Question

Consider the following payoff matrix for Firm A and Firm B. Firm A sells ski equipment and Firm B sells ski clothing (complementary goods). These two firms are choosing the location of their stores in a mall and will increase profits if they choose to locate in the same corner. There are two available spots in both the NW corner and the SW corner of the shopping mall. Determine whether Firm A and Firm B have a dominant strategy. Work through the equilibrium mixed strategy and find the expected payoffs.  

		Firm B
		NW Corner (q)	SW Corner (1–q)
Firm A	NW Corner (p)	50,30	20,15
	SW Corner (1-p)	20,15	35,45

Answer 1

Answer.

Dominant strategy is strategy for a player having best response to all strategy profile of other player. In the above game both the firms do not have a dominant strategy. For firm A, at NW corner 50 is greater than 20 but at SW corner for firm A , 20 is less than 35. Whereas for firm B , at NW corner 30 is greater than 15 but at SW corner , 15 is less than 45. Thus, both the firms don’t have the dominant strategy.

Mixed strategy is in which player makes a random choice between two or more possible actions based on the set of probabilities.

So, the expected pay off for firm A = 50*p*q + p*(1-q)*20 + (1-p)*q*20 + (1-p)*(1-q)*35

A = 50*p*q + 20p - 20p*q + 20q - 20p*q + 35 + 35p*q - 35p - 35q

A = 45p*q - 15p - 15q + 35

Partial differentiating with respect to p and then equating to zero, we get

A = 45q -15

45q -15 = 0

45q = 15

q = 15/45

q = 1/3

probability is firm A will choose 1/3 for NW corner and 2/3 for SW corner.

Expected pay off for firm B = 30*p*q + (1-q)*(p)*15 + (q)*(1-p)*15 + (1-p)*(1-q)*45

B = 30p*q + 15p - 15p*q + 15q - 15p*q + 45 - 45p - 45q + 45p*q

B = 45p*q - 30p - 30q + 45

partial differentiating with respect to q and equate it to zero, we get

B = 45p - 30

45p = 30

p = 30/45

p = 2/3

probability is firm B will choose 2/3 for NW corner and 1/3 for SW corner.