Question

There are two players. First, Player 1 chooses Yes or No. If Player 1 chooses No, the game ends and each player gets a payoff of 1.5. If Player 1 chooses Yes, then the following simultaneous-move battle of the sexes game is played:

			Player 2
		O		F
Player 1	O	(2,1)		(0,0)
	F	(0,0)		(1,2)

Using backward induction to find the Mixed-Strategy Subgame-Perfect Equilibrium.

Answer 1

Let Player 1 will play (O,F) with strategy p and 1-p respectively

and Player 2 will play strategy O & F with strategy q and 1-q

Now we will try to find vlues of p and q such that both are indfferent between O & P

E(O) for Player 1=2*q+0(1-q) and E(P) for Player 1=0*q+2*(1-q)

2q+0(1-q)=0q+2(1-q)

2q=2-2q

q=1/2

Similalry we will find p for player 2 and mixed strategy for player 1 is (1/2,1/2) and mixed strategy for player 2 is (1/2,1/2)

Expected payoff for player 1 and 2 is "1" if they play this simulataneous game

Hence they will not choose this simultaneous game to play therfore using backward induction they will choose Yes