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