Question

Venus and Serena are playing a tennis match. Each of them uses two strategies: Hit left or hit right. The payoffs from each strategy combination are given below (The rows correspond to Venus's strategies, and the columns correspond to Serena's strategies. The first number in each payoff combination (x,y) is Venus's payoff, and the second number is Serena's payoff. )

	Left	Right
Left	30,70	80,20
Right	90,10	20,80

11. What is Venus's dominant strategy?

12. What is Serena's best response, if Venus plays left?

13. What is Venus's best response, if Serena plays right?

14. What is the pure strategy Nash equilibrium of this game?

15. What is the mixed strategy Nash equilibrium of this game (of the form (p,q), where p=probability that Venus plays left, q=probability that Serena plays left).

Answer 1

11. There is no dominant strategy for Venus

12. if Venus plays left, Serena's best response is Left

13. If Serena plays right, Venus's best reponse is Left.

14. The pure strategy Nash is empty. There exists no pure Nash(Since the best response of both does not complement.)

15. Using Indifference Lemma, we can find the mixed strategy for both the players which makes them indifferent to each possible strategy.

For Serena: 30p + 80(1-p) = 90p + 20(1-p) => 80 -50p = 70p+20 => 60 = 120p   ==> p = 1/2.

For Venus: 70q + 10(1-q) = 20q + 80(1-q) => 60q+ 10= 80-60q ==> 70 = 120q ==> q = 7/12.

Thus (1/2,7/12) is mixed strategy Nash.