Game Theory: Nash Equilibria and Dominant Strategies in Different Industries, Assignments of Economics

Download Game Theory: Nash Equilibria and Dominant Strategies in Different Industries and more Assignments Economics in PDF only on Docsity! Economics 841 – Problem Set #5 Spring 2009 Mike Conlin SOLUTIONS 1. There are two firms (Firm 1 and Firm 2) in an industry and the firms produce a homogeneous good. Let Q1 be the output of Firm 1 and Q2 be the output of Firm 2. Let the market demand curve for the good be as depicted below where Q is total output such that Q= Q1+Q2. Let Firm 1 have a constant marginal cost of 10 and total fixed costs of 200. Firm 2 has constant marginal costs of 20 and total fixed costs of 100. 0 5 10 15 20 25 30 35 40 45 50 0 25 50 75 10 0 12 5 15 0 17 5 20 0 22 5 25 0 D MR a) Suppose the firms select quantity at the same time (simultaneously). This is the Cournot model. Depict Firm 1’s and Firm 2’s reaction curves on the graph below. Solve for the likely (equilibrium) outcome. Basically, solve for the likely output of Firm 1 and Firm 2. (It is difficult to precisely determine these outputs using the graph below so estimate.) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 0 10 20 30 40 50 60 70 80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 21 0 22 0 23 0 24 0 25 0 Q1 Q2 The easiest way to graph Firm 1’s reaction curve is to figure out what output maximizes Firm 1’s profits if Firm 1 expects Firm 2 to produce Q2=0. From the graph above, you can see this output for Firm 1 would be 100. From the graph above, also note that if Firm 1 expects Firm 2 to produce Q2=200, Firm 1’s best response is to produce 0. Plot these two points on the graph and draw a straight line between them and you have Firm 1’s reaction curve. You can do a similar thing for Firm 2 taking into account that Firm 2’s MC is 20. WHILE YOU CAN DETERMINE THE REACTION CURVES IN THIS WAY, YOU SHOULD UNDERSTAND HOW TO CALCULATE FIRM 1’S PROFIT MAXIMIZING OUTPUT FOR DIFFERENT AMOUNTS OF FIRM 2’S OUTPUT. What are Firm 1’s profits and Firm 2’s profits? Show your calculations. Firm 1 produces an output of 83.3 and Firm 2 produces an output of 33.3. Therefore, total output is 116.7 and price is 26.7. Firm 1’s profits are 26.7(83.3)-10(83.3)-200=1191 and Firm 2’s profits are 26.7(33.3)-20(33.3)- 100=123. MC2 MC1 Firm 1’s reaction curve Firm 2’s reaction curve 33.3 83.3 2. Below is an editorial entitled “Why FTC Opposes The Staples Merger” which appeared in the Wall Street Journal. For simplicity, assume Staples and Office Depot have no fixed costs. Assume that the marginal cost, average variable cost and average total cost for staplers are constant at $0.50. The market demand for staplers in Washington D.C. is depicted on the graph below. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 20 40 60 80 100 120 140 160 180 200 220 240 D Q MR Suppose Staples and Office Depot in Washington D.C. are each charging the same price for staplers and each making profits of $90. Also assume that they are the only two stores that sell office supplies in Washington D.C. How much will the price of a stapler increase if the merger occurs? EXPLAIN. If both Staples and Office Depot are making profits of $90, then each must be charging a price of $1.50 and selling 90 staplers each. You can determine this price by just trail and error. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 20 40 60 80 100 120 140 160 180 200 220 240 MC MC=AVC=ATC Firm 1’s Profits Firm 2’s Profits 4. Here is a story many of you have heard. There are two friends taking MBA814 this semester. Both had done pretty well on all of the homeworks and the midterm, so that going into the final they had a solid 4.0. They were so confident the weekend before the final that they decided to go to a party in Chicago. The party was so good that they overslept all day Sunday, and got back too late to study for the final that was scheduled for Monday morning. Rather than take the final unprepared, they went to Prof. Conlin with a sob story. They said they had gone to Chicago and had planned to come back in plenty of time to study for the final but had had a flat tire on the way back. Because they did not have a spare, they had spent most of the night looking for help. Now they were really tired, so could they please have a makeup final the next day? Prof. Conlin thought it over and agreed. The two studied all of Monday evening and came well prepared on Tuesday morning. Prof. Conlin placed them in separate rooms and handed the test to each. The first question on the first page, worth 10 points, was very easy. Each of them wrote a good answer, and greatly relieved, turned the page. It had just one question, worth 90 points. It was: “Which tire?” Suppose that each student’s “payoff” is 100 (because they receive an 4.0 in the class) if they answer the second question the same and each student’s “payoff” is 30 (because they receive a 2.0 in the class) if they answer the second question differently. a) Depict the above situation as a normal form game. Student 2 Front-Left Front- Right Back-Left Back-Right Front-Left 100,100 30,30 30,30 30,30 Front-Right 30,30 100,100 30,30 30,30 Back-Left 30,30 30,30 100,100 30,30 Student 1 Back-Right 30,30 30,30 30,30 100,100 b) Does either student have a dominant strategy? If so, please identify the dominant strategy. Neither student has a dominant strategy. The best strategy for Student 1 depends on the strategy of Student 2 and the best strategy for Student 2 depends on the strategy of Student 1. c) Identify all pure strategy Nash Equilibria. There are 4 pure strategy Nash Equilibria. For both students to select Front-Left, for both students to select Front- Right, for both students to select Back-Left, or for both students to select Back-Right. Do you really believe that one of these 4 will likely happen and which one do you expect is more likely? It probably depends on the student’s past experiences with flat tires. d) Identify one mixed strategy Nash Equilibrium. One mixed strategy Nash Equilibrium is for Student 1 to say Front-Left, Front-Right, Back-Left and Back- Right each with probability .25 and for Student 2 to say Front-Left, Front-Right, Back-Left and Back- Right each with probability .25. Student 1’s expected payoff no matter what she says is .25*100+.75*30=47.5. Student 2’s expected payoff no matter what she says is .25*100+.75*30=47.5. Therefore, a best response for each student is to randomize over the four different strategies. Another mixed strategy Nash Equilibrium is for Student 1 to say Front-Left and Front-Right each with probability .5 and for Student 2 to say Front-Left and Front-Right each with probability .5. Student 1’s expected payoff if she says Front-Left or Front-Right is .5*100+.5*30=65 while her expected payoff from saying Back-Left or Back-Right is 30 given Stiudent 2’s strategy . A similar story holds for Student 2. There are obviously many mixed strategy Nash Equilibria. 5. Mr. Clemens and Mr. McNamee are partners in a drug company (called Hall of Fame Results Inc.) that produces B-12 vitamin supplements (wink-wink). Mr. Clemens is going on vacation the second week of January and Mr. McNamee is going on vacation the first week of January. At the end of the second week in January, Hall of Fame Results Inc. is going to introduce a new “supplement” targeting high school and college athletes. The likely success of the new supplement will depend on how hard Mr. Clemens and Mr. McNamee work prior to its introduction. While their payoffs increase with the likelihood of success (holding the amount they work constant), their payoffs decrease with the amount they work (holding the likelihood of success constant). Mr. Clemens and Mr. McNamee will not change their vacation plans but must decide whether to work hard or not work hard during the week they are in the office. Because Mr. Clemens is in the office the first week of January and Mr. McNamee is in the office the second week of January, Mr. McNamee observes whether Mr. The extensive form of the game is depicted below. Mr. Clemens Work Hard Don’t Work Hard Mr. McNamee Mr. McNamee Don’t Don’t Work Hard Work Hard Work Hard Work Hard Clemens 100 70 80 60 McNamee 80 90 65 50 a) What is (are) the pure strategy Nash Equilibrium (ia) of the above game? (Note: An equilibrium identifies the strategy of each player and a strategy tells a player what to do at each of his decision points.) (Clemens strategy is Work Hard ; McNamee strategy is Don’t Work Hard if Clemens selects Work Hard and Don’t Work Hard if Clemens selects Don’t Work Hard) (Clemens strategy is Don’t Work Hard ; McNamee strategy is Don’t Work Hard if Clemens selects Work Hard and Work Hard if Clemens selects Don’t Work Hard) b) What is (are) the Subgame Perfect Equilibrium (ia) of the above game? The green lines above are obtained by backward induction and represent the following Subgame Perfect Equilibrium. (Clemens strategy is Don’t Work Hard ; McNamee strategy is Don’t Work Hard if Clemens selects Work Hard and Work Hard if Clemens selects Don’t Work Hard)