4. Consider the extensive form game depicted in the following figure (where the

Question

4. Consider the extensive form game depicted in the following figure (where the payoff of player 1 is written on top, and the payoff of 2 is on the bottom) commitment freedom 2 Figure 1: An extensive Form Game (a) Suppose we were to write this game in normal form as an M × N game. (Hint: a player's strategy is a complete action plan that tells him what to play at each of his information sets.) What would M and N be? (b) Find all subgame perfect Nash equilibria using backwards induction (c) Show that your solution to part (b) is also a valid Nash Equilibrium of the normal form game (d) Can you find another Nash Equilibrium of the normal form game that is different than the solution from part (b)

Explanation / Answer

This is normal form of given extensive game as it has M=8 rows and N=3 columns

b)

When player 1 plays "Commitment" player 2 will always play strategy"a" as player 1 has higher payoff after playing "A" following "commitment" Nash Equilibrium using Backward induction (CAA,a) ; (CAB,a)

When player 1 plays "Freedom" player 2 will always play strategy"b" as player 1 has higher payoff after playing "A" following "commitment" Nash Equilibrium using Backward induction (FAB,b);(FAA,b);(FBB,b)

As when player 1 plays Commitment player 2 will always play strategy "a' as he will have highest payoff than playing "b" hence strategy'b' will be easily dominated by "a" hence eliminated. Similar logic we can work out with when player 2 plays Freedom

c)

Looking at Normal form game

Best response function of player 1 for player 2's strategy a is CAA & CAB similarly FAA,FAB & FBB for player 2's strategy "b"

hence Nash Equilibrium for normal gane are (CAA,a);(CAB,a);(CBA,b);(CBB,b);(FAA,a);(FAB,b);(FBA,a);(FBB,b)

Above NE comprise of all NE we got usinng backward induction.

D)

In solution c) we have all the NE other than part of solution b)

(CBA,b);(CBB,b);(FAA,a);(FAB,b);(FBA,a)

a b (C,A,,A) (2,1) (0,0) (C,A,B) (2,1) (0,0) (C,B,A) (0,0) (1,2) (C,B,B) (0,0) (1,2) (F,A,A) (2,1) (0,0) (F,A,B) (0,0) (1,2) (F,B,A) (2,1) (0,0) (F,B,B) (0,0) (1,2)

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