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Solutions to Chapter 7 Exercises
SOLVED EXERCISES
S1. (a) The game most resembles an assurance game because the two Nash equiibria
occur when the players use symmetric moves. (Here they both use the same moves to arrive at the
(b) The two pure-strategy Nash equilibria for this game are (Risky, Risky) and (Safe,
S2. (a) There is no pure-strategy Nash equilibrium here, hence the search for an
equilibrium in mixed strategies. Rowena’s p-mix (probability p on Up) must keep Colin
(b) Rowena’s expected payoff is 2.5. Colin’s expected payoff = 17.33.
(c) Joint payoffs are larger when Rowena plays Down, but the highest possible
payoff to Rowena occurs when she plays Up. Thus, in order to have a chance of getting 4,
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
S3. The two pure-strategy Nash equilibria are (Don’t Help, Help) and (Help, Don’t Help).
The mixed-strategy Nash equilibrium has the following equilibrium mixtures:
That is, each player helps two-thirds of the time and doesn’t help one-third of the time.
S4. (a) On the one hand, Evert does worse when using DL than she did before. On the
So the mixed-strategy Nash equilibrium occurs when Evert plays 7/10(DL) + 3/10(CC) and
(c) Compared with the previous game, Evert plays DL with the same proportion,
whereas Navratilova plays DL less, going from 3/5 to 2/5. Navratilova’s q-mix changes because
The mixed-strategy Nash equilibrium is
Batter plays 2/3(Anticipate fastball) + 1/3(Anticipate curveball) and Pitcher plays
1/2(Throw fastball) + 1/2(Throw curveball).
(c) The mixed-strategy Nash equilibrium is now
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
greater than his previous expected payoff. The pitcher can increase his expected payoff because
the batter is forced to adjust his equilibrium strategy in a way that favors the pitcher.
In the mixed-strategy Nash equilibrium James plays 9/10(Swerve) + 1/10(Straight), and Dean
plays 9/10 (Swerve) + 1/10(Straight). James and Dean play Straight less often than in the
previous game.
(c) If James and Dean collude and play an even number of games where they
These expected payoffs are much worse than the collusion example or the mixed-strategy
S7. Sally’s expected payoff from choosing Starbucks when Harry is using his p-mix is p; her
expected payoff from choosing Local Latte when Harry is mixing is 2 – 2p. Similarly, Harry’s
Sally’s best response to Harry’s p-mix is to choose Local Latte for values of p below 2/3
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Expected payoffs for Sally and Harry are 2/3 each. Both players would prefer either of
S8. (a) When x < 1, No is a dominant strategy for both players, so (No, No) is the unique
(b) There is a mixed-strategy Nash equilibrium when x > 1. In that MSE, Yes will be
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) This is an example of an assurance-type game because there are two Nash
S9. (a)
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) Revolver yields a higher expected payoff than Knife when
(c) Revolver yields a higher expected payoff than Wrench when
(d) Professor Plum will use only Knife and Wrench in his equilibrium mixture,
(e) Eliminating Revolver from consideration, we get the two-by-two table:
Professor Plum
Knife Wrench
Let q be the probability that Professor Plum plays Knife. The mixed-strategy Nash
equilibrium occurs where
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
S10. (a) To find Bart’s equilibrium mix, set Lisa’s payoffs from each of her pure
strategies, against Bart’s p-mix, equal to one another. In his p-mix, Bart will play Rock with
probability p1, Scissors with probability p2, and Paper with probability 1 – p1 – p2. Lisa’s payoff
Rearranging and substituting 1/3 for p1 yields 30p2 = 10, or p2 = 1/3 also. Then we get 1 – p1 – p2 =
(b) Check Bart’s payoffs from each of his pure strategies against Lisa’s mix:
Bart’s expected payoff from using the pure-strategy Paper exceeds his expected payoffs from his
other two strategies. He should play only Paper when Lisa uses the mix described. Presumably,
Lisa has chosen a nonequilibrium mix, so Bart is not indifferent among his available pure
strategies.
S11. (a)
Vendor 2
A B C D E
Vendor 1 A 85, 85 100, 170 125, 195 150, 200 160, 160
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) For both vendors, locations A and E are dominated. Thus, for a fully mixed
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Vendor 2
B C D q-mix
B 110, 110 150, 170 175, 175 175 – 65qB – 25qC,
175 – 65qB – 5qC
(c) To find the equilibrium p, set Vendor 2’s payoffs equal:
One way to explain why A and E are unused in the equilibrium is to point out that they
are (as noted above) dominated. This also implies that A and E are unused because they result in a
payoff against the opponent’s equilibrium mixture that is lower than produced by choices B, C,
and D. Specifically, when Vendor 2 uses the equilibrium mixture probabilities of (6/23, 11/23,
6/23), Vendor 1’s payoff from choosing
A is 100(6/23) + 125(11/23) + 150(6/23) = 2,875/23.
Clearly, A and E are inferior choices.
An alternative possibility is a partially mixed equilibrium in which one player plays pure
C and the other player mixes using strategies B and D with probabilities 1/13 = 0.076 and 12/13 =
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Vendor 1 chooses B with probability p and D with probability (1 – p). To make this an
equilibrium, pure C should be Vendor 2’s best response to this mixture. A and E are clearly bad
for Vendor 2, as we established above. Vendor 2 does not switch to B if
Similarly, Vendor 2 does not switch to D if
Thus, there is really a whole continuum of mixed-strategy equilibria, in which Vendor 2 plays
pure C and Vendor 1 mixes between B and D in any proportions between 1/13 and 12/13. The
answer just above describes the equilibrium that results at the extreme points of this range.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
