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Solutions to Chapter 6 Exercises
S10. (a) The game tree is shown below:
Frieda’s has two actions at one node, so it has two strategies.
(b) The strategic form is given below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Frieda’s
Urban Rural
Big Giant Big Giant
Titan UU UR RU RR UU UR RU RR
UUUU 5, 5, 1 5, 5, 1 5, 2, 5 5, 2, 5 5, 5, 2 3, 4, 4 5, 5, 2 3, 4, 4
UUUR 5, 5, 1 5, 5, 1 5, 2, 5 5, 2, 5 5, 5, 2 4, 4, 4 5, 5, 2 4, 4, 4
UURU 5, 5, 1 5, 5, 1 5, 2, 5 5, 2, 5 4, 3, 4 3, 4, 4 4, 3, 4 3, 4, 4
The eight pure-strategy Nash equilibria are indicated by shaded cells.
(c) Strategy UUUR for Titan weakly dominates strategy UUUU, and it also weakly
Note that the possible equilibria produce two possible outcomes. The first four equilibria produce
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
S11. (a) In the simultaneous version of the game each store has only two strategies: Urban and
Rural. The payoff table, with best responses underlined, follows:
Frieda’s = U Frieda’s = R
Big Giant Big Giant
U R U R
The U, U, R equilibrium (with payoffs of 5, 5, 2) are likely focal for Titan and Big Giant. Frieda’s would
The R, R, R equilibrium (with payoffs of 4, 4, 4) may be more attractive to a big store if it is risk
(b) When all three stores request Urban there is a one-third chance that Titan and Big Giant
The payoff table is the same as in part (a), with the exception of the Urban, Urban, Urban cell:
Frieda’s = U Frieda’s = R
Big Giant Big Giant
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U R U R
The Nash equilibria are U, U, U (with expected payoffs of 4, 4, 11/3) and R, R, R (with payoffs of 4,
(c) The change in the payoff table causes an important change in the equilibria of the games
found in parts (a) and (b). The randomized allocation of the two Urban slots when all three stores choose
S12. (a) As seen in Exercise S10 of Chapter 5, Nancy’s best response to Monica’s choice m is
Monica knows Nancy’s best-response rule, so when she chooses her m to maximize her profit she can
plug in (1 + m/4) for n:
(b) When m = 12/7 and n = 10/7 the profits are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
In Exercise S10 of Chapter 5, when Monica and Nancy choose their effort levels simultaneously their
profits are
Both Monica and Nancy make higher profits when Monica commits to an effort level first, but Nancy
experiences a greater increase in her profits. This game thus has a second-mover advantage.
S13. (a) The game table for the first-stage game (with best responses underlined) follows:
Nancy
Yes No
(b) There are two pure-strategy Nash equilibria: (Yes, No) and (No, Yes).
UNSOLVED EXERCISES
U1. (a) The game tree is shown below.
The subgame-perfect equilibrium is (Down, If Down then Right).
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) The strategic form of this game is shown below:
Player B
If Down,
then Left
If Down, then
Right
Here we see that there are two Nash equilibria: (Up, Left) yields a payoff of (2, 2) and (Down, Right)
yields a payoff of (3, 1). The (Down, Right) equilibrium is subgame perfect—see answer to part (a)—it
(c) To find the subgame-perfect equilibrium from the strategic form of the game, start with
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U2. (a) The strategic form of the game follows:
Minerva
aaa aab aba abb baa bab bba bbb
There is only Nash equilibrium: (N, bab) with payoffs of (2, 1).
(b) There is no credibility problem for the only Nash equilibrium (N, bab); it is subgame
perfect.
U3. (a) The strategic form of the game follows:
Minerva
aa ab ba bb ca cb
N N 3, 3 3, 3 5, 2 5, 2 0, 4 0,4
The Nash equilibria are (SN, ca) and (SS, ca), each with payoffs of (2, 2).
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) The subgame-perfect equilibrium (SPE) is (SN, ca). The equilibrium (SS, ca) is not
credible because Albus gets a lower payoff from playing S instead of N at his second node.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
