# An Answer for the Self-test on Mixed Strategies

GAME 4: HAWK-DOVE
```                                         BIRD 2
Hawk         Dove
Hawk           -2, -2        4,0
BIRD 1
Dove              0,4        1,1

```
This is like the Hawk-Dove Game we looked at in class, except I have changed the numbers so that while the ordinal rankings are the same, Hawks do much better when they meet Doves. (I changed 0,2 to 0,4; and 2,0, to 4,0).

4_1 What is the mixed-strategy Nash equilibrium probability of Hawk for Bird 1?
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 Bird 1 chooses too low a mixing probability for Hawk, then the payoffs for Bird 2 would be greater for Hawk than for Dove. Thus, Bird 2 would not be willing to play a mixed strategy-- he would play pure Hawk-- and we would not end up with the mixed-strategy equilibrium we were trying to calculate. (Bird 1 would not want to mix either, if Bird 2 did not.)

If, for example, Bird 1 chose to play Hawk with probability .1, then Bird 2 would always play Hawk:

Payoff of Bird 2 (Hawk) =.1 (-2) + (1-.1) (4) =3.4 > Payoff of Bird 2 (Dove) =.1 (0) + (1-.1) (1) = .9.

To find the Nash equilibrium, find a mixing probability X that equates the payoffs from the two pure strategies.

Payoff of Bird 2 (Hawk) =X (-2) + (1-X) (4) = Payoff of Bird 2 (Dove) =X (0) + (1-X) (1).