978-0393919684 Chapter 4 Solution Manual Part 1

Solutions to Chapter 4 Exercises

SOLVED EXERCISES

S1. (a) For Rowena, Up strictly dominates Down, so Down may be eliminated. For Colin, Right

(c) There are no dominated strategies for Rowena. For Colin, Left dominates Middle and

(d) Beginning with Rowena, Straight dominates Down, so Down is eliminated. Then for

S2. (a) Zero-sum or constant-sum game (payoffs in all cells sum to 4)

S3. (a) (i) The minima for Rowena’s strategies are 3 for Up and 1 for Down. The minima for

(c) (i) The minima for Rowena’s three strategies are 1 for Up, 2 for Straight, and 1 for Down.

The minima for Colin’s strategies are 4 for Left, 2 for Straight, and 1 for Right. (ii) Rowena wants the

(d) (i) The minima for Rowena’s strategies are 1 for Up, 1 for Straight, and 0 for Down. The

S4. (a) Rowena has no dominant strategy, but Right dominates Left for Colin. After eliminating

(b) Down and Right are weakly dominant for Rowena and Colin, respectively, leading to a

(d) There are no dominant or dominated strategies. Use best-response analysis to find the

S5. (a) Neither Rowena nor Colin has a dominant strategy, because neither has one action that is

(b) For Colin, East dominates South, so South may be eliminated. Then, for Rowena, Fire

Colin

East West

Rowena

Water 2, 3 1, 1

Fire 1, 1 2, 2

(c) The game is not dominance solvable, because a unique solution cannot be attained

(d) There are two pure-strategy Nash equilibria, which are (Water, East) and (Fire, West).

S6. False. A dominant strategy yields the highest payoff available to you against each of your

S7. The payoff matrix is given below. Best-response analysis shows there are two pure-strategy Nash

equilibria: (Help, Not Help) with payoffs (2, 3) to (I, You) and (Not Help, Help) with payoffs (3, 2).

You

Help Not

S8. (a) Best-response analysis shows that there are two pure-strategy Nash equilibria: (Lab, Lab)

(b) The textbook gives numerous multiple-equilibria games, so we shall examine each. The

S9. (a) The Nash equilibria are (1, 1), (2, 2), and (3, 3). You could argue that (1, 1) is a focal

(b) Expected (average) payoff from flipping a (single) coin to decide whether to play 2 or 3

S10. (a) The payoff tables follow:

Carlos Yes

Bernardo

Yes No

Carlos No Bernardo

Yes No

(c) A natural focal point is where Arturo and Bernardo write No and Carlos writes Yes,

S11. There are three ticket buyers, and each ticket buyer can do three things: not purchase a ticket

(represented as $0), purchase a $15 ticket, and purchase a $30 ticket. To represent this game, we need

Moe $0 Curly

$0 $15 $30

Moe $15 Curly

$0 $15 $30

–15

Moe $30 Curly

$0 $15 $30

Larry

–15

–20, –20,

Best-response analysis shows that there are no pure-strategy Nash equilibria for when any player

S12. (a) The strategic-form game may be described as a zero-sum game, but for clarity, we have

included both payments:

Bruce

1 2

(b) Best-response analysis shows that no combination of actions is a pure-strategy Nash

equilibrium.

S13. (a) With only two men, two brunettes, and one blonde, the payoffs are as follows:

Young Man 2

Approach

blonde

Approach

brunette

Approach

brunette

There are two Nash equilibria in which one young man approaches the blonde and one the

(b) For three young men with three brunettes and one blonde, the payoff table is given

below:

Young Man 3

Approach blonde Approach brunette

Young Man 2 Young Man 2

Approach

blonde

Approach

brunette

Approach

blonde

Approach

brunette

Approach

This time there are three Nash equilibria. Each has the same characteristics: one young man

(c) With four young men, four brunettes, and one blonde, there will be four Nash equilibria.

(d) For n young men, with n brunettes and 1 blonde, there will be n Nash equilibria. Let k be

the number of other men approaching the blonde. If you are one of the young men and k = 0, you get a