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Unformatted text preview: 244 [CH. 7] SIMULTANEOUS-MOVE GAMES WITH MIXED STRATEGIES I (e) What are the expected payoffs for each team when the home team runs the 20—yard play on third down while the rival team anticipates the 20—yard play? (Use your answer to part (b) and remember that there is a 50% success rate for the 20-yard play when the rival team anticipates the 20-yard play.) (f) What are the expected payoffs to each team when the home team runs the 10—yard play on third down and the rival team anticipates the 10-yard play on that down? (Use your answer in part ((1) and remem- ber that there is an 80% success rate for the 10-yard play when the rival team anticipates the 10-yard play.) (g) What are the expected payoffs to each team when the home team runs the 10-yard play on third down while the rival team anticipates the 20-yard play? (h) Now construct the game table for third down with 20 yards to go. (Use your answers from parts (e), (f), and (g). (i) What are the equilibrium p—mix and q-mix for each team on third down? (j) What is the expected payoff to the home team for the overall two-stage game? 811. The recalcitrant James and Dean are playing their more dangerous vari- ant of chicken again (see Exercise 86). They’ve noticed that their payoff for being perceived as “tough" varies with the size of the crowd. The larger the crowd on hand, the more glory and praise each receives from driving straight when his opponent swerves. Smaller crowds, of course, have the opposite effect. Let k > 0 be the payoff for appearing “tough." The game may now be represented as: JAMES (a) Expressed in terms of k, with what probability does each driver play Swerve in the mixed~strategy Nash equilibrium? Do James and Dean play Swerve more or less often as k increases? (b) In terms of k, what is the expected value of the game to each player when both are playing the mixed—strategy Nash equilibrium found in part (a)? (c) At what value of k do both James and Dean mix 50-50 in the mixed—strategy equilibrium? 812. 813. EXERCISES 245 (d) How large must k be for the average payoff to be positive under the al- ternating scheme discussed in part (c) of Exercise 86? Consider the following zero-sum game: The entries are the Row player's payoffs, and the numbers A, B, and C are all positive. What other relations among these numbers (for example, A < B < C) must be valid for each of the following cases to arise? (a) At least one of the players has a dominant strategy. (b) Neither player has a dominant strategy, but there is a Nash equilibrium in pure strategies. (c) There is no Nash equilibrium in pure strategies, but there is one in mixed strategies. (d) Given that case (c) holds, write a formula for Row’s probability of choos— ing Up. Call this probability p, and write it as a function of A, B, and C. (Optional) Return to the game between Evert and Navratilova as shown in Figure 7.1. Suppose that Evert is risk-averse, as discussed in Appendix 2 to this chapter, so that she dislikes uncertainty in outcomes. In particu— lar, suppose that Evert has a square—root utility function, so that her util~ ity equals the square root of the payoff listed in the table, and suppose that Navratilova remains risk neutral, so that her utility equals her payoff. Sup- pose that the players know each other's utility functions, and each player wishes to maximize her expected utility. (a) Find the mixed»strategy Nash equilibrium of this game. (b) How did the two players' mixing proportions change, relative to the original case where both players were risk neutral? Explain why this might have happened. ...
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