978-0393919684 Chapter 6 Solution Manual Part 1

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Solutions to Chapter 6 Exercises

SOLVED EXERCISES

S1. Second-mover advantage. In a sequential game of tennis, the second mover can best respond to

the first mover’s chosen action. Put another way, the second mover can exploit the information she learns

S2. The strategic form, with best responses underlined, is shown below:

Player 2

LL LR RL RR

There are two Nash equilibria: (D, LL) with payoffs of (3, 3) and (U, LR) with payoffs of (2, 4). Only the

S3. The strategic form is shown below:

Boeing

If In, then Peace If In, then War

There are two Nash equilibria: (In; If In, then Peace) and (Out; If In, then War). Only the first of these,

S4. (a) The strategic form follows:

Tinman

If N, then t If N, then b

(b) The only Nash equilibrium is (S; If N, then t) with payoffs of (1, 0).

S5. (a) The strategic form follows. The initials of the strategies indicate which action each player

would take at his first, second, and third nodes, respectively.

Tinman

nnn nns nsn nss snn sns ssn sss

NNN 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1

NNS 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1

Pure-strategy Nash equilibria are indicated by double borders. The unique subgame-perfect Nash

equilibrium is (SSN, nns), with payoffs of (4, 5).

(b) The remaining Nash equilibria are not subgame perfect because a player cannot credibly

threaten to make a move that will give himself a lower payoff than he would otherwise receive. The

S6. (a) The strategic form follows. The initials of the Scarecrow’s strategies indicate which

action he would take at his first, second, and third nodes, respectively.

Lion = u Lion = d

Tinman Tinman

t b t b

NNN 1, 1 , 1 2 , 3 , 2

NNS 1, 1 , 1 2 , 3 , 2 NNS 1, 1 , 1 2 , 3 , 2

(b) Nash equilibria (SNS, t, u), (SNS, b, u), (SSS, t, u), and (SSS, b, u) are not subgame

S7. (a) The game tree is shown below:

( b)

The rollback equilibrium for the game above is (Fast, Guess Fast/Guess Curve).

PITCHER

BATTER,

BATTER

PITCHER

0.30 , 0.70

0.15 , 0.85

0.20 , 0.80

0.35 , 0.65

It can also be represented as the tree in part (c) with an information set between the pitcher’s two nodes.

(e) The game table, with best responses underlined, follows:

Batter

Guess fast Guess curve

There is no cell where both players are mutually best responding. There is no Nash equilibrium in pure

strategies.

S8. (a) See table below. Best responses are underlined. There are 36 Nash equilibria shown by

shading in the cells of the table:

Emily

Contribute Don’t

Nina Nina

Talia CC CD DC DD CC CD DC DD

1: CCCC 3, 3, 3 3, 3, 3 3, 4, 3 3, 4, 3 3, 3, 4 1, 2, 2 3, 3, 4 1, 2, 2

2: CCCD 3, 3, 3 3, 3, 3 3, 4, 3 3, 4, 3 3, 3, 4 2, 2, 2 3, 3, 4 2, 2, 2

3: CCDC 3, 3, 3 3, 3, 3 3, 4, 3 3, 4, 3 2, 1, 2 1, 2, 2 2, 1, 2 1, 2, 2

(b) Working with the normal form of the game, use iterated dominance of weakly dominated

strategies to find the subgame-perfect equilibrium. For Talia, strategy 1 is weakly dominated by strategy

The set of strategies that leads to the subgame-perfect equilibrium is the only set in which all

three women use strategies that entail choosing “rationally” (that is, choosing the action that leads to the

S9. The larger tree follows: