Self-test on Mixed Strategies

For each game, click on the answer you think is right. That will take you to an explanation of why it is right or wrong. Since this is not a real test, also click on any answers you think might might be right, so you can understand why they are wrong.

In fact-- click on all of them. You should be able to understand why wrong answers are wrong as well as why right answers are right. A useful exercise, and possible test question, is to ask yourself how you would explain why a given answer is wrong.

After doing a few of these, you will find it easy to make up your own examples.

GAME 1: UNFORGIVING TENNIS
```                                     SERVER
Forehand      Backhand
Forehand            9,1         0,10
Backhand           0,10         6,4

```
This game is like the Tennis Game in Dixit and Nalebuff, except that if the Receiver is surprised, he never succeeds in returning the serve.

1_1 What is the Nash equilibrium probability of Forehand for the Server?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

1_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

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_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

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

GAME 3: THE WELFARE GAME
```                                         PAUPER
Work         Loaf
Aid           3,2           -1,3
GOVERNMENT
No  Aid       -1,1           0,0

```
``The Welfare Game'' models a government that wishes to aid a pauper if he searches for work but not otherwise, and a pauper who searches for work only if he cannot depend on government aid. This is a well-known problem in public policy, called ``The Samaritan's Dilemma'' by Tullock (1983, p. 59), who attributes it to James Buchanan. I use it in Chapter 3 of Games and Information . The same problem arises on a private level when parents decide how much to help a lazy child.

3_1 What is the Nash equilibrium probability of Aid for the Government?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

3_2 What is the Nash equilibrium probability of Work for the Pauper?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

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

4_2 What is the mixed-strategy Nash equilibrium probability of Hawk for Bird 2?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7