Suggested Solutions to Questions in Topic 6 1. 9 Introduction of advertisement game between Firm A and B The payoff matrix for an advertising game a) Is there any dominant strategy b) Find the Nash Equilibrium Observe from this payoff matrix that if both firms decide to advertise Firm A makes a profit of 10 and Firm B will make a profit of 5. If Firm A advertises and Firm B does not advertise Firm A will earn 15 and Firm B will earn zero. First consider Firm A. Firm A should advertise no matter what Firm B does because Firm A does best by advertising. If Firm B advertises A earns a profit of 10 if it advertises but only 6 if it doesn’t. Thus Advertising is a dominant strategy for Firm A 1 2 a) Find the Nash Equilibrium in the Introduction of Home Banking Services Games Introduction of Home Banking Services (HB) Game Between Afriland First Bank and Atlantic Bank Atlantic Bank Introduce HB Do Not introduce HB Introduce HB 24, 8 40, 6 Do Not Introduce HB 30, 12 36, 10 Afriland First Bank (payoffs are in millions of FCFA) Solution The Game Tree for Introduction of Home Banking Services (HB) Between Afriland First Bank and Atlantic Bank AFFB payoff AMB payoff (payoffs are in Millions of FCFA) HB 24 8 NHB 40 6 HB 30 12 NHB 36 10 HB AM AF NHB AM In this game, African First Bank does not have a dominant strategy. It is better off not introducing home banking if Atlantic Bank introduces one, but it prefers introducing home banking if Atlantic bank does not introduce one. Despite the absence of dominant strategy for the Afriland First Bank, 2 there is still a Nash equilibrium: Atlantic introduces home banking, and Afriland First bank doesn’t. It is interesting to see how Afriland First Bank might figure out which strategy to choose in this game. If it envisions this payoff matrix, it should realise that while it does not have a dominant strategy, Atlantic does “introduce home banking”. 3. Table 1 gives the payoff matrix for two rival holiday companies, who intend to expand their business by offering additional vacations to either Limbe or to Kribi. Firm B Limbe Kribi Limbe 100, 100 200, 200 Kribi 200, 200 100, 100 Firm A (payoffs are in Thousand of FCFA) Find the Nash equilibrium Solution Firm A payoff Firm B payoff (Payoff is in 000 FCFA) Limbe 100 100 Kribi 200 200 Limbe 200 200 Kribi 100 100 Limbe B A Kribi B Here we have two Nash equilibrium (Limbe – Kribi) or (Kribi – Limbe) 3 4. Afriland First Bank and Atlantic Bank are competing in an oligopolistic industry. Afriland First Bank, the larger of the two banks, is contemplating its capacity strategy, which we might broadly characterise as “aggressive” and “passive”. The aggressive strategy involves a large increase in capacity aimed at increasing the firm’s market share, while the passive strategy involves no change in the Afriland First Bank’s capacity. Atlantic Bank, the smaller competitor, is also pondering its capacity expansion strategy, it will also chose between an “aggressive strategy” or a “passive strategy”. The table below shows the profits associated with each pair of choices made by the two firms: Atlantic Bank Passive Aggressive Passive 0, 0 100, 300 Aggressive 300, 100 50, 50 Afriland First Bank (payoffs are in millions of FCFA) If Afriland First Bank could move first and Atlantic bank reacts what will be the best strategies? Solution AFFB payoff AMB payoff (Payoffs are in Millions of FCFA) Passive 0 0 Aggressive 100 300 Passive 300 100 Aggressive 50 50 Passive AL AFB Aggressive AL 4 If AFB can choose first, then the best sequential move strategy will be “Aggressive” for AFB and “passive” for AB. AFB will make the latter choice given that it can anticipate the decision of its rival. 5. Two banks, Afriland First Bank and Atlantic Bank, dominate the financial industry of Cameroon, with Afriland First Bank producing a new product called Flash Cash and Atlantic Bank producing Flash Cash Plus. Each bank has investigated the possibility of undertaking a more aggressive advertising campaign in order to maintain its market share in a competitive environment. The possible outcomes of such strategies are shown in the payoff matrix (Table 1) showing yearly profit figures (FCFA m). Atlantic Bank proctor lever Advertise No Advertise Advertise 11, 6 16, 0 No Advertise 7, 9 11, 3 Afriland First Bank 22, 2 (payoffs are in millions of FCFA) a) Use the payoff matrix to explain what it meant by the term “dominant strategy” for the respective banks. b) Suppose the payoff values in the bottom-right square now change from 11, 3 respectively to 22, 2, how will this influence the dominant strategy of each bank? Will the Nash equilibrium change? Solution a) A dominant strategy for Afriland First Bank would be one that is optimal for that company irrespective of Atlantic bank strategy. Similarly, the dominant strategy for Atlantic Bank would be the one that is optimal for it, irrespective of Afriland First Bank strategy. On the other hand, the Nash equilibrium is where Afriland First Bank is carrying out its optimal strategy giving what Atlantic bank is doing likewise. The dominant strategy for Afriland First Bank is to advertise, because whatever Atlantic Bank does, advertising still proves to be the most profitable strategy for Afriland First Bank. The best strategy of Atlantic Bank 5 is also to advertise. So (advertise, advertise is the dominant strategy for both firms. b) In this case Afriland First Bank does not have a dominant strategy because advertising is not the best course of action taking both Atlantic Bank’s strategies into consideration. Therefore, the Nash equilibrium will not change since Afriland First Bank will know that Atlantic Bank will have a dominant strategy of advertising whatever it does. 6. Find a Nash equilibrium: UNICS Finance Versus Western Union Commission charges per money transfer of 5000 FCFA or less UNICS Western Union 10.25 7.25 8.25 6.25 10.5 11.5 12.5 110.5 66; 190 79; 201 82; 212 75; 223 68; 199 82; 211 86; 224 80; 237 70; 198 85; 214 90; 229 85; 244 73; 191 89; 208 95; 225 81; 205 Solution With four strategies for each firm, this looks like a complicated game. But we can greatly simplify it by searching for dominated strategy. To see why, compare Western Union’s payoffs among the four rows in the table. You will see that Western Union’s payoff is always higher in row 1 – a money transfer commission charge of 8.25 FCFA – than in any other row. Thus, a commission charge of 8.25 FCFA for Western Union is a dominant strategy and the other three prices are dominated strategies. If UNICS assumes that Western Union will follow its dominant strategy, UNICS should expect Western Union to set a commission charge per money transfer of 8.25 FCFA. Given this commission charge, UNICS best response is to set a price of 12.5 FCFA. The Nash equilibrium in this game is for Western Union to set a commission charge of 8.25 FCFA and UNICS to set a commission charge of 12.5 FCFA. 7 If Western Union could move first and UNICS reacts find the Nash equilibrium. Commission charges per money transfer UNICS 6 Western Union 6.25 7.25 8.25 9.25 10.5 11.5 12.5 110.5 85; 190 79; 201 82; 212 75; 223 87; 199 82; 211 86; 224 80; 237 93; 198 85; 214 90; 229 85; 244 97; 191 89; 208 95; 225 81; 205 Solution With four strategies for each firm, this looks like a complicated game. But we can greatly simplify it by searching for dominated strategy. To see why, compare Western Union’s payoffs among the four rows in the table. You will see that Western Union’s payoff is always higher in row 1 – a money transfer commission charge of 6.25 FCFA – than in any other row. Thus, a commission charge of 6.25 FCFA for Western Union is a dominant strategy and the other three prices are dominated strategies. If UNICS assumes that Western Union will follow its dominant strategy, UNICS should expect Western Union to set a commission charge per money transfer of 6.25 FCFA. Given this commission charge, UNICS best response is to set a price of 11.5 FCFA. The Nash equilibrium in this game is for Western Union to set a commission charge of 6.25 FCFA and UNICS to set a commission charge of 11.5 FCFA. 8. Afriland First Bank and Amity Bank are the most reliable banks in Cameroon. These two banks compete head-to-head for expansion in Equatorial Guinea. The following table shows the profit in millions of FCFA that each firm earns when it makes the expansion decision in Equatorial Guinea. Find the Nash Equilibrium in the Modified Capacity Expansion Game between Afriland First Bank (AFB) and Amity Bank (AB). AB Large Small Do not Build AFB Large Small Do not Build 0; 0 80; 120 90; 180 120; 80 160; 160 150; 200 150; 90 170; 180 180; 180 Payoffs are in millions of FCFA. Solution 7 Figure 2 Modified Capacity Expansion Game between Afriland First Bank (AFB) and Amity Bank AB) AFB payoff AB payoff Payoffs are in millions of FCFA Large Large FCFA0 FCFA0 Small FCFA120 FCFA80 Do Not Build FCFA150 FCFA90 Large FCFA80 FCFA120 Small FCFA160 FCFA160 Do Not Build FCFA170 FCFA180 Large FCFA90 FCFA180 Small FCFA150 FCFA200 Do Not Build FCFA180 FCFA180 AB Small AFB AB Do Not Build AB Neither player has a dominant strategy, and with three strategies rather than two, the task of finding a Nash equilibrium is daunting. No matter what Amity bank does, Afriland First Bank is always better off by choosing “Small” and the same conclusion holds the other way round. In this simultaneous-move game, Amity Bank cannot observe Afriland First Bank’s decision beforehand, and therefore Afriland First Bank Cannot force Amity Bank hand. 9. Afriland First Bank and Amity Bank are competing in an oligopolistic industry. Afriland First Bank, the larger of the two banks, is contemplating its capacity strategy, which we might broadly characterise as “aggressive” and “passive”. The aggressive strategy involves a large increase in capacity aimed at increasing the firm’s market share, while the passive strategy involves no change in the Afriland First Bank’s capacity. Amity Bank, the smaller competitor, is also pondering its capacity expansion strategy, it will also chose between an “aggressive strategy” or a “passive strategy”. The table below shows the profits associated with each pair of choices made by the two firms: Amity Bank 8 Aggressive Passive Aggressive 25, 9 33, 10 Passive 30, 13 36, 12 Afriland First Bank (payoffs are in millions of FCFA) a) If both banks decide their strategies simultaneously, what is the Nash equilibrium? b) If Afriland First Bank could move first and Amity bank reacts what will be the best strategies? Solution AFFB payoff AMB payoff (payoffs are in Millions of FCFA) Aggressive 25 9 Passive 33 10 Aggressive 30 13 Passive 36 12 Aggressive AM AFB Passive AM a) If both banks choose simultaneously, then Afriland First bank will choose its dominant strategy, “Passive”. Amity bank knowing the Afriland First Bank has a dominant strategy, will assume AFB will play this strategy and choose “aggressive”. The Nash equilibrium, therefore, has AFB selecting Passive and AB selecting “Aggressive”. b) If AFB can choose first, then the new best sequential move strategy will be “Aggressive” for AFB and “passive” for AB. 9 10 This game is set in the South Pacific in 1943. Admiral Imamura must transport Japanese troops from the port of Rabaul in New Britain, across the Bismarck Sea to New Guinea. The Japanese fleet could either travel north of New Britain, where it is likely to be foggy, or south of New Britain, where the weather is likely to be clear. U.S. Admiral Kenney hopes to bomb the troop ships. Kenney has to choose whether to concentrate his reconnaissance aircraft on the Northern or the Southern route. Once he finds the convoy, he can bomb it until its arrival in New Guinea. Kenney’s staff has estimated the number of days of bombing time for each of the outcomes. The payoffs to Kenney and Imamura from each outcome are shown in the box below. The game is modeled as a“zero-sum game.” For each outcome, Imamura’s payoff is the negative of Kenney’s payoff. a) b) c) d) Is there any dominant strategy for Kenney? Is there any dominant strategy for Imamura? Is there any equilibrium, Nash equilibrium? Explain your result. Solution a) This game does not have a dominant strategy equilibrium, since there is no dominant strategy for Kenney. His best choice depends on what Imamura does. b) No c) The only Nash equilibrium for this game is where Imamura chooses the northern route and Kenney concentrates his search on the northern route. To check this, notice that if Imamura goes North, then Kenney gets an expected two days of bombing if he (Kenney) chooses North and only one day if he (Kenney) chooses South. Furthermore, if Kenney concentrates on the north, Imamura is indifferent between going north or south, since he can be expected to be bombed for two days either way. Therefore if both choose “North,” then neither has an incentive to act differently. You can verify that for any other combination of choices, the admiral or Kenney would want to change. As things actually worked out, Imamura chose the Northern route and Kenney concentrated his search on the North. After about a day’s search the Americans found the Japanese 10