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Solutions to Chapter 7 Exercises
S12. (a) The kicker’s expected payoffs playing against the goalie’s mixed strategy of L
42.2%, R 42.2%, and C 15.6% are as follows:
HL 42.2(0.50) + 15.6(0.85) + 42.2(0.85) = 70.23
(b) Thus, the kicker should be using low, side shots (LL and LR) and high, centered
(c) For the goalie playing against the kicker using LL 37.8%, HC 24.4%, and LR
37.8% of the time, payoffs from each of his possible strategies are
(d) All three strategies give almost the same expected payoff, so the goalie could
(e) These mixed strategies are Nash equilibria, because each player’s strategy is a
(f) The equilibrium payoff to the kicker is 71.77.
S13. (a) Suppose Rowena plays Up with probability p and Down with probability (1 – p).
(b) Colin must keep Rowena indifferent between her two pure strategies, Up and
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) In each case, the player’s equilibrium mixture probabilities are totally
(d) To guarantee that a mixed-strategy Nash equilibrium exists, the values for p and
S14. (a) Consider any one of the young men. If he chooses to go after a brunette, he gets
a guaranteed payoff of 5. If he chooses to go after the blonde, he gets 10 if none of the other (n –
(b) The same logic can be applied to the m young men who are choosing Blonde, so
Consider any one of the (n – m) choosing Brunette. If he switched to Blonde, his payoff will be
10 if all the m-mixers happen to choose Brunette, and 0 otherwise. Therefore, his expected payoff
is
So this pure Brunette chooser does not want to switch to a pure Blonde strategy if
But we already have the condition of the mixed-strategy equilibrium
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
UNSOLVED EXERCISES
U1. (a) Best-response analysis shows that no pure-strategy Nash equilibrium exists.
The mixed-strategy Nash equilibrium is
(c) The offense uses a different mixture than the defense because its opponent has
different payoffs. It thus requires a different mixture to keep its opponent indifferent.
The mixed-strategy Nash equilibrium is
U3. (a) If Anne plays 1 with probability p, Bruce’s expected payoff from playing Odd is
1 – p:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Bruce
Odd Even
(d) The mixed-strategy Nash equilibrium of the game is
The expected payoff to both Anne and Bruce is 1/2.
The mixed-strategy Nash equilibrium is
(b) Since her payoff for playing DL decreased from 50 to 25 (because of
Evert’s expected payoff is lower now (it used to be 62) because of Navratilova’s improved
defense.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U5. (a) We would expect Harry’s equilibrium p to increase (meaning that his play for
The new mixed-strategy Nash equilibrium is
Before, the equilibrium was
(d) Despite the fact that Sally’s expected payoff in the mixed-strategy equilibrium
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) Dean’s probability of playing Straight is higher now. James’s probability of
U7. The graph of the best-response curves follows:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Nash equilibria occur where the best-response curves intersect: (0, 1), (1/2, 1/2), and (1, 0). These
U8. (a) The two pure-strategy Nash equilibria are (D, L), yielding payoffs of (2, 5), and
(b) The graph below shows the payoffs of Colin’s four strategies when Rowena
When p < 0.5, Colin will play L, and when p > 0.5, Colin will play N. When p = 0.5, Colin is
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
!
!
L
M
N
R
Payoff
to
Colin
U9. (a) The graph of the expected payoffs of Professor Plum’s strategies is shown below:
(
b
)
P
r
o
f
e
s
s
or Plum will use both Revolver and Knife in his mixture because the intersection of those lines is
the lowest point on the upper envelope of his payoffs.
(c) The mixed-strategy Nash equilibrium of the game occurs when Mrs. Peacock
(d) The only difference between this game and that in Exercise S9 is that Professor
Plum’s payoff from playing Revolver when Mrs. Peacock plays Ballroom is now 2 instead of 1.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Peacock’s equilibrium value of p changes. Also, since Professor Plum plays Revolver instead of
U10. (a) The procedure here is similar to that in Exercise S10. Now Lisa’s payoffs from
First, equate Lisa’s payoffs from Rock and Paper and find 20p2 + 10p1 – 10 = 10p1 – 10p2 or 30p2
(b) In this question, Bart has an “advantage” when he plays Rock and Lisa plays
Scissors, but the equilibrium mixtures entail Bart using relatively less Rock (3/12) and Lisa also
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
