is a rm and a worker. In this game, nature chooses the "type"of the rm (player 1). With probability p, the rm is of high quality(H) and, with probability 1 p, the rm is of low quality (L).The rm chooses either to o er a job to the worker (O) or not too er a job (N).If no job is o ered, the game ends and both parties receive 0.If the rm o ers a job, the worker either accepts (A) or rejects (R) theo er.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20142 / 33

Watson, Ch. 29 Exercise 3Firm:The worker’s e ort on the job brings the rm a pro t of 2.If the worker rejects an o er of employment, then the rm gets apayo of -1 (associated with being jilted).Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20143 / 33

Watson, Ch. 29 Exercise 3Worker:Rejecting an o er yields a payo of 0 to the worker.Accepting an o er yields the worker a payo of 2 if the rm is of highquality, and -1 if the rm is of low quality.The worker does not observe the quality of the rm directly.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20144 / 33

Watson, Ch. 29 Exercise 3Let us nd all PBEs of this game.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20145 / 33

Watson, Ch. 29 Exercise 30,00,0Félix Muñoz-García (WSU)NH1qOHp1R2(1-p)NLAOL(1-q)EconS 424 - Recitation 10AR2,2-1,02,-1-1,0April 28, 20146 / 33

Watson, Ch. 29 Exercise 3First strategy pro le candidate to be PBE: N H O L : Let us startwith the separating equilibrium where the informed player 1 (the rm)makes a job o er to the worker (O) only when the rm is of lowquality, O L , but not to make a job o er otherwise, i.e., N H .The gure below shades the branches corresponding to this strategy.0,00,0Félix Muñoz-García (WSU)NH1OHp1qR2(1-p)NLAOLEconS 424 - Recitation 10(1-q)AR2,2-1,02,-1-1,0April 28, 20147 / 33

Watson, Ch. 29 Exercise 3Given this strategy from the rm, the worker’s beliefs after observinga job o er must beq p 0 0p 0 (1 p) 1Intuitively, this implies that after observing O, the workerconcentrates all his beliefs on being in the lower node of theinformation set, i.e., low-quality rm.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20148 / 33

Watson, Ch. 29 Exercise 3Given these beliefs, it is optimal for the worker to reject the job o er,since 0 1.For future reference, we shade the branch in which the worker rejectsthe rm’s job o er, an action that is independent on the rm’s type(something that the worker cannot observe).If the worker rejects, then the high-quality rm prefers to not make ajob o er, N H , since 0 1, as prescribed by this strategy pro le.However, the low-quality rm prefers to not make job o ers, N L , whichcontradicts the initially proposed strategy pro le N H O L .Then, the separating strategy pro le N H O L cannot be sustained as aPBE of this game.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 20149 / 33

Watson, Ch. 29 Exercise 3Second strategy pro le candidate to be PBE: O H N L Let uscontinue with the separating equilibrium where the informed player( rm) makes a job o er to the worker (O) only when the rm is ofhigh quality, O H , but not to make a job o er otherwise, i.e., N L .The gure below shades the branches corresponding to this strategy.0,00,0Félix Muñoz-García (WSU)NH1OHp1qR2(1-p)NLAOLEconS 424 - Recitation 10(1-q)AR2,2-1,02,-1-1,0April 28, 201410 / 33

Watson, Ch. 29 Exercise 3Given this strategy from the rm, the worker’s beliefs after observinga job o er must beq p 1 1p 1 (1 p) 0Intuitively, this implies that after observing O, the workerconcentrates all his beliefs on being in the upper node of theinformation set, i.e., high-quality rm.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201411 / 33

Watson, Ch. 29 Exercise 3Given these beliefs, it is optimal for the worker to accept the job o er,since 2 0.For future reference, we shade the branch in which the worker acceptsthe rm’s job o er, an action that is independent on the rm’s type(something that the worker cannot observe).If the worker accepts, then the high-quality rm prefers to make a jobo er, OH, since 2 0, as prescribed by this strategy pro le.The low-quality rm also prefers to make a job o er, O L , whichcontradicts the initially proposed strategy pro le O H N L .Then, the separating strategy pro le O H N L cannot be sustained as aPBE of this game either.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201412 / 33

Watson, Ch. 29 Exercise 3Third strategy pro le candidate to be PBE: O H O L . Let uscontinue with the pooling equilibrium where both types of rm (highand low-quality) make a job o er to the worker.The gure below shades the branches corresponding to this strategy forplayer 1.0,00,0Félix Muñoz-García (WSU)NH1OHp1qR2(1-p)NLAOLEconS 424 - Recitation 10(1-q)AR2,2-1,02,-1-1,0April 28, 201413 / 33

Watson, Ch. 29 Exercise 3Given this strategy from the rm, the worker’s beliefs after observinga job o er must beq p 1 pp 1 (1 p) 1Intuitively, this implies that after observing O, the worker cannot inferadditional information about the rm’s quality, since all rms makejob o ers. (Note that this result radically di ers from that in the twoseparating strategy pro les described above.)Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201414 / 33

Watson, Ch. 29 Exercise 3Given these beliefs, it is optimal for the worker to accept the job o erif EU2 (A) EU2 (R ), whereEU2 (A) 2p (1p )( 1) 3p1EU2 (R ) 0 0! 3p 1 01p for A R31p for A R31p for A R3Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201415 / 33

Watson, Ch. 29 Exercise 3Hence, if p 13 , the worker accepts the job o er and we can shadebranch A in the gure.Otherwise, the worker rejects the job o er and we must shade branchR.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201416 / 33

Watson, Ch. 29 Exercise 3Let us divide our analysis according to the value of p:If p 13 , the worker accepts the job o er. The high-quality rm prefersto make a job o er (as prescribed) since 2 0, while the low-quality rm also wants to make a job o er (as prescribed) since 2 0.Hence, the pooling strategy pro le O H O L can be sustained as PBE ofthis game when p 13 .If p 13 , the worker rejects the job o er. The high-quality rm doesnot want to make a job o er since making it yields a payo of -1 (sincethe o er is rejected) whereas not making such an o er yields a payo of 0.Hence, the pooling strategy pro le O H O L cannot be sustained as PBEof this game when p 13 .Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201417 / 33

Watson, Ch. 29 Exercise 3Fourth strategy pro le candidate to be PBE: N H N L . Let uscontinue with the pooling equilibrium where no type of rm (high orlow-quality) make a job o er to the worker.The gure below shades the branches corresponding to this strategy forplayer 1.0,00,0Félix Muñoz-García (WSU)NH1OHp1qR2(1-p)NLAOLEconS 424 - Recitation 10(1-q)AR2,2-1,02,-1-1,0April 28, 201418 / 33

Watson, Ch. 29 Exercise 3Given this strategy from the rm, the worker’s beliefs after observinga job o er must beq 0p 0 p 0 (1 p) 0 0Hence, q can be any number in the [0, 1] interval.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201419 / 33

Watson, Ch. 29 Exercise 3Given these beliefs, it is optimal for the worker to accept the job o erif EU2 (A) EU2 (R ), whereEU2 (A) 2q ( 1)(1q)EU2 (R ) 0! 2q ( 1)(1q) 01for A R31q for R A31q for R A3q Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201420 / 33

Watson, Ch. 29 Exercise 3Hence, if q 13 , the worker accepts the job o er and we can shadebranch A in the gure.Otherwise, the worker rejects the job o er and we must shade branchR.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201421 / 33

Watson, Ch. 29 Exercise 3Let us divide our analysis according to the value of q:If q 31 , the worker accepts the job o er. The high-quality rmprefers to make a job o er since 2 0, and the low-quality rm alsowants to make a job o er since 2 0.Hence, the pooling strategy pro le N H N L cannot be sustained as PBEof this game when q 13 .If q 31 , the worker rejects the job o er. The high-quality rm doesnot want to make a job o er: making it yield a payo of -1 (since theo er is rejected) whereas not making such an o er yields a payo of 0.Similarly for the low-quality rm.Hence, no type of rm makes job o ers and the pooling strategy pro leN H N L can be sustained as PBE of this game when q 13 .Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201422 / 33

Harrington, Ch. 11 Exercise 5Consider a two-player game between a prospective employee, whomwe’ll refer to as the applicant, and an employer.The applicant’s type is her intellect, which may be low, medium orhigh, with probability 13 , 21 , and 16 , respectively.After the applicant learns her type, she decides whether or not to goto college. The personal cost in gaining a college degree is higherwhen the applicant is less intelligent, because a less smart student hasto work harder if she is to graduate.Assume that the cost of gaining a college degree is 2, 4, and 6 for anapplicant who is of high, moderate, and low intelligence, respectively.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201423 / 33

Harrington, Ch. 11 Exercise 5The employer decides whether to o er the applicant a job as amanager or as a clerk.The applicant’s payo to being hired as a manager is 15, while thepayo to being a clerk is 10. These payo s are independent of theapplicant’s type.The employer’s payo from hiring someone as a clerk is 7 (and is thesame regardless of intelligence and whether or not the person has acollege degree).If the applicant is hired as a manager, then the employer’s payo increases with the applican’t intellect, from 4, to 6, to 14, dependingon whether the applicant has low, moderate, or high intellect,respectively.Note that the employer’s payo does not depend on whether or not theapplicant has a college degree.The extensive form of this game is shown on the next slide.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201424 / 33

Harrington, Ch. 11 Exercise 5NatureLowHighModerateApplicantApplicantNo collegeCollegeApplicantNo collegeCollegeNo agerApplicant 9Employer4Clerk4715Félix Muñoz-García (WSU)4Clerk107116Clerk67156EconS 424 - Recitation 10Clerk1071314Clerk871514April 28, 2014Clerk10725 / 33

Harrington, Ch. 11 Exercise 5Find a PBE in which students of low intellect do not go to collegeand those of moderate and high intellect do.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201426 / 33

Harrington, Ch. 11 Exercise 5The gure on the next slide shades the branches corresponding to thestrategy pro le we need to check, where the applicant goes to collegeboth when his innate ability is moderate and high, but does not go tocollege when his ability is low.Félix Muñoz-García (WSU)EconS 424 - Recitation 10April 28, 201427 / 33

Harrington, Ch. 11 Exercise 5NatureLowHighModerateApplicantApplicantNo collegeCollegeApplicantNo collegeCollegeNo agerApplicant 9Employer4Clerk4715Félix Muñoz-García (WSU)4Clerk107116Clerk67156EconS 424 - Recitation 10Clerk1071314Clerk871514April 28, 2014Clerk10728 / 33

Harrington, Ch. 11 Exercise 5Step 1 - Employer’s beliefs:If the student did not go to college, then he is low intellect withprobability 1If the student did go to college, then he is low intellect with probabilityzero.µ2 moderate intellect with probability:1Félix Muñoz-García (WSU)µ2 high intellect with probability:EconS 424 - Recitation 1012121211 161611 1611 34 14April 28, 201429 / 33