# 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_2 What is the Nash equilibrium probability of Forehand for the Receiver?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

A. Try again. If the Receiver chooses such a low probability of Forehand, the Server can take advantage of that by serving to his Forehand all the time.

If the Receiver chooses the a probability of .1, for example, then

Server's Payoff (Forehand) = .1 (1) + .9(10) = 9.1 > Server's Payoff (Backhand) =.1(10) + .9 (4) = 4.6,

so the Server will serve to his Forehand all the time.

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

Server's Payoff (Forehand) = Y (1) + (1-Y)(10) = Server's Payoff (Backhand) =Y(10) + (1-Y)(4).