978-0393123982 Chapter 29 Solution Manual Part 1

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Chapter 29 NAME

Game Theory

Introduction. In this introduction we offer two examples of two-person

games. The first game has a dominant strategy equilibrium. The second

game is a zero-sum game that has a Nash equilibrium in pure strategies

that is not a dominant strategy equilibrium.

Example: Albert and Victoria are roommates. Each of them prefers a

clean room to a dirty room, but neither likes housecleaning. If both clean

the room, they each get a payoff of 5. If one cleans and the other doesn’t

clean, the person who does the cleaning has a utility of 2, and the person

who doesn’t clean has a utility of 6. If neither cleans, the room stays a

mess and each has a utility of 3. The payoffs from the strategies “Clean”

and “Don’t Clean” are shown in the box below.

Clean Room–Dirty Room

Albert

Victoria

Clean Don’t Clean

Clean 5,5 2,6

Don’t Clean 6,2 3,3

In this game, whether or not Victoria chooses to clean, Albert will get

a higher payoff if he doesn’t clean than if he does clean. Therefore “Don’t

Clean” is a dominant strategy for Albert. Similar reasoning shows that

no matter what Albert chooses to do, Victoria is better off if she chooses

“Don’t Clean.” Therefore the outcome where both roommates choose

“Don’t Clean” is a dominant strategy equilibrium. This is true despite

the fact that both persons would be better off if they both chose to clean

the room.

Example: This game is set in the South Pacific in 1943. Admiral Imamura

must transport Japanese troops from the port of Rabaul in New Britain,

across the Bismarck Sea to New Guinea. The Japanese ﬂeet could either

travel north of New Britain, where it is likely to be foggy, or south of

New Britain, where the weather is likely to be clear. U.S. Admiral Ken-

352 GAME THEORY (Ch. 29)

The Battle of the Bismarck Sea

Kenney

Imamura

North South

North 2,−2 2,−2

South 1,−1 3,−3

This game does not have a dominant strategy equilibrium, since there

is no dominant strategy for Kenney. His best choice depends on what Ima-

mura does. The only Nash equilibrium for this game is where Imamura

chooses the northern route and Kenney concentrates his search on the

northern route. To check this, notice that if Imamura goes North, then

Kenney gets an expected two days of bombing if he (Kenney) chooses

North and only one day if he (Kenney) chooses South. Furthermore, if

Kenney concentrates on the north, Imamura is indifferent between go-

ing north or south, since he can be expected to be bombed for two days

either way. Therefore if both choose “North,” then neither has an incen-

tive to act differently. You can verify that for any other combination of

choices, one admiral or the other would want to change. As things actually

worked out, Imamura chose the Northern route and Kenney concentrated

his search on the North. After about a day’s search the Americans found

the Japanese ﬂeet and inﬂicted heavy damage on it.∗

29.1 (0) This problem is designed to give you practice in reading a game

matrix and to check that you understand the definition of a dominant

strategy. Consider the following game matrix.

A Game Matrix

Player A

Player B

Left Right

Top a, b c, d

Bottom e, f g, h

∗This example is discussed in R. Duncan Luce and Howard Raiffa’s

Games and Decisions, John Wiley, 1957, or Dover, 1989. We recommend

this book to anyone interested in reading more about game theory.

NAME 353

(a) If (top, left) is a dominant strategy equilibrium, then we know that

(b) If (top, left) is a Nash equilibrium, then which of the above inequalities

29.2 (0) In order to learn how people actually play in game situations,

economists and other social scientists frequently conduct experiments in

which subjects play games for money. One such game is known as the

voluntary public goods game. This game is chosen to represent situations

in which individuals can take actions that are costly to themselves but

that are beneficial to an entire community.

In this problem we will deal with a two-player version of the voluntary

public goods game. Two players are put in separate rooms. Each player

is given $10. The player can use this money in either of two ways. He can

keep it or he can contribute it to a “public fund.” Money that goes into

the public fund gets multiplied by 1.6 and then divided equally between

the two players. If both contribute their $10, then each gets back $20 ×

1.6/2 = $16.If one contributes and the other does not, each gets back

$10 ×1.6/2 = $8 from the public fund so that the contributor has $8

at the end of the game and the non-contributor has $18–his original $10

plus $8 back from the public fund. If neither contributes, both have their

original $10. The payoff matrix for this game is:

Voluntary Public Goods Game

Player A

Player B

Contribute Keep

Contribute $16,$16 $8,$18

Keep $18,$8 $10,$10

(a) If the other player keeps, what is your payoff if you keep? $10.

354 GAME THEORY (Ch. 29)

(b) If the other player contributes, what is your payoff if you keep?

29.3 (1) Let us consider a more general version of the voluntary public

goods game described in the previous question. This game has Nplayers,

each of whom can contribute either $10 or nothing to the public fund.

All money that is contributed to the public fund gets multiplied by some

number B>1 and then divided equally among all players in the game

(including those who do not contribute.) Thus if all Nplayers contribute

$10 to the fund, the amount of money available to be divided among the

Nplayers will be $10BN and each player will get $10BN/N = $10Bback

from the public fund.

(a) If B>1, which of the following outcomes gives the higher payoff to

each player? a) All players contribute their $10 or b) all players keep their

(b) Suppose that exactly Kof the other players contribute. If you

keep your $10, you will have this $10 plus your share of the public

fund contributed by others. What will your payoff be in this case?

game? All keep their $10. Explain your answer. If K

other players contribute, a player’s payoff

will be $10 + 30K/5if he keeps the $10 and

NAME 355

(d) In general, what relationship between Band Nmust hold for “Keep”

to be a dominant strategy? We ask when the payoff

from keeping is larger than the payoff from

(e) Sometimes the action that maximizes a player’s absolute payoff, does

not maximize his relative payoff. Consider the example of a voluntary

public goods game as described above, where B=6andN= 5. Suppose

that four of the five players in the group contribute their $10, while the

fifth player keeps his $10. What is the payoff of each of the four contribu-

tors? $60×4/5 = $48.What is the payoff of the player who keeps

would be the payoff to the fifth player if instead of keeping his $10, he

contributes, so that all five players contribute. $60 ×5/5 = $60.

If the other four players contribute, what should the fifth player to max-

(f) If B=6andN= 5, what is the dominant strategy equilib-

rium for this game? All contribute. Explain your answer.

Where B=6and N=5,ifKother

players contribute, a player’s payoff will

be $10 + 60K/5if he keeps the $10 and

356 GAME THEORY (Ch. 29)

29.4 (1) The Stag Hunt game is based on a story told by Jean Jacques

Rousseau in his book Discourses on the Origin and Foundation of In-

equality Among Men (1754). The story goes something like this: “Two

hunters set out to kill a stag. One has agreed to drive the stag through

the forest, and the other to post at a place where the stag must pass. If

both faithfully perform their assigned stag-hunting tasks, they will surely

kill the stag and each will get an equal share of this large animal. During

the course of the hunt, each hunter has an opportunity to abandon the

stag hunt and to pursue a hare. If a hunter pursues the hare instead of the

stag he is certain to catch the hare and the stag is certain to escape. Each

hunter would rather share half of a stag than have a hare to himself.”

The matrix below shows payoffs in a stag hunt game. If both hunters

hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one

hunts stag and the other hunts hare, the stag hunter gets 0 and the hare

hunter gets 3.

The Stag Hunt Game

Hunter A

Hunter B

Hunt Stag Hunt Hare

Hunt Stag 4,4 0,3

Hunt Hare 3,0 3,3

(a) If you are sure that the other hunter will hunt stag, what is the best

(b) If you are sure that the other hunter will hunt hare, what is the best

NAME 357

(d) This game has two pure strategy Nash equilibria. What are they?

(e) Is one Nash equilibrium better for both hunters than the other?

(f) If a hunter believes that with probability 1/2 the other hunter will

hunt stag and with probability 1/2 he will hunt hare, what should this

29.5 (1) Evangeline and Gabriel met at a freshman mixer. They want

desperately to meet each other again, but they forgot to exchange names

or phone numbers when they met the first time. There are two possible

strategies available for each of them. These are Go to the Big Party or

Stay Home and Study. They will surely meet if they both go to the party,

and they will surely not otherwise. The payoff to meeting is 1,000 for

each of them. The payoff to not meeting is zero for both of them. The

payoffs are described by the matrix below.

Close Encounters of the Second Kind

Evangeline

Gabriel

Go to Party Stay Home

Go to Party 1000,1000 0,0

Stay Home 0,0 0,0

(a) A strategy is said to be a weakly dominant strategy for a player if the

payoff from using this strategy is at least as high as the payoff from using

any other strategy. Is there any outcome in this game where both players