Economics EC460 Mike Conlin SOLUTIONS Game Theory 1. Software Developers Inc. and AMC Applications Inc. are two companies developing statistical software programs. These companies must simultaneously decide whether to develop software that operates on the Unix operating system, Microsoft s Windows operating system or an objectorientated operating system. The table below indicates Software Developers Inc. and AMC Applications Inc. profits depending on the operating system they develop their statistical software for. AMC Applications Inc. Unix Windows Object- Orientated Unix 40, 50 30, 40 30, 30 Software Developers Inc. Windows 45, 40 80, 90 60, 20 Object-Orientated 25, 60 60, 75 70, 80 a) Does Software Developers Inc. or AMC Applications Inc. have a dominant strategy? Explain. No, the best strategy of Software Developers Inc. depends on the strategy of AMC Applications Inc. and the best strategy of AMC Applications Inc. depends on the strategy of Software Developers Inc.. b) Identify all pure strategy Nash Equilibria. The first strategy listed identifies Software Developers Inc. strategy and the second is AMC Applications Inc. strategy. Two Nash Equilibria: (Windows, Windows) (Object-Orientated, Object-Orientated) For the (Windows, Windows) Nash Equilibrium, Software Developers Inc. cannot increase their payoff of 80 by changing their strategy of choosing Windows given that AMC Applications Inc. chooses Windows. In addition, AMC Applications Inc. cannot increase their payoff of 90 by changing their strategy of choosing Windows given that Software Developers Inc. chooses Windows. Therefore, (Windows, Windows) is a Nash Equilibrium. The same logic can be used for (Object-Orientated, Object-Orientated). c) Can you explain why profits may be greater for both firms when they develop software for the same operating system? There are likely to be benefits associated with having a standardized operating system. For example, using a standardized operating system may increase the overall demand for software (which is very likely). I did not ask you to identify mixed strategy Nash Equilibria. However, I thought you might be interested First note that in any mixed strategy Nash Equilibria Software Developers Inc. would play Unix with probability zero because no matter what AMC Applications Inc. strategy is, Software Developers Inc. obtains a higher payoff from choosing Windows. Therefore, Software Developers Inc. will never choose Unix as part of any Nash Equilibria. If this is the case, AMC Applications Inc. best response to Software Developers Inc. s strategy cannot be to choose Unix because AMC Applications Inc. s payoff from choosing Windows is higher than from choosing Unix given that Software Developers Inc. does not choose Unix. Therefore, define p S as the probability Software Developers Inc chooses Windows and (1-p s ) as the probability Software Developers Inc chooses Object-Orientated. Define p A as the probability AMC Applications Inc chooses Windows and (1-p A ) as the probability AMC Applications Inc chooses Object-Orientated. Have each player randomize to make the other player indifferent between choosing Windows and Object-Orientated and then solve for p s and p A.
2. 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 Student 1 Front-Right 30,30 100,100 30,30 30,30 Back-Left 30,30 30,30 100,100 30,30 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.
3. Holiday Inn and Choice Hotels must select how many hotels to build on South Padre Island. Suppose they select the number of hotels at the same time. The following table provides Holiday Inn s and Choice s profits based on the number of hotels they decide to build. Suppose both Holiday Inn and Choice are deciding between 0, 1 or 2 hotels. Let M denote millions of dollars. Choice Hotels 0 Hotels 1 Hotel 2 Hotels 0 Hotels 0 M, 0 M 0 M, 9 M 0 M, 10 M Holiday Inn 1 Hotel 8 M, 0 M 7 M, 8 M 5 M, 11 M 2 Hotels 7 M, 0 M 5 M, 6 M 3 M, 4 M a) Does either Holiday Inn or Choice Hotels have a dominant strategy? If so, please identify the dominant strategy. In either case, please provide an EXPLANATION. A dominant strategy is a strategy that results in the highest payoff to a player regardless of the opponent s strategy. Holiday Inn s payoff is greatest by choosing 1 hotel whether Choice chooses 0, 1, or 2 hotels. Therefore, Holiday Inn s dominant strategy is to choose 1 hotel. Choice s payoff is never greatest from choosing 0 hotels no matter what Holiday Inn chooses. Choice s payoff is greatest from choosing 1 hotel if Holiday Inn chooses 2 hotels but not when Holiday Inn chooses 0 or 1 hotel. Choice s payoff is greatest from choosing 2 hotels if Holiday Inn chooses 0 or 1 hotel but not when Holiday Inn chooses 2 hotels. Therefore, Choice does not have a dominant strategy. The strategy that provides the largest payoff for Choice depends on the strategy of Holiday Inn. b) Identify the Pure Strategy Nash Equilibrium (ia). Holiday Inn selecting 1 hotel and Choice selecting 2 hotels is a Nash Equilibrium. 4. East Lansing and Lansing are two cities located right next to each other. Each city is deciding whether to allow casinos in their city. If both cities allow casinos, each city s payoff will be 30. (The reason for the negative payoff is that the casinos will compete against each other and, therefore, will not be very profitable. The city s payoff will be a function of the casinos profitability because it will affect the tax revenue the city collects. In addition, the more casinos that exist, the busier are local law enforcement personnel.) If East Lansing allows casinos and Lansing does not allow casinos, then East Lansing s payoff is 60 and Lansing s payoff is 10. If East Lansing does not allows casinos and Lansing allows casinos, then East Lansing s payoff is -20 and Lansing s payoff is 50. If neither city allows casinos, each city s payoff is 0. Suppose East Lansing and Lansing make their decision of whether to allow casinos at the same time. (Hint: It may be useful to depict the normal form game (i.e., the table with payoffs and strategies).) East Lansing Lansing Allow Don t Allow Allow -30, -30 60, -10 Don t Allow -20, 50 0, 0 a) Does either East Lansing or Lansing have a dominant strategy? EXPLAIN IN DETAIL. Neither East Lansing nor Lansing has a dominant strategy. If Lansing Allows, East Lansing prefers to Not Allow and if Lansing Doesn t Allow, East Lansing prefers to Allow. Similar logic holds with Lansing. b) Identify all Pure Strategy Nash Equilibria. East Lansing strategy is specified first and then Lansing s strategy is specified. There are two Pure Strategy Nash Equilibria and they are : (Allow, Don t Allow) and (Don t Allow, Allow)
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 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)
6. Suppose Gang A and Gang B sell drugs in the same area, but their customers can choose to buy drugs in other areas the gangs don t operate in. If there are conflicts between the two gangs, people will be less likely to come buy drugs from them so they will have to lower their prices to sell their drugs, hurting their profitability. The gangs profits are represented by their payoffs in the extensive form game below. Gang A Fight Don t Fight Gang B Gang B Retaliate Don t Retaliate Attack Don t Attack GANG A: -50 70 20 65 GANG B: -40-30 10 50 What is the Subgame Perfect Equilibrium of the above game? Explain. Gang A Fight: Gang B Don t Retaliate if Gang A Fights and Gang B Don t Attack if Gang A Don t Fights It is depicted above by the red lines.
7. Lafollete Kitchen and Design is providing Stacy Dickert-Conlin a quote (i.e., specify a price) for a kitchen renovation. For simplicity, assume Lafollete can either quote a high price of $30,000 or a low price of $20,000. After Lafollete quotes a price, Stacy Dickert-Conlin decides whether to have Lafollete renovate the kitchen or not to have the kitchen renovated. (Assume there is no negotiating over the quoted price.) If Stacy Dickert-Conlin decides to have Lafollete renovate the kitchen, Lafollete then decides whether to do a high quality job or a low quality job. Note that Lafollete decides on the quality of the job after Stacy makes her decision. The maximum Stacy Dickert- Conlin is willing to pay for a high quality job is $35,000 and the maximum she is willing to pay for a low quality job is $29,000. It costs Lafollete $18,000 to do a high quality job and $17,000 to do a low quality job. a) Depict the extensive form of the game described above (i.e., the game tree). Lafollete High Price of $30k Low Price of $20k Stacy Dickert Conlin Stacy Dickert Conlin Don t Renovate Renovate Don t Renovate Renovate Lafollete Lafollete s Payoff: 0 0 Stacy Dickert Conlin s Payoff: 0 0 Lafollete High Quality Low Quality High Quality Low Quality Lafollete s Payoff: 30k-18k=12k 30k-17k=13k 20k-18k=2k 20k-17k=3k Stacy Dickert Conlin s Payoff: 35k-30k=5k 29k-30k=-1k 35k-20k=15k 29k-20k=9k b) What is the Subgame Perfect Equilibrium of the game? BE PRECISE AND MAKE SURE YOU IDENTIFY A STRATEGY FOR EACH PLAYER. The red lines above represent the Subgame Perfect Equilibrium. This equilibrium is: Lafollete bids a price of $20k and selects a low quality job if Stacy Dickert-Conlin selects to Renovate no matter what Lafollete initially bids. Stacy Dickert-Conlin chooses not to renovate if Lafollete bids $30k and chooses to renovate if Lafollete bids $20k. This equilibrium results in a payoff for Lafollete of $3k and a payoff for Stacy Dickert-Conlin of $9k.
8. Both Magic Johnson and Lou Anna Simon are interested in opening separate Chester s in East Lansing. Chester s, a quick service chicken sandwich restaurant, is a fast growing franchise restaurant. For simplicity, suppose Chester s (the franchisor) charges a fixed monthly fee and does not obtain a percent of total revenue. As for the fixed monthly fee, Chester s is willing to charge different franchisees different fixed fees. If Magic Johnson has the only Chester s in East Lansing, his monthly demand for chicken sandwiches would be as depicted below. Assume Magic Johnson cannot price discriminate. If Magic Johnson has only Chester s in East Lansing 10 9 8 7 6 5 4 3 2 1 0 0 10 20 30 40 50 D 60 70 MR 80 90 100 MC If Lou Anna Simon has only Chester s in East Lansing 9 8 7 6 5 4 3 2 1 0 0 10 20 30 40 D 50 60 MR 70 MC 80 Assume Magic Johnson and Lou Anna Simon cannot price discriminate. Suppose the marginal cost of each chicken sandwich is $2 and monthly fixed costs are $10 for both Magic and Lou Anna Simon (not including the franchisor s fixed monthly fee). If both Magic Johnson and Lou Anna Simon become Chester s franchisees, they open up separate Chester s restaurants in East Lansing. In this case, Magic s monthly profits (prior to the franchisor s fixed fee) would be $70 and Lou Anna Simon s profits (prior to the franchisor s fixed fee) would be $60. Note that these profits do not include the franchisor s monthly fixed fee. Suppose Chester s decides to first offer a franchise to Magic Johnson by making a take-it-or-leaveit fixed monthly fee offer. Magic Johnson then decides whether or not to accept this fixed monthly fee offer. Whether or not Magic Johnson accepts the offer, Chester s then makes a take-it-or-leaveit offer to Lou Anna Simon. Chester s take-it-or-leave-it offer to Lou Anna Simon can be different than the offer to Magic Johnson. Lou Anna Simon then accepts or rejects this offer. a) Depict the extensive form game and identify the subgame perfect equilibrium. b) Would Chester s be better off if, when making the offer to Magic Johnson, Chester s could commit not to offer a franchisee to Lou Anna Simon? Explain.
If Magic has the only Chester s, his profits are 6*40-2*40-10-FF MJ = 150- FF MJ. If Lou Anna Simon has the only Chester s, her profits are 5*30-2*30-10-FF MJ = 80- FF MJ. The extensive form below represents the game. Chester s FF MJ Accept MJ Reject Chester s Chester s FF LAS FF LAS LAS LAS Accept Reject Accept Reject Chester: FF MJ +FF LAS FF MJ FF LAS 0 MJ: 70- FF MJ 150- FF MJ 0 0 LAS: 60- FF LAS 0 80- FF LAS 0 Using backward induction to determine Subgame Perfect Equilibrium. If Magic Johnson accepted Chester s offer, Lou Anna Simon would accept FF LAS if FF LAS <60. If Magic Johnson rejected Chester s offer, Lou Anna Simon would accept FF LAS if FF LAS <80. Chester should offer Lou Anna Simon a fixed fee of FF LAS =60 if Magic accepted and a fixed fee of FF LAS =80 if Magic Johnson rejected the offer of FF MJ. Magic Johnson should accept FF MJ <70 (realizing that if he accepts, Chester s will then offer a fixed fee of 60 to Lou Anna Simon which she will accept). Chester s should offer a fixed fee of FF MJ =70 to Magic Johnson which he accepts.. In the end, Chester s offers FF MJ =70. Magic Johnson accepts the offer. Chester s offers FF LAS =60 and Lou Anna Simon accepts this offer. PART B) There is a benefit from being able to commit to Magic Johnson. By committing not to offer a franchisee to Lou Anna Simon, Magic Johnson would then be willing to accept FF MJ <150 and Chester s would offer FF MJ =150 while offering this commitment to Magic Johnson. Chester s payoff would be $20 more than when they did not commit (part a).
9. Bill owns a warehouse in Lansing that Jane operates her exporting business from. Suppose Jane is leasing the warehouse and the lease agreement stipulates that, if Bill decides to sell the warehouse, Jane would be allowed to submit a bid on the warehouse before other potential buyers. Let Fred be the only other potential buyer and assume Bill is thinking about selling the warehouse. The minimum Bill is willing to accept is $3 million ($3M). If Bill decides to sell, Jane would make a bid (denote as B J ) which Bill would either accept or initially reject. If Bill accepts, then the warehouse would be sold to Jane for B J. The warehouse is worth $6M to Jane. If Bill initially rejects Jane s bid, Fred then decides whether or not to bid on the warehouse. Fred incurs a cost of $500,000 when putting together a bid. If Fred decides to put together a bid, Fred must then decide exactly what to bid (denote as B F ). The warehouse is worth $5M to Fred. If Fred does not put together a bid, Bill then decides whether or not to accept Jane s bid (B J ). If Fred does put together a bid, Bill then decides whether to reject both bids, accept Jane s bid (B J ) or accept Fred s bid (B F ). a) Depict the extensive form of the game (i.e., draw the game tree). Bill Don t Sell Sell Bill: 3 Jane Jane: 0 Fred: 0 BJ Bill Reject Initially Fred Accept Bid Don t Bid BJ Bill 6-BJ Fred 0 Accept BJ Reject BF BJ 3 Bill 6-BJ 0 0 0 Reject Accept BJ Accept BF Bill: 3 BJ BF Jane: 0 6-BJ 0 Fred -.5 -.5 5- BF -5 b) What is the subgame perfect equilibrium of this game? (I suspect this would be the likely outcome to this game.) Backward Induction Bill: Accept B J if B J >B F and B J >3, Accept B F if B F >B J and B F >3, Reject if B F <3 and B J <3. (doesn t really matter where the weak inequalities are put) Fred: Offer B F =B J if 3<B J <4.5, Offer B F =3 if 3>B J, otherwise bid less than B J. Bill (if Fred Doesn t Bid): Accept if B J >3.
Fred: Don t Bid if B J >4.5 and Bid if B J <4.5. Bill: Accept if B J >4.5, Reject Initially if B J <4.5. Jane: Offer B J =4.5 Bill: Sell In the end, Bill sells and accepts Jane s offer of 4.5M (you do not have to go thru the entire logic above to get full credit). c) How would the outcome of the above game change if Jane could make a second offer (perhaps a matching offer) after Fred makes an offer? In other words, what would you expect to happen if Jane had a Right of First Refusal? (This is very similar to the example done in lecture when I discussed the Right of First Refusal in the context of the NBA. Based on the discussion in class, provide the intuition on how a Right of First Refusal would change the outcome. You do not have to draw the corresponding game tree but you can if it helps.) As noted in class, if Jane had a Right of First Refusal, Jane would match whatever bid Fred submits. Knowing this, Fred would not bid because he would incur the 500,000 cost and he would not win anyways. Knowing this, Jane would offer Bill (a penny over) $3M and Bill would accept this offer. In the end, the Right of First Refusal would hurt Bill and help Jane. d) How would the outcome of the game change if Jane s offer was taken off the table if Bill rejected it? What this means is, what would happen if Bill could not accept Jane s offer after Fred decides whether not to make an offer or make an offer of B F? (You do not have to draw the corresponding game tree but you need to explain the intuition.) If Jane s offer is taken off the table, Fred would only make an offer of $3M if Bill rejects Jane s offer. Knowing this, Jane would only offer $3M (perhaps a penny over that) and Bill would accept it. Notice that Jane benefits from being able to take her offer off the table.
10. Suppose it is 2008 and American International Group (AIG) executives are deciding whether to make a very risky investment. They can either choose to make this risky investment or choose not make this risky investment. If they choose not to make the risky investment, AIG s expected 2008 payoff is $14B ($14 Billion) and the United States government s 2008 payoff is 0. If they make this risky investment and the economy takes a dive, they have a chance of becoming insolvent. AIG realizes that if this occurs, the United States government would then decide whether or not to bailout AIG. Suppose AIG s expected 2008 payoff is $12B if they make the risky investment thinking that the government would not bail them out and $15B if they make the risky investment thinking that the government would bail them out. The United States government s expected 2008 payoff from choosing a bailout if AIG becomes insolvent is minus $8B and the United States government s expected 2008 payoff from choosing no bailout if AIG becomes insolvent is minus $6B. Suppose this is the game the United States government and AIG play in 2008 and it is the exact same game that is played in 2009 irrespective of what happens in 2008. Suppose the annual interest rate is 10% for AIG as well as the United States government. a) Depict below the extensive form of the above game (include both the 2008 and 2009 decisions along with the payoffs). AIG Make risky investment Don t make risky investment in 2008 in 2008 US Gov t AIG bailout No bailout Make risky Don t make risky AIG AIG investment investment in 2009 in 2009 Make risky Don t Make Make risky Don t make Investment risky investment Investment risky investment in 2009 in 2009 in 2009 in 2009 US Gov t US Gov t US Gov t No No No bailout bailout bailout bailout bailout bailout AIG: 15+15/1.1 15+14/1.1 12+15/1.1 12+14/1.1 14+15/1.1 14+14/1.1 US: -8-8/1.1-8 -6-8/1.1-6 -8/1.1 0 AIG: 15+12/1.1 ` 12+12/1.1 14+12/1.1 US: -8-6/1.1-6-6/1.1-6 b) What is the subgame perfect equilibrium of this game? Depict on the graph above. (You can also obtain partial credit by describing the subgame perfect equilibrium.) See above. In the end, AIG does not make the risky investment in both 2008 and 2009. The reason is that the US government would not bail them out if they did make the risky investment (since -8<- 6). If the US government s payoff was greater from bailing AIG out than from not bailing them out, then the US government would bail them out and AIG would then have incentive to take the risky investment since 14<15. c) Now suppose the United States government could credibly commit to either a bailout or no bailout prior to AIG s investment decision. Would the outcome of this game change and would this outcome be preferred by the United States government? Explain. (A game tree is not necessary.) The outcome of the game would not change because the US government can already credibly commit to not bailing AIG out. See the answer to part b).