Question

2 36 8 LM 2 2 2 4 2 5

Consider the extensive form game above. The game has _______ proper subgames. The strategy profile (AGJKM, CF) leads to a payoff of _______ for Player 1 and _________ for Player 2. In the backward induction equilibrium in pure strategies. Player 2 gets a payoff of ____________ (Please, enter only numerical values like: 0, 1, 2, 3,....).

Answer 1

1) 6 proper subgames

2) 6 for Player 1

3) 7 for Player 2

4) Player 2 gets a payoff of 3

Explanation

1)There are a total of 7 subgames (each one starting at a node). One of them is the game itself. So, the number of proper subgames is 7-1=6

2&3) Suppose Player 1 plays strategy A, Player 2 plays strategy C, Player 1 plays G, Player 2 plays F, Player 1 plays M. Then the payoff is (6,7). That is, Player 1 gets a payoff of 6 and Player 2 gets a payoff of 7.

4) Among K and L, Player 1 will choose L as it has a higher payoff for him. This will give Player 2 a payoff of 6. So, Player 2 knows that of he choose E, he will get a payoff of 6 and Player 1 will get a payoff of 3.

With a similar argument, if Player 2 choose F, he will get a payoff of only 1 (and Player 1 gets 8). So, player 2 will choose E over F to maximise his payoff.

So, if Player 1 choose G, (Player 2 will choose E and Player 1 will choose L), he will get a payoff of 3. Whereas if Player 1 choose H, he will get a payoff of 7. So, Player 2 will choose H giving Player 1 a payoff of 3. So, if Player 2 choose C, instead of D, he will get a payoff of 3. Whereas, by similar argument, if he choose D, he will get a payoff of only 2. Hence Player 2 will choose C over D. So Player 1 will be getting a payoff of 7 of he choose A while he will get only 5 is he choose B.

Hence the choices will be A, C, H. And player 2 will get a payoff of 3.