# An Answer for the Self-test on Mixed Strategies

GAME 2
```                                      SERVER
Forehand      Backhand
Forehand            90,1         0,10
Backhand           0,10         6,4

```
The idea in this game is to see what happens to Game 1 when one payoff- (9,1)-- is drastically changed-- to (90,1). It's not tennis anymore, but let's see what happens.

2_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. Try again. If the Receiver chooses such a high probability of Forehand, the Server can take advantage of that by serving to his Backhand all the time.

If the Receiver chooses the a probability of .5, for example, then

Server's Payoff (Forehand) = .5 (1) + .5(10) = 5.5 < Server's Payoff (Backhand) =.5(10) + .5 (4) =7,

so the Server will serve to his Backhand 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).