1 Consider the following simultaneous move game where Player

1. Consider the following simultaneous move game, where Players 1 and 2 choose between actions X or Y Player 2 Player X c.d a,a b.b a. Suppose that the payoffs to the two players are such that a>b>cd and that b. Which equilibrium is payoff dominant? Which equilibrium is risk dominant? c. Redo parts a-b of this question assuming that a>b>c>d but that a+dkb+c. a+d>b+c. What are the pure strategy Nash equilibria for this game? Carefully explain your reasoning. What changes and why?

Solution

1. (a) If a>b>c>d and a + d > b+c.

For example : a= 5 , b= 3 , c = 2 and d=1.

This satisfies the condition : a+d > b+c

Then , the matrix looks like this:

When player 1 chooses X, then player 2 has more pay off in choosing X. Similarly,when player 1 chooses Y , then player 2 has more pay off in choosing Y.

Similarly, when player 2 chooses X ,then player 1 has more pay off in choosing X. And when player 2 chooses Y, then player 1 has more pay off in choosing Y.

So, the Nash equilibrium is (3,3) and (5,5).

i.e Nash equilibrium is (b,b) and (a,a).

(b) Pay off dominant equilibrium = (5,5) i.e (a,a)

And risk dominant equilibrium = (3,3) i.e (b,b).

(c) Now, if a= 9 , b=8 , c=7 and d=1 .

Then this satisfies the condition a+d < b+c  

And by plotting the values in matrix. The equilibrium would not change .

		Player 2
		X	Y
Player 1	X	3,3	2,1
	Y	1,2	5,5