# 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

C. CORRECT. If the Receiver chooses the equilibrium probability of .4 for Forehand, the Server cannot take advantage of that.

If the Receiver chooses the a probability of .4, then

Server's Payoff (Forehand) = .4 (1) + .6(10) = 6.4 = Server's Payoff (Backhand) =.4(10) + .6 (4) =6.4,

so the Server will be indifferent about his choice of where to serve-- he cannot take advantage of the Receiver.

To get that 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). This solves out for Y=.4