ECO 3305 Sec 001 Spring 2014 Dr. K. Becker QUIZ #13 Monday, April 14th In Figure 1, a “game” between Apple and three App Companies is depicted. Figure 1: IOS game Apple Company 1 Company 2 Company 3 0 0 0 0 Do not develop IOS 8 Apple Develop IOS 8 Company 1 Develop Do not develop Company 2 D Company 2 DND D DND Company 3 D DND 5 2 2 2 3 1 1 0 D DND 3 1 0 1 D is “Develop” DND is “Do not develop” -2 -1 0 0 Company 3 D 3 0 1 1 DND D DND -2 0 -1 0 -2 0 0 -1 -3 0 0 0 If Apple develops IOS 8, software company #1 moves first in deciding whether to develop an application and, after its choice has been revealed, then companies #2 and #3 act simultaneously in deciding whether to develop an app or not. Derive all subgame perfect Nash equilibria Solution: Consider the subgame between companies 2 and 3 associated with apple having developed IOS 8 and company 1 having developed an application. The strategic form of the game is shown in Figure SOL9.2.1. Figure SOL9.2.1 Company 3 Develop Do not develop Company 2 Develop Do not develop 2,2 0,1 1,0 0,0 Develop is a dominant strategy for each company so there is a unique Nash equilibrium of (Develop, Develop) for this subgame. Next consider the subgame associated with Aplple having developed IOS 8 and company 1 having not developed an application. The strategic form of the game is shown in Figure SOL9.2.2. Figure SOL9.2.2 Company 3 Develop Do not develop Company 2 Develop Do not develop 1,1 0,-1 -1,0 0,0 This has two Nash equilibria: (Develop, Develop) and (Do not develop, Do not develop). Move up the tree to the subgame initiated by Apple having developed IOS 8 where company 1 has to decide whether to develop an application. First suppose that the Nash equilibrium for the subgame in which company 1 does not develop an application is (Develop, Develop). Replacing the two final subgames with the Nash equilibrium payoffs, the situation is as depicted in Figure SOL9.2.3. If company 1 develops an application then its payoff is 2, while its payoff is 0 from not doing so. Hence, it chooses Develop. Figure SOL9.2.3 IBM Company 1 Company 2 Company 3 Do not develop OS/2 0 0 0 0 Apple Develop IOS 8 Company 1 Develop 5 2 2 2 Do not develop 3 0 1 1 Now suppose the Nash equilibrium when company 1 does not develop an application is (Do not develop, Do not develop). Replacing the two final subgames with the Nash equilibrium payoffs, the situation is as depicted in Figure SOL9.2.4. If company 1 develops an application then its payoff is 2, while its payoff is 0 from not doing so. Hence, it chooses Develop. Figure SOL9.2.4 Apple Company 1 Company 2 Company 3 Do not develop IOS 8 0 0 0 0 IBM Develop IOS 8 Company 1 Develop Do not develop 5 2 2 2 -3 0 0 0 Thus, regardless of which Nash equilibrium is used in the subgame in which company 1 chooses Do not develop, company 1 optimally chooses Develop. Now we go to the subgame which is the game itself. If Apple chooses to develop IOS 8 then, as previously derived, company 1 develops an application and this induces both companies 2 and 3 to do so as well. Hence, Apple's payoff is 5. It is then optimal for Apple to develop IOS 8. There are then two subgame perfect Nash equilibria (where a strategy for company 2, as well as 3, is an action in response to company 1 choosing Develop and an action in response to Company 1 choosing Do not develop): (Develop IOS 8, Develop, (Develop,Develop), (Develop,Develop)), (Develop IOS 8, Develop, (Develop,Do not develop), (Develop,Do not develop)) Both equilibria result in the same outcome path.
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