Problem Set 1 | Quizerry

a) Only x

b) Only y

c) Only z

d) Both y and z

5.There are 2 players who have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive

s_1

and

s_2

, with

0≤s_1

s_2≤1

. If

s_1+s_2≤1

, then the players receive the shares they named and if

s_1+s_2>1

, then both players fail to achieve an agreement and receive zero. This game is known as `Nash Bargaining’.

Which of the following is a strictly dominant strategy?

1 point

a)1;

b) 0.5;

c) 0;

d) None of the above.

6.There are 2 players who have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive

s_1

and

s_2

, with

0≤s_1

s_2≤1

. If

s_1+s_2≤1

, then the players receive the shares they named and if

s_1+s_2>1

, then both players fail to achieve an agreement and receive zero.

Which of the following strategy profiles is a pure strategy Nash equilibrium?

1 point

a) (0.3, 0.7);

b) (0.5, 0.5);

c) (1.0, 1.0);

d) All of the above

Each firm sets a nonnegative price ( $p_1$ and $p_2$ ).

All consumers buy from the firm with the lower price, if $p_1≠p_2$ . Half of the consumers buy from each firm if $p_1=p_2$ .

D is the total demand.

Profit of firm $i$ is:

0 if $p_i>p_j$ (no one buys from firm $i$ );
$D\frac{p_i−c}{2}$ if $p_i=p_j$ (Half of customers buy from firm $i$ );
$D(p_i−c)$ if $p_i<p_j$ (All customers buy from firm $i$ )

Find the pure strategy Nash equilibrium:

1 point

a) Both firms set

$p = 0$ .

b) Firm 1 sets

$p = 0$ , and firm 2 sets $p = c$ .

c) Both firms set

$p = c$ .

d) No pure strategy Nash equilibrium exists.

Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B.
When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B.
The candidate getting 2 or more votes is the winner (majority rule).

Find all very weakly dominant strategies (click all that apply: there may be more than one, or none).

1 point

a) Voter 1 voting for A.

b) Voter 1 voting for B.

c) Voter 2 (or 3) voting for A.

d) Voter 2 (or 3) voting for B.

Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B.
When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B.
The candidate getting 2 or more votes is the winner (majority rule).

Find all pure strategy Nash equilibria (click all that apply)? Hint: there are three.

1 point

a) 1 voting for A, and 2 and 3 voting for B.

b) All voting for A.

c) All voting for B.

d) 1 and 2 voting for A, and 3 voting for B.

1 \ 2	x	y	z
a	1,2	2,2	5,1
b	4,1	3,5	3,3
c	5,2	4,4	7,0
d	2,3	0,4	3,0

1 \ 2	x	y	z
a	1,2	2,2	5,1
b	4,1	3,5	3,3
c	5,2	4,4	7,0
d	2,3	0,4	3,0

1 \ 2	x	y	z
a	1,2	2,2	5,1
b	4,1	3,5	3,3
c	5,2	4,4	7,0
d	2,3	0,4	3,0

1 \ 2	x	y	z
a	1,2	2,2	5,1
b	4,1	3,5	3,3
c	5,2	4,4	7,0
d	2,3	0,4	3,0

Problem Set 1 >> Game Theory

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