1. Working Some Problems

2. A Duel Does this game have any pure strategy equilibria? Yes No
Pitcher throws
Fastball
Curveball
.35,.65
.3,.7
.2,.8
.5,.5
Batter prepares for
Does this game have any pure strategy equilibria?
Yes No

3. Finding Mixed strategy equilibrium
In N.E, Batter will play each strategy with positive probability only if the two strategies have the same expected payoff for him.
Suppose Pitcher throws a fastball with probability p and a curveball with probability 1-p.
Batter’s expected payoff from “Prepare for Fastball” is .35p+.30(1-p)
Batter’s expected payoff from “Prepare for curveball” is .2p+.5(1-p).
These payoffs are equal if .35p+.30(1-p)=.2p+.5(1-p).
Solve this equation for p.

4. A Duel In a mixed strategy Nash equilibrium what is the
Pitcher throws
Fastball
Curveball
.35,.65
.3,.7
.2,.8
.5,.5
Batter prepares for
In a mixed strategy Nash equilibrium what is the
probability that Batter prepares for a fastball?
4/7
3/7
6/7
4/5

5. Does this game have a Nash equilibrium in which Kicker mixes left
and right but does not kick to center?

6. If there is a Nash equilibrium where kicker
never kicks middle but mixes between left and
right, Goalie will never play middle but will
mix left and right (Why?)
If Goalie never plays middle but mixes left
and right, Kicker will kick middle. (Why?)
So there can’t be a Nash equilibrium where
Kicker never kicks Middle. (See why?)

7. Problem 4: For what values of x is there a mixed strategy Nash equilibrium in which the victim might resist or not resist and the Mugger assigns zero probability to showing a gun?

8. Problem 7.7, Find mixed strategy Nash equilibria

9. A mixed strategy N.E. strategy does not give positive probability
To any strictly dominated strategy
c dominates a and y dominates z
Look at reduced game without these strategies

10. Problem 7.7 Find mixed strategy Nash equilibia
For player 1, Bottom strictly dominates Top. Throw out Top
Then for Player 2, Middle weakly dominates Right. Therefore if
Player 1 plays bottom with positive probability, player 2 gives zero
Probability to Right.
There is no N.E. in which Player 1 plays Bottom with zero probability, (Why?)
(If he did, what would Player 2 play? Then what would 1 play?)

11. Problem 8, Chapter 7
A Nash equilibrium is any strategy pair in which the defense defends
against the outside run with probability .5 and the offense runs up the
middle with probability No matter what the defense does,
The offense gets the same payoff from wide left or wide right,
So any probabilities pwl and pwr such that pwl+pwr=.25
will be N.E. probabilities for the offense.

12. Problem 10.
Each of three players is deciding between the pure strategies go and stop. The payoff to
go is 120, where m is the number of players that choose go, and the payoff to stop is 55 m
(which is received regardless of what the other players do). Find all Nash equilibria in mixed strategies.
Let’s find the “easy ones”.
Are there any symmetric pure strategy equilibria?
How about asymmetric pure strategy equilibria?
How about symmetric mixed strategy equilibrium?
Solve 40p^2+60*2p(1-p)+120(1-p)2=55
40p2-120p+65=0

13. What about equilibria where one guy is in for sure and other
two enter with identical mixed strategies?
For mixed strategy guys who both
Enter with probability p, expected payoff from entering is
(120/3)p+(120/2)(1-p). They are indifferent about entering or not if
40p+60(1-p)=55. This happens when p=1/4.
This will be an equilibrium if when the other two guys enter with
Probability ¼, the remaining guy is better off entering than not.
Payoff to guy who enters for sure is:
40*(1/16)+60*(3/8)+120*(9/16)=92.5>55.

14. Advanced Rock-Paper-Scissors
0,0
-1,1
2,-2
1,-1
-2,2
Are there pure strategy Nash equilibria?
Is there a symmetric mixed strategy Nash equilibrium?
What is it?

15. Remember: Midterm on Thursday
Chapters 2-5 and 7
No need to bring bluebooks.
Calculators and phones not allowed