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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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(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
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.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
