978-0393919684 Chapter 7 Solution Manual Part 1

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,

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

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

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

(c) This is an example of an assurance-type game because there are two Nash

S9. (a)

(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

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

(b) For both vendors, locations A and E are dominated. Thus, for a fully mixed

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 =

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.