# 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_1 What is the Nash equilibrium probability of Forehand for the Server?
A. Zero.
B. More than 0 but less than or equal to .2
C. Greater than .2 but less than .5.
D. Between .5 and .7, inclusive
E. Greater than .7

A. Try again. The Server must choose an extremely low probability of Forehand, but not a probability of zero, or the Receiver will take advantage of him.

If the server chooses a probability of 0, then

Receiver's Payoff (Forehand) = 0 (90) + 1(0) = 0 < Receiver's Payoff (Backhand) =0(0) + 1 (6) = 6.

The Receiver can never use his Forehand, but it is scary enough to the Server that the Server is fooled into making himself predictable. (But of course the Server could go on to try answers b,c,d, and e. )

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 (90) + (1-X)(0) = Receiver's Payoff (Backhand) =X(0) + (1-X) (6).