Unformatted text preview: 244 [CH. 7] SIMULTANEOUSMOVE 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 20yard play when the rival team anticipates
the 20yard 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
10yard play on that down? (Use your answer in part ((1) and remem
ber that there is an 80% success rate for the 10yard play when the rival
team anticipates the 10yard play.) (g) What are the expected payoffs to each team when the home team runs
the 10yard play on third down while the rival team anticipates the
20yard 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 qmix for each team on third
down? (j) What is the expected payoff to the home team for the overall twostage
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 5050 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 zerosum 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 riskaverse, 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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 Fall '08
 Charness,G

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