Question

Suppose we have a game where S 1 = { U,D } and S 2 = { L,R } . If Player 1 plays U, then her payoff is ? regardless of Player 2’s choice of strategy. Player 1’s other payoffs are u 1 ( D,L ) = 0 and u 1 ( D,R ) = 12. You may choose any numbers you like for Player 2 because we are only concerned with Player 1’s payoff.

a.) Draw the normal-form matrix for this game. For the following questions, assume Player 1’s belief is ? 2 = (1 / 4 , 3 / 4).

b.) Given ? 2 , what is Player 1’s expected payoff from playing U?

c.) Given ? 2 , what is Player 1’s expected payoff from playing D?

d.) Given ? 2 , what value of ? makes Player 1 indifferent between playing U and D?

Answer 1

Answer: We have game where S 1 = { U,D } and S 2 = { L,R } . If Player 1 plays U, then her payoff is ? regardless of Player 2’s choice of strategy. Player 1’s other payoffs are u 1 ( D,L ) = 0 and u 1 ( D,R ) = 12.

I will assign payoffs a, b, c, and d randomly for player 2.

a)

		Player2 (1/4)	(3/4)
player 1		L	R
	U	, a	, b
	D	0, c	12, d

b) Expected payoff for player 1 if he plays U.

S₁(U,(1/4, 3/4)) = *1/4 +*3/4=

c) Expected payoff for player 1 if he plays D.

S₁(D,(1/4, 3/4)) = 0*1/4 +12*3/4= 9

d) To be indifferent

S₁(U,(1/4, 3/4)) = S₁(D,(1/4, 3/4))

*1/4 +*3/4=  0*1/4 +12*3/4

=> = 9 .... from part b and c.