978-0393919684 Chapter 6 Solution Manual Part 2

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:

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

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

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

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).

(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

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).

(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.