Question

Q1. Consider the following game. Two players simultaneously and independently choose one of three venues. They would like to choose the same venue (i.e. meet), but their favorite venues are different:

Football cafe ballet

Football (3,2) (1,0) (1,1)

cafe (0,0) (2,2) (0,1)

ballet (0,0) (0,0) (2,3)

a. What are the pure-strategy Nash equilibria of this game?
b. Derive a mixed-strategy Nash equilibrium in which players 1 and 2 mix over Football

and Cafe only?

Now suppose that player 1 is macho and would not consider going to the ballet, while player 2 is averse to contact sports and would not consider going to the football? The payoffs are as above, but player 1 faces the restricted strategy set (Football,Cafe), while player 2 faces the restricted strategy set (Cafe,Ballet).

c. Write down the reduced payoff matrix and solve for the pure strategy Nash equilibria of the reduced game. Describe the new equilibrium that emerges. Briefly interpret this equilibrium and provide some intuition for it.

(Remember to show all working)

Answer 1

Answer for a)

When Player 1 plays Football then best response of player 2 is Football

When Player 1 plays Cafe then best response of player 2 is Cafe

When Player 1 plays Ballet then best response of player 2 is Ballet

Simlarly

When Player 2 plays Football then best response of player 1 is Football

When Player 2 plays Cafe then best response of player 1 is Cafe

When Player 2 plays Ballet then best response of player 1 is Ballet

Therfore Pure NE will be (Football,Footbll); (Ballet,Ballet),(Cafe,Cafe)

Answer for b)

Let Mixed Strategy Payoff be (p,q) for player 2 and (r,s) for player 1

Player 1 is indifferent of Football and Cafe

E(F)=3p+1*q=E(C)=0p+2*q

3p+q=2q

3p=q=1-p

4p=1, p=1/4 and q=3/4 Hence Mixed Strategy for player 2(1/4,3/4)

Similarly for player 2

E(F)=2r+0(1-r)=E(C)=0*r+2(1-r)

2r=2-2r

4r=2

r=0.5 and s=1-r=0.5

Mixed Payoff (0.5,0.5)

Answer for C)

Reduced PAyoff MAtrix is

Cafe Ballet

Football (1,0) (1,1)

Cafe (2,2) (0,1)

Best response for Player 1 Football is Ballet from Player 2 and Best Response for Player 2 Cafe is Cafe from Payer 1

Hence NE (Cafe,Cafe)

For instance BR1(Cafe)=Cafe and BR2(Cafe)=Cafe

Hence Nash Equilibria is (Cafe,Cafe)