# An Answer for the Self-test on Mixed Strategies

```                                      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

A. Try again. If the Receiver chooses such a low probability of Forehand, the Server will respond by serving to his Forehand all the time. But then the Receiver would also choose Forehand all the time, and if that happened the Server would switch to the pure strategy of Backhand. This cycle would never end, so a low probability of Forehand chosen by the receiver would not be part of a Nash equilibrium.

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).