Question

Q3. Serena and Venus are playing Tennis. Consider a single point in which Serena is serving. She has to decide whether to serve to Venus’s forehand or backhand. Venus has to anticipate either a forehand or a backhand. The following table gives the success percentage for Serena.

Venuse Anticipate Anticipate backhand forehand Serve to 80* 20 Serena forehand 40 90* Serve to backhand

For instance, if Serena serves to the forehand and Venus anticipates a backhand, Serena is likely to win 80 percent of the time. Serena wants to maximize the probability that she wins, while Venus wants to minimize it.

a. What is Venus’s best response if Serena is serving to the forehand?

b. What is Serena’s best response if Venus is anticipating a forehand?

c. Is there a dominant strategy for Serena? If yes, what is it? If no, explain why not.

d. Is there a pure strategy Nash equilibrium in this game? If yes, what is it? If no, explain why not.

e. Find the mixed strategy Nash equilibrium of this game.

Answer 1

Answer a

If Serena is serving forehand, Venus can anticipate backhand which will make her winning percentage to 100-80 =20% or Venus can anticipate forehand which will make her winning percentage to 100-20=80%. Hence Best Response for Venus is to Anticipate Forehand as payoff is higher

Answer b

If Venus is Anticipating forehand, Serena can serve backhand which will make her winning percentage to 90% or Serena can serve forehand which will make her winning percentage to 20%. Hence Best Response for Serena is to Serve Backhand as payoff is higher

Answer c

If Venus is Anticipating Backhand Serena will serve Backhand as payoff is higher and if Venus is anticipating Forehand, Serena will serve Backhand as payoff is higher. Hence Serena strategy is dependent on Venus anticipation. Hence there is no dominant strategy for Serena.

Answer d

Let’s analyse the response of Serena. If Venus is anticipating Backhand, Serena will serve Forehand as payoff is higher and if Venus is Anticipating Forehand, serena will serve Backhand as payoff is higher. Best Responses of Serena is shown with S

		Venus
		Anticipate Backhand	Anticipate Forehand
Serena	Serve Forehand	80 S	20
	Serve Backhand	40	90 S

Let’s analyse the response of Venus. If Serena is Serving Forehand, Venus will anticipate Forehand as payoff is higher and if Serena is Serving Backhand, Venus will anticipate Backhand as payoff is higher. Best Responses of Venus is shown with V

		Venus
		Anticipate Backhand	Anticipate Forehand
Serena	Serve Forehand	80 S	20 V
	Serve Backhand	40 V	90 S

We can see that Best Responses do not intersect on any outcome. Hence there is no Pure Strategy Nash Equilibrium

Answer e

To find Mixed Strategy Equilibrium, Let us assume that Serena is serving forehand with probability p and serving backhand with probability (1-p)

Hence payoff of Venus for Anticipating Backhand when Serena is serving forehand will be p*(100-80) =20p and when Serena is serving backhand will be  (1-p)* (100-40)= 60*(1-p) = 60-60p. Hence total payoff = 20p +60-60p = 60-40p

And payoff of Venus for Anticipating Forehand when Serena is serving forehand will be p*(100-20) =80p and when Serena is serving backhand will be  (1-p)* (100-90)= 10*(1-p) = 10-10p. Hence total payoff = 80p +10-10p = 10+70p

For a mixed strategy equilibrium Total payoff for Venus should be equal in both the actions. Hence, 60-40p = 10+70p

or, 60-10 = 70p+40p

or 50=110p

or, p = 50/110

or p= 5/11 and hence (1-p) = 1-5/11 = (11-5)/11=6/11

Hence Serena will serve forehand with probability 5/11 and serve backhand with probability 6/11

Similarly Let us assume that Venus is anticipating backhand with probability q and anticipating forehand with probability (1-q)

Hence payoff of Serena for Serving forehand when Venus is anticipating backhand will be q*80 =80q and when Venus is anticipating forehand will be  (1-q)* (20)= 20*(1-q) = 20-20q. Hence total payoff = 80q +20-20q = 60q+20

And payoff of Serena for serving backhand when Venus is anticipating backhand will be q*(40) =40q and when Venus is anticipating forehand will be  (1-q)* (90)= 90*(1-q) = 90-90q. Hence total payoff = 40q +90-90q = 90-50q

For a mixed strategy equilibrium Total payoff for Serena should be equal in both the actions. Hence, 60q+20 = 90-50q

or, 60q+50q = 90-20

or 110q=70

or, q = 70/110

or q= 7/11 and hence (1-q) = 1-7/11 = (11-7)/11=5/11

Hence Venus will anticipate backhand with probability 7/11 and anticipate forehand with probability 5/11

Hence mixed Strategy Nash Equilibrium = {(5/11,6/11), (7/11,5/11)} where (5/11,6/11) is mixed strategy of Serena and (7/11,5/11) is mixed strategy of Venus.