Question

2 Consider the following normal form game. Bill's payoffs are given first. Find all pure strategy Nash equilibrium. Show your steps. (20 points Tony A 10, 30 0, 20 20, 30 Bill B 15, 35 10, 40 10, 40 С 25,25 5,25 5,25

Answer 1

		Tony
		X	Y	Z
Bill	A	(10,30)	(0,20)	(20,30)
	B	(15,35)	(10,40)	(10,40)
	C	(25,25)	(5,25)	(5,25)

Tony      
X    Y    Z
Bill   A   (10,30)   (0,20)   (20,30)
   B   (15,35)   (10,40) (10,40)
   C   (25,25) (5,25) (5,25)

Now we have two player (Bill, Tony) where strategy set for each player is as follows

Bill={A,B,C}

Tony={X,Y,Z}

When Bill choose to play A then best response of Tony would be either X or Z because in this scenario Tony will have higher payoff of 30

When Tony chooses to play X then best response of Bill is C as in this case Bill have higher payoff from strategy C payoff of 25

When Tony chooses to play Z then best response of Bill is A as in this case Bill have higher payoff from strategy A payoff of 20

Hence combination of (A,Z) is Pure strategy Nash Equilibrium because it completes the cycle such as if Bill chooses A then Tony responds with Z and if Tony chooses Z then Bill responds with A.

When Bill choose to play B then best response of Tony would be Y or Z because in this scenario Tony will have higher payoff of 40

When Tony chooses to play Y then best response of Bill is B as in this case Bill have higher payoff from strategy B payoff of 10

Hence combination of (B,Y) is Pure strategy Nash Equilibrium because it completes the cycle such as if Bill chooses B then Tony responds with Y and if Tony chooses Y then Bill responds with B.

When Bill choose to play C then best response of Tony would be either X or Y or Z because in this scenario Tony will have higher payoff of 25

When Tony chooses to play X then best response of Bill is C as in this case Bill have higher payoff from strategy C payoff of 25

When Tony chooses to play Y then best response of Bill is B as in this case Bill have higher payoff from strategy A payoff of 10

When Tony chooses to play Z then best response of Bill is A as in this case Bill have higher payoff from strategy A payoff of 20

Hence combination of (C,X) is Pure strategy Nash Equilibrium because it completes the cycle such as if Bill chooses C then Tony responds with X and if Tony chooses X then Bill responds with C.

Therefore in total we have 3 Pure strategy Nash Equilibrium and they are as follows

(C,X); (B,Y); (A,Z)