Exercises

1. Determine whether the following game is dominance solvable.

Player B
Player A		s1	s2
	S1	2;4	6;-2
	S2	3;-3	-5;-4

2. Is the following game dominance solvable? Does it have any Nash equilibria?

Player B
Player A		s1	s2	s3
	S1	2;1	-2;0	4;-1
	S2	0;0	3;1	5;4
	S3	1;5	4;2	4;2

3. Does Matching Pennies have a Nash equilibrium in pure strategies?

4. Chicken

Two people drive their cars directly at each other until one (or both) swerves off the road or they crash into each other. Payoffs: swerving while other does not: 0; both swerve: 5; neither swerves: -10; continuing on while the other swerves: +10.

1. Construct the strategic matrix.
2. Is this a zero-sum game?
3. Is the game dominance solvable?
4. Is there any Nash equilibrium in pure strategies?

5. Grab the Dollar

A dollar is placed on the table between A and B who must each decide whether to grab it or not. If both grab, then they are both fined 1 dollar; if only one grabs, he keeps the dollar.

a. Construct the strategic matrix.

b. Determine the Nash equilibria in pure strategies

6. Verify that the following game does not have a Nash equilibrium in pure strategies.

Player B
Player A		s1	s2
	S1	4,2	-5,6
	S2	-1,5	0,-2

1. Determine a Nash equilibrium in mixed strategies.
2. Which are the most common outcomes?
3. What are the expected payoffs?

7.

Consider the following game:

Player B
Player A		s1	s2
	S1	1;2	3;0
	S2	2;1	1;3

Table 5

Let Pr(S1)=1/4.

What’s the EP of :

1. pure strategy s1 against mixed strategy S?
2. pure strategy s2 against mixed strategy S?
3. mixed strategy S against pure strategy s1?
4. mixed strategy S against pure strategy s2?

Now let Pr(S1)=1/4 and Pr(s1)=1/3.

What’s the EP of

5. Mixed strategy s against mixed strategy S?
6. Mixed strategy S against mixed strategy s?

Answers