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IKT 415 - SPRING 2016 - Quiz 2 Ayça Özdog̃an Sequential-move (Extensive-Form) Games with Complete-Information 1. Sequential dove-hawk game - 30/40 points Aşag̃ıda guvercin-şahin oyununun sequential bir versiyonu verilmiştir. 1 G N 1 (1, 1) H D 2 H (-1, -1) 2 D H (4, 0) (0, 4) (a) Oyuncaların strateji kümelerini yazin. (6 puan) (b) Nash dengelerini bulun. (10 puan) 1 D (1, 1) (c) Subgame perfect Nash dengelerini bulun. (14 puan) (d) BONUS (10 puan) Weak perfect Bayesian Nash dengelerini bulun. 2 2. Wages and Employment in a Unionized Firm - 20 points Bir sendikaya bagli firma (F) ile bu ekonomide emegin tek saticisi olan (monopoly seller of labor to the firm) sendika (S) arasindaki iliskiyi asagidaki oyunla modellemeye calisalim. Bu ekonomide sendika ucretleri belirliyor (S chooses w); firma ise ne kadar emek gucu kullanilacagini (F chooses labor L) belirliyor. Sendikanin fayda fonksiyonu ve firmanin kar fonksiyonu asagidaki gibi verilsin: US (w, L) = 4w + L πF (w, L) = ln L − w.L Once sendika bir ucret talebinde bulunuyor. Bunu gozlemleyen ve kabul eden firma ne kadar labor employ edecegini belirliyor. (a) Bu oyun perfect information mi imperfect information midir? (3 puan) (b) Kac tane subgame vardir? (2 puan) (c) Bu ekonomideki subgame perfect denge (w∗ , L∗ ) nedir? (15 puan) 3 3. Education and contracts (50 puan): Suppose that there is an employee (Player 1) who decides whether to be content with his undergraduate degree (U) or get a graduate degree (G). Getting a graduate degree is costly and the cost to the employee is c = 2. After observing the degree (U or G) of the employee, the employer (Player 2) offers either the manager job (M) or the cashier job (C) to the employee. The wage employer pays to the employee for the manager job is wM = 10 and that for the cashier job is wC = 6. The profit employer makes when the employee works at the manager job is 10 and it is 5 at the cashier job. (a) Draw the game tree. Write down the strategy sets for each player. (13 points) (b) Is this extensive-form game a perfect-information or an imperfect-information game? (2 points) (c) How many subgames does it have? (2 points) (d) Find the set of Nash equilibria (by writing the normal-form representation). (8 points) (e) Find the set of Subgame Perfect Nash Equilibria. (8 points) 4 (f) Now, suppose that the employer cannot observe the degree chosen by the employee. Draw the game tree. How many subgames does it have? Find the set of Nash equilibria and Subgame Perfect Nash equilibria in this situation. (17 points) 5 4. BONUS Market games (10 puan): Suppose that there are three firms in a market that compete by choosing an output level qi ≥ 0. Firm 1 (market leader) chooses q1 first. After observing its choice, Firm 2 and Firm 3 (the followers) simultaneously choose q2 and q3 , respectively. The inverse demand function is given by P = 250 − Q where Q is the total output produced (i.e. Q = q1 + q2 + q3 ). The marginal cost of each firm is c = 10. (a) Is this extensive-form game a perfect-information or an imperfect-information game? (1 points) (b) How many subgames does it have? (2 points) (c) Find the subgame perfect Nash equilibrium.(7 points) 6