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Answer Hints HUL311 - APPLIED GAME THEORY MAJOR EXAMINATION (Nov 21, 2017), IITD SEM-I, AY 2017-18, Time Allowed: 2 Hours. (ANSWER ALL, Max marks=30) ANSWER HINTS Q1: Short Note [5 3 = 15 marks]. You must describe the problem and highlight its solu- × tion with possible realistic implications. (a) Inefficient trade and adverse selection. Your answer must contain (a) problem description (b) properly de- fined strategies of players (c) equilibrium solution and (d) realistic im- plications - e.g., application of `Lemon Market' in international trade, insurance (with healthy and unhealthy type), second hand car market, etc. (b) Cournot competition with cost uncertainty. Your answer must contain (a) problem description (b) properly de- fined strategies of players (q [0; )) (c) showing the best response i 2 1 function in graph with cH and cL and to show respective quantities and profits (d) realistic implications - e.g., credible cost/type revela- tion through advertersing, etc. (c) Effort choice in study groups. Your answer must contain (a) problem description (b) properly de- fined strategies of players (c) equilibrium solution and (d) realistic im- plications - e.g., voluntary contribution with type dependent donors - people donates beyond a cost threshold, jumping into the river to save a child - who will jump depend on cost threshold, etc. 1 Q2 [2:5 2 = 5 marks]. ∗ Consider two firms that play a Cournot competition game with demand p = 100 q; and costs 2 − for each firm given by ci(qi) = 10qi (it is known that, q = i=1 qi). Imagine that before the two firms play the Cournot game, firm 1 can invest in cost reduction.P If it invests, the costs of firm 1 will drop to c1(q1) = 5q1. The cost of investment is F > 0. Firm 2 does not have this investment opportunity. (a). Find the value F ∗ for which the unique subgame perfect equilibrium involves firm 1 in- vesting. (b). Assume that F > F ∗. Find a Nash equilibrium of the game that is not subgame perfect. See next page... 2 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 8. Credibility and Sequential Rationality 77 Answer: If firm 1 does not invest then they are expected to play the Cournot Nash equilibrium where both firms have costs of 10.Each firm solves, max (100 ( + ) 10) − − which leads to the first order condition 90 2 =0 − − yielding the best response function 90 ()= − 2 and the unique Cournot Nash equilibrium is 1 = 2 =30with profits 1 = 2 =900.Iffirm 1 does invest then for firm 1 the problem becomes max (100 (1 + 2) 5)1 1 − − which leads to the best response function 95 2 1(2)= − 2 For firm 2 the best response function remains the same as solved earlier with costs 102,sotheuniqueCournotNashequilibriumisnowsolved using both equations, 90 1 95 − = − 2 1 2 100 85 1 which yields 1 = 3 , 2 = 3 ,andprofits are 1 =1 111 9 while 7 2 =8029 .Hence,theincreaseinprofits from the equilibrium with 1 1 investment for player 1 are ∗ =1 111 900 = 211 ,whichisthemost 9 − 9 that player 1 would be willing to pay for the investment anticipating that they will play the Cournot Nash equilibrium after any choice of player 1regardinginvestment.If∗ then the unique subgame perfect equilibrium is that first, player 1 invests, then they players choose 1 = 100 85 3 , 2 = 3 ,andifplayer1didnotinvestthepayerschoose1 = 2 =30 (Note that if ∗ then the unique subgame perfect equilibrium is that first, player 1 does not invest, then they players choose 1 = 2 = 100 85 30,andifplayer1didinvestthepayerschoose1 = 3 , 2 = 3 .) ¥ © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 78 8. Credibility and Sequential Rationality (b) Assume that ∗.FindaNashequilibriumofthegamethatisnot subgame perfect. Answer: We construct a Nash equilibrium in which player 1 will invest 85 despite ∗.Player2’sstrategywillbe,play2 = 3 if player 1 invests, and 2 =100if he does not invest. With this belief, if player 1 does not invest then he expects the price to be 0, and his best response is 1 =0leading to profits 1 =0.Ifheinveststhenhisbestresponse 85 100 to 2 = 3 is 1 = 3 ,whichtogetherareaNashequilibriuminthe 1 Cournot game after investment. For any 1 111 9 this will lead to 1 positive profits, and hence, for ∗ 1 111 9 the strategy of player 2 100 described above, together with player 1 choosing to invest, play 1 = 3 if he invests and 1 =0if he does not is a Nash equilibrium. It is not subgame perfect because in the subgame following no investment, the players are not playing a Nash equilibrium. ¥ 7. 8. Entry Deterrence 1: NSG is considering entry into the local phone market in the Bay Area. The incumbent S&P, predicts that a price war will result if NSG enters. If NSG stays out, S&P earns monopoly profits valued at $10 million (net present value, or NPV of profits), while NSG earns zero. If NSG enters, it must incur irreversible entry costs of $2 million. If there is a price war, each firm earns $1 million (NPV). S&P always has the option of accom- modating entry (i.e., not starting a price war). In such a case, both firms earn $4 million (NPV). Suppose that the timing is such that NSG first has to choose whether or not to enter the market. Then S&P decides whether to “accommodate entry” or “engage in a price war.” What is the subgame perfect equilibrium outcome to this sequential game? (Set up a game tree.) Answer: Letting NSG be player 1 and S&P be player 2, Q3 [6 marks]. Assume that the (inverse) market demand for iPhone in Delhi is p = 36 Q. The manufacturer − of iPhone licenses a single dealer to sell this brand in Delhi. Therefore, the dealer acts as a monopoly in the Delhi market. The manufacturer sells each iPhone to the dealer for d c, ≥ where c = 20 is the marginal cost of producing one iPhone. In addition, the manufacturer may levy a fixed fee of φ on the dealership. Consider a two-stage game in which in Stage I the man- ufacturer sets the per-unit price charged to the dealer d, and the fixed fee φ that maximizes its profit (πm). In Stage II the dealer determines the quantity sold as to maximize the dealership's profit (πd). (i) Solve this game for equilibrium price (both for dealer's and manufacturer's), quantity sold in the market and profits that each party earns, possibly as function of φ. (5) (ii) Can the manufacturer set a possible contract with the dealer so that both, the manufacturer and the dealer, make a higher profit compared to (i)? (1) See next page... 5 Solutions to Problem Sets Industrial Organization Oz Shy What happens here is that a lower per-unit dealer price increases sales, as opposed to a high dealer price d =$80resulting from a double markup. Higher sales ex- tract more revenue from consumers and the distribution between the dealer and the manufacturer is done via the fixed fee φ =$200. (b) (i) In the second stage, the dealer takes d as given and chooses output level Q to solve max ⇡d =(p d)Q φ = [(36 Q) d]Q p − − − − d⇡d 36 d = 0= =36 2Q d = Q = − . ) dQ − − ) 2 Therefore, 36 + d (36 d)2 p = and ⇡d = − φ. 2 4 − Note that φ =0for this part of the problem. (ii) In the first stage, the manufacturer selects a dealer price d to solve 36 d d⇡m 56 2d max ⇡m =(d 20)Q =(d 20) − = 0= = − = d =28. d − − 2 ) dd 2 ) ✓ ◆ Hence, 36 d 36 + d Q = − =4,p= =32, 2 2 (36 d)2 36 d ⇡d = − φ =16 φ, and ⇡m =(d 20) − + φ =32+φ. 4 − − − 2 ✓ ◆ (iii) Consider the following contract between the manufacturer and the dealer: d = c = 20 and φ =40. Then, (36 20)2 ⇡d = − 40 = 24 > 16 and ⇡m =40> 32. 4 − THE END Draft=io-solutions090619.tex 2009/06/19 16:52 Page 44 of 44 Downloaded from: www.ozshy.com Q4 [4 marks]. Suppose that two firms, A and B, are competing in prices over infinitely many time periods, t = 1; 2; ::::. For simplicity suppose that they can only set 2 prices in each period, pH and pL, with pH > pL > 0. The payoffs from possible price combinations are given in the table below. The first payoff in brackets is of firm A, and the second one is of firm B. B pL pH pL (100,100) (500,0) A pH (0, 300) (200,250) Assume that they started with price pH in the first period. Let the grim-trigger strategy be pH , if the other firm has played pH until period t 1: pt = − 8p , otherwise. < L (a) Find the minimum: threshold discount factor for firm A to play the grim-trigger strategy assuming that the firm B is playing grim-trigger strategy. (2) (b) Find the minimum threshold discount factor for firm B to play the grim-trigger strategy assuming that the firm A is playing grim-trigger strategy. (2) Too easy to solve yourself... 7.
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