Question

1. Consider the following extensive form game with perfect information 1 Out 2 2 In 3 3 a) (Level A) Write down the normal form associated with this extensive form game (b) (Level A) First suppose -0. Find a subgame perfect equilibrium for this game (c) (Level B) Again suppose α-0. Find a pure strategy ash equilibrium of this extensive form game that is not subgame perfect (d) (Level B) Now suppose a-3. Find all pure strategy subgame perfect equi- libria. (Hint: there are two

Answer 1

(a). Let us write normal form of this game.

Number of strategies of Player 1 = 4
Number of strategies of Player 2 = 3
Thus, payoff table will be 4x3

P1/P2	L	C	R
InT	(4,2)	(3,3)	(0,a)
InB	(4,2)	(1,1)	(0,a)
OutT	(2,2)	(2,2)	(2,2)
OutB	(2,2)	(2,2)	(2,2)

(b) a = 0 and we are asked to find the perfect sub-game equilibrium.

One cannot apply backward induction in this game because it is not a perfect information game. One can compute the subgame perfect equilibrium, however. This game has two subgames: one starts after Player 1 plays In; the second one is the game itself. The subgame perfect equilibria are computed as follows.

First compute a Nash equilibrium of the subgame, then fixing the equilibrium actions as they are (in this subgame), and taking the equilibrium payoffs in this subgame as the payoffs for entering the subgame, compute a Nash equilibrium in the remaining game.

Player 1 plays In, after which Player 2 plays L or C. When L is selected, Player 1 gets utility 4 and Player 2 gets utility 2.
If Player 2 plays C, Player 1 will play T, getting utility of 3 and Player 2 getting utility of 3 as well. These are the subgames.

(c) a = 0 and we are asked to find the Nash equilibrium that is not subgame perfect.
In this case, both try to maximise their payoffs. Player 2 gets maximum utility if it selects C and Player 1 gets maximum utility in case of InT. This is a Nash equilibria.

(d) If a = 3, equilibria become In/R/T or In/R/B.