978-0393123982 Chapter 30 Solution Manual Part 1

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Chapter 30 NAME

Game Applications

Introduction. As we have seen, some games do not have a “Nash equi-

librium in pure strategies.” But if we allow for the possibility of “Nash

equilibrium in mixed strategies,” virtually every game of the sort we are

interested in will have a Nash equilibrium.

The key to solving for such equilibria is to observe that if a player is

indifferent between two strategies, then he is also willing to choose ran-

domly between them. This observation will generally give us an equation

that determines the equilibrium.

Example: In the game of baseball, a pitcher throws a ball towards a batter

who tries to hit it. In our simplified version of the game, the pitcher can

pitch high or pitch low, and the batter can swing high or swing low. The

ball moves so fast that the batter has to commit to swinging high or

swinging low before the ball is released.

Let us suppose that if the pitcher throws high and batter swings low,

or the pitcher throws low and the batter swings high, the batter misses

the ball, so the pitcher wins.

If the pitcher throws high and the batter swings high, the batter

always connects. If the pitcher throws low and the batter swings low, the

batter will connect only half the time.

This story leads us to the following payoff matrix, where if the batter

hits the ball he gets a payoff of 1 and the pitcher gets 0, and if the batter

misses, the pitcher gets a payoff of 1 and the batter gets 0.

Simplified Baseball

Pitcher

Batter

Swing Low Swing High

Pitch High 1,0 0,1

Pitch Low .5,.5 1,0

This game has no Nash equilibrium in pure strategies. There is no

combination of actions taken with certainty such that each is making the

best response to the other’s action. The batter always wants to swing

the same place the pitcher throws, and the pitcher always wants to throw

to the opposite place. What we can find is a pair of equilibrium mixed

strategies.

366 GAME APPLICATIONS (Ch. 30)

In a mixed strategy equilibrium each player’s strategy is chosen at

random. The batter will be willing to choose a random strategy only if

the expected payoff to swinging high is the same as the expected payoff

to swinging low.

The payoffs from swinging high or swinging low depend on what the

pitcher does. Let πPbe the probability that the pitcher throws high and

1−πPbe the probability that he throws low. The batter realizes that if

he swings high, he will get a payoff of 0 if the pitcher throws low and 1

if the pitcher throws high. The expected payoff to the batter is therefore

πP.

If the pitcher throws low, then the only way the batter can score is

if pitcher pitches low, which happens with probability 1 −πP.Eventhen

the batter only connects half the time. So the expected payoff to the

batter from swinging low is .5(1 −πP).

These two expected payoffs are equalized when πP=.5(1 −πP). If

we solve this equation, we find πP=1/3. This has to be the probability

that the pitcher throws high in a mixed strategy equilibrium.

Now let us find the probability that the batter swings low in a mixed

strategy equilibrium. In equilibrium, the batter’s probability πBfrom

swinging low must be such that the pitcher gets the same expected payoff

from throwing high as from throwing low The expected payoff to the

pitcher is the probability that the batter does not score.

If the pitcher throws high, then the batter will not connect if he

swings low, but will connect if he swings high, so the expected payoff to

the pitcher from pitching high is πB.

If the pitcher throws low, then with probability (1 −πB), the batter

will swing swing high, in which case the pitcher gets a payoff of 1. But

when the pitcher throws low, the batter will swing low with probability πB

and connect half the time, giving a payoff to the pitcher of .5πB. Therefore

the expected payoff to the pitcher from throwing low is (1 −πB)+.5πB=

1−.5πB. Equalizing the payoff to the pitcher from throwing high and

throwing low requires πB=1−.5πB. Solving this equation we find that

in the equilibrium mixed strategy, πB=2/3.

Summing up, the pitcher should throw low two-thirds of the time,

and the batter should swing low two-thirds of the time.

Calculus 30.1 (2) Two software companies sell competing products. These prod-

ucts are substitutes, so that the number of units that either company sells

is a decreasing function of its own price and an increasing function of the

other product’s price. Let p1be the price and x1the quantity sold of

product 1 and let p2and x2be the price and quantity sold of product

2. Then x1= 1000 90 −1

2p1+1

4p2and x2= 1000 90 −1

2p2+1

4p1.

Each company has incurred a fixed cost for designing their software and

writing the programs, but the cost of selling to an extra user is zero.

Therefore each company will maximize its profits by choosing the price

that maximizes its total revenue.

NAME 367

(a) Write an expression for the total revenue of company 1, as a function

(b) Company 1’s best response function BR1(·) is defined so that BR1(p2)

is the price for product 1 that maximizes company 1’s revenue given

that the price of product 2 is p2. With the revenue functions we have

specified, the best response function of company 1 is described by the

with respect to p1and solve for the revenue-maximizing p1given p2.)

(e) Suppose that company 1 sets its price first. Company 2 knows the

price p1that company 1 has chosen and it knows that company 1 will

not change this price. If company 2 sets its price so as to maximize its

revenue given that company 1’s price is p1, then what price will company

2choose? p2=$90 + p1/4If company 1 is aware of how company

2 will react to its own choice of price, what price will company 1 choose?

30.2 (1) Here is an example of the “Battle of the Sexes” game discussed

in the text. Two people, let us call them Michelle and Roger, although

they greatly enjoy each other’s company, have very different tastes in

entertainment. Roger’s tastes run to ladies’ mud wrestling, while Michelle

prefers Italian opera. They are planning their entertainment activities for

next Saturday night. For each of them, there are two possible actions,

go to the wrestling match or go to the opera. Roger would be happiest

if both of them went to see mud wrestling. His second choice would be

for both of them to go to the opera. Michelle would prefer if both went

to the opera. Her second choice would be that they both went to see the

mud wrestling. They both think that the worst outcome would be that

they didn’t agree on where to go. If this happened, they would both stay

home and sulk.

368 GAME APPLICATIONS (Ch. 30)

Battle of the Sexes

Michelle

Wrestling Opera

(a) Is the sum of the payoffs to Michelle and Roger constant over all

outcomes? No. Does this game have a dominant strategy equilibrium?

No.

(b) Find two Nash equilibria in pure strategies for this game. Both

chooses opera with probability 2/3and

30.3 (1) This is an example of the game of “Chicken.” Two teenagers

in souped-up cars drive toward each other at great speed. The first one

to swerve out of the road is “chicken.” The best thing that can happen

to you is that the other guy swerves and you don’t. Then you are the

hero and the other guy is the chicken. If you both swerve, you are both

chickens. If neither swerves, you both end up in the hospital. A payoff

matrix for a chicken-type game is the following.

Chicken

Leroy

Swerve Don’t Swerve

NAME 369

(a) Does this game have a dominant strategy? No. What are the two

(b) Find a Nash equilibrium in mixed strategies for this game. Play

30.4 (0) I propose the following game: I ﬂip a coin, and while it is in the

air, you call either heads or tails. If you call the coin correctly, you get

to keep the coin. Suppose that you know that the coin always comes up

(a) Suppose that the coin is unbalanced and comes up heads 80% of

the time and tails 20% of the time. Now what is your best strategy?

(b) What if the coin comes up heads 50% of the time and tails 50% of the

time? What is your best strategy? It doesn’t matter.

(where a coin’s type is the probability that it comes up heads), and that

you will know my choice. What type of coin should I choose to minimize

(d) What is the Nash mixed strategy equilibrium for this game? (It may

help to recognize that a lot of symmetry exists in the game.) I

30.5 (0) Ned and Ruth love to play “Hide and Seek.” It is a simple

game, but it continues to amuse. It goes like this. Ruth hides upstairs or

downstairs. Ned can look upstairs or downstairs but not in both places.

If he finds Ruth, Ned gets one scoop of ice cream and Ruth gets none. If

370 GAME APPLICATIONS (Ch. 30)

he does not find Ruth, Ruth gets one scoop of ice cream and Ned gets

none. Fill in the payoffs in the matrix below.

Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 1,0 0,1

Downstairs 0,1 1,0

(a) Is this a zero-sum game? Yes. What are the Nash equilibria in

(b) Find a Nash equilibrium in mixed strategies for this game.

Ruth hides upstairs and Ned searches

liven it up a little. Now if Ned finds Ruth upstairs, he gets two scoops of

ice cream, but if he finds her downstairs, he gets one scoop. If Ned finds

Ruth, she gets no ice cream, but if he doesn’t find her she gets one scoop.

Fill in the payoffs in the graph below.

Advanced Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 2,0 0,1

Downstairs 0,1 1,0

NAME 371

(d) Are there any Nash equilibria in pure strategies? No. What mixed

time will Ned find Ruth? 1/2.

30.6 (0) Perhaps you have wondered what it could mean that “the meek

shall inherit the earth.” Here is an example, based on the discussion in

the book. In a famous experiment, two psychologists∗put two pigs—a

little one and a big one—into a pen that had a lever at one end and a

trough at the other end. When the lever was pressed, a serving of pigfeed

would appear in a trough at the other end of the pen. If the little pig

would press the lever, then the big pig would eat all of the pigfeed and

keep the little pig from getting any. If the big pig pressed the lever, there

would be time for the little pig to get some of the pigfeed before the big

pig was able to run to the trough and push him away.

Let us represent this situation by a game, in which each pig has two

possible strategies. One strategy is Press the Lever. The other strategy

is Wait at the Trough. If both pigs wait at the trough, neither gets any

feed. If both pigs press the lever, the big pig gets all of the feed and the

little pig gets a poke in the ribs. If the little pig presses the lever and

the big pig waits at the trough, the big pig gets all of the feed and the

little pig has to watch in frustration. If the big pig presses the lever and

the little pig waits at the trough, then the little pig is able to eat 2/3

of the feed before the big pig is able to push him away. The payoffs are

as follows. (These numbers are just made up, but their relative sizes are

consistent with the payoffs in the Baldwin-Meese experiment.)

Big Pig–Little Pig

Little Pig

Big Pig

Press Wait

Press −1,9−1,10

Wait 6,4 0,0

∗Baldwin and Meese (1979), “Social Behavior in Pigs Studied by

Means of Operant Conditioning,” Animal Behavior

372 GAME APPLICATIONS (Ch. 30)

(a) Is there a dominant strategy for the little pig? Yes, Wait. Is

there a dominant strategy for the big pig? No.

(b) Find a Nash equilibrium for this game. Does the game have more than

(Incidentally, while Baldwin and Meese did not interpret this experiment

as a game, the result they observed was the result that would be predicted

by Nash equilibrium.)

30.7 (1) Let’s have another look at the soccer example that was discussed

in the text. But this time, we will generalize the payoff matrix just a little

bit. Suppose the payoff matrix is as follows.

The Free Kick

Goalie

Kicker

Kick Left Kick Right

Jump Left 1,0 0,1

Jump Right 1-p,p 1,0

Now the probability that the kicker will score if he kicks to the left

and the goalie jumps to the right is p. Wewillwanttoseehowthe

equilibrium probabilities change as pchanges.

(a) If the goalie jumps left with probability πG, then if the kicker kicks

right, his probability of scoring is πG.

(b) If the goalie jumps left with probability πG, then if the kicker kicks

left, his probability of scoring is p(1 −πG).

to the same probability of scoring for the kicker. (Your answer will be a

function of p.) πG=p

1+p.