# 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

A. Try again. If the Server chooses such a low probability of Forehand, the Receiver will take advantage of that by choosing a high probability of Backhand.

For example, if the server chooses a probability of .1, then

Receiver's Payoff (Forehand) = .1 (9) + .9(0) = .09 < Receiver's Payoff (Backhand) =.1(0) + .9 (6) = .54.

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).