# 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

B. CORRECT. If the Server chooses that probability of Forehand, the Receiver cannot advantage of his choice.

If the server chooses the equilibrium probability of .4, then

Receiver's Payoff (Forehand) = .4 (9) + .6(0) = 3.6 < Receiver's Payoff (Backhand) =.4(0) + .6 (6) = 3.6.

To derive that value of .4, 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).