Question

1. Consider the following game. Player 1’s payoffs are listed first:

		Player 2
		X	Y
Player 1	A	90 , 1	10 , 0
	B	10 , 0	50 , 1
	C	100 , 0	80 , 1

1. Imagine that player 1 makes a decision first and Player 2 makes a decision after observing player 1’s choice. What is the subgame-perfect equilibrium of this game?

2. Imagine that player 2 makes a decision first and Player 1 makes a decision after observing player 2’s choice. What is the subgame-perfect equilibrium of this game?

3. Are your answers to (a) and (b) above the same or different? Explain.

Answer 1

a. If player 1 were to move first, he would check Player 2's response to his moves before he makes the first move. If Player 1 were to move A, player 2 would respond with X with payoffs (90, 1). For B, player 2 would choose Y (payoffs 50, 1), and for C, player 2 would move Y (payoffs  80, 1). As we see from the payoffs, Player 1 gains most in the first move (90>80>50). Therefore he would move A and Player 2 would follow with X.

Thus, (A, X) is the Nash equilibrium with payoffs (90, 1).

b. If Player 2 were to move first, he too would check Player 1's response before going first. If he chose X, Player 1 would choose C with payoffs (100, 0). If he chose Y, Player 1 would choose C with payoffs (80, 1). Between the two options, Player 2 gains more in the second option. So he would choose Y and Player 1 would choose C. Therefore (C, Y) is the Nash equilibrium with payoffs (80, 1).

c. Yes. answers to (a) and (b) are different. This is caused by first mover's advantage. Each player checks the other player's reaction before choosing his strategy. For player 1, C is the optimum strategy against any stratgey of Player 2. Similrly, for Player 2, Y is better against Player 1's B and C, and X against player 1's A. Thus, whoever moves first gains by moving according to his advantage.