# An Answer for the Self-test on Mixed Strategies

GAME 1: UNFORGIVING TENNIS
```                                     SERVER
Forehand      Backhand
Forehand            9,1         0,10
Backhand           0,10         6,4

```
This game is like the Tennis Game in Dixit and Nalebuff, except that if the Receiver is surprised, he never succeeds in returning the serve.

1_1 What is the Nash equilibrium probability of Forehand for the Server?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

C. Try again. If the Server chooses such a high probability of Forehand, the Receiver can advantage of that by choosing Forehand all the time.

If the server chooses the a probability of .5, then

Receiver's Payoff (Forehand) = .5 (9) + .5(0) = 4.5 > Receiver's Payoff (Backhand) =.5(0) + .5 (6) = 3.

To get the correct answer, you need to choose a mixing probability X for the Server such that the Receiver does no better from Forehand than from Backhand. To do that, you solve

Receiver's Payoff (Forehand) =X (9) + (1-X)(0) = Receiver's Payoff (Backhand) =X(0) + (1-X) (6).