# 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

B. 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

Did you notice something odd here? The equilibrium mixing probability for the Receiver is exactly the same as in Game 1! One of the Receiver's payoffs is changed to 90 in Game 2, but that does not affect the Receiver's choice of strategy. This is a well-known peculiarity of mixed strategy equilibria: if one player's payoff parameters change, it is the other player's mixing probability that changes. The reason is that the Receiver is choosing his mixing probability to avoid giving the Server an incentive to use a pure strategy, which means he is making his calculations using the Server's payoff parameters, not his own.