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nicholson_snyder_--_ch_8_14_15 - Chapter 8 Strategy and...

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Chapter 8: Strategy and Game Theory 55 CHAPTER 8 STRATEGY AND GAME THEORY These problems cover a variety of different concepts introduced in the chapter. They range in difficulty from the simplest exercise of finding the Nash equilibrium in a two- by-two matrix to characterizing equilibrium when players have continuous actions and payoffs with general functional forms. Practice with problems may be the primary way for students to master the material on game theory. Comments on Problems 8.1 Provides practice in finding pure- and mixed-strategy Nash equilibria using a simple payoff matrix. The three-by-three payoff matrix makes the problem slightly harder than the simplest case of a two-by-two matrix. Although this problem points the student where to look for the mixed-strategy equilibrium, in other cases there may be many possibilities that need to be checked for mixed- strategy equilibria. In a game represented by a three-by-three matrix, each player has four combinations of two or more actions, and so there are 16 possible types of mixed-strategy equilibria to check. Software, called Gambit, has been developed that can solve for all the Nash equilibria of games the user specifies in extensive or normal form. Gambit is freely available on the Internet. It is easy to use, almost functioning as a “game-theory calculator.” One useful classroom exercise would be have students solve some of the problems on a game-theory problem set using Gambit, either alone or in teams. McKelvey, R.D.; A.M. McLennan; and T. L. Turocy (2007) Gambit: Software Tools for Game Theory , Version 0.2007.01.30. http://econweb.tamu.edu/gambit 8.2 A slight generalization of payoffs in the Battle of the Sexes provides students with further practice in computing mixed-strategy Nash equilibria. 8.3 Provides practice in converting the payoff matrix for a simultaneous game into one for a sequential game. Illustrates the application of subgame-perfect equilibrium in the simple case of the famous Chicken game. 8.4 The problem provides practice in computing the Nash equilibrium in a game with continuous actions (similar to the Tragedy of the Commons in this chapter and in Chapter 15 with the Cournot game, except in this problem the best-
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56 response functions are upward-sloping). Players’ best responses are computed using calculus, and the resulting equations are then solved simultaneously. 8.5 Asks students to solve for the mixed-strategy Nash equilibrium with a general number of players n . The “punchline” to the problem that the blond is less likely to be approached as the number of males increases is a paradoxical result characteristic of such games. The problem is based on a scene in the Academy Award winning movie, A Beautiful Mind , about the life of John Nash, in which the Nash character discovers his equilibrium concept (the one scene in the movie that involves any game theory). If the classroom facilities allow, it is worthwhile to show students this scene (Scene 5: “Governing Dynamics”) when covering this problem.
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