Describe the normal form of the game players strategies

Question

Assignment

Question 1 - Warm Up:

For each of the following games:

a)

	A	B
X	2 , 1	0 , 0
Y	0 , 0	1 , 2

b)

	A	B
X	2 , 1	0 , 0
Y	0 , 0	1 , 2

c)

	A	B
X	2 , 2	-1 , 1
Y	1 , -1	-1 , -1

i) Describe the normal form (players, strategies, payoffs). (You can do this only for game a, the rest are very similar)
ii) Do players have strictly dominant strategies? Explain.
iii) Does the game have a Dominant Strategy Equilibrium? If so, find it.

Question 2 - Battle of the sexes

A husband and wife want to go out together. They have two options - Opera and Boxing. The husband prefers going to the Opera while the wife prefers Boxing. If they can coordinate on going to the same place, the person that goes to their preferred event gets 2 utils (utility units) and the other person gets 1. If they cannot coordinate they are both disappointed and get 0.

i) Describe the normal form of the game (players, strategies, payoffs).
ii) Do players have strictly dominant strategies? Explain.
iii) Does the game have a Dominant Strategy Equilibrium? If so, find it.

Question 3 - Hotelling's Model

Two political parties, Left wing and Right wing compete in electoral elections. There is a continuum of voters uniformly distributed on the interval [0, 1] where 0 represents the leftmost view (extreme left) and 1 the rightmost. Each party can declare that their view is either 0, 1/4, 1/2, 3/4 or 1 and each voter votes to the party that declare that their views are closest to his views. For example, if a voter's view is represented by 0.3, the left party chose 1/4 and the right party chose 1/2, this voter votes for the left party as 1/4 is closer to 0.3. If both parties choose the same spots they will split the votes between them.

Each party's objective is to have the highest percentage of voters voting for it.

i) Describe the normal form of the game (players, strategies, payoffs).

ii) Solve the game using Iterated Elimination of Strictly Dominated Strategies.

iii) Optional (Bonus, harder): Repeat ii) for the general case in which the parties can choose any of the spots 0, 1/m, 2/m, ... , (m-1)/m, 1. What is the set of surviving strategies? Does it depend on m and how?

Question 4- Weakly Dominated Strategies

Iterated Deletion of Weakly Dominated Strategies (IDWDS) is a similar concept to Iterated Deletion of Strictly Dominated Strategies that was taught in class. The difference between the two is that in IDWDS a strategy s of player i can be deleted there exists a strategy s' that weakly dominates s.

Weakly Dominated Strategy: A strategy s is a weakly dominated strategy for player i if exists a strategy S' ≠ S of player i such that:

u_i(S_i', S_-i) ≥ u_i(S_i, S_-i) ∀ S_-i

Show that in the two player game below, in which one player chooses a row and the other a column different order of elimination of weakly dominated strategies can lead to different outcomes.

	L	R
U	3,1	4,1
M	2,0	0,1
L	0,1	2,1

L R
U 3,1 4,1
M 2,0 0,1
L 0,1 2,1