Question

O3. Serena and Venus are 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. playing Tennis. Consider a single point in which Serena is serving. She Venus Anticipate Anticipate backhand forehand Serve to forehand 80 20 Serena Serve to backhand 40 90 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.

1. Find the mixed strategy Nash equilibrium of this game, please explain and show your work.

Answer 1

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.