January 20 Lecture
Econ 171
The game of matching pennies has
A) two pure strategy Nash equilibria
B) One pure strategy Nash equilibrium
C) One mixed strategy Nash equilibrium and no
pure strategy Nash equilibria
D) Two mixed strategy Nash equilibria and no
pure strategy Nash equilibria
E) One mixed strategy Nash equilibrium and
two pure strategy Nash equilibria.
If a game has at least one pure
strategy Nash equilibrium
A) It must also have at least one mixed strategy
Nash equilibrium that is not a pure strategy Nash
equilibrium
B) It will not have a mixed strategy Nash equilibrium
C) It might or might not have a mixed strategy Nash
equilibrium that is not a pure strategy equilibrium
D) It must have a dominant strategy for each player.
Field Goal or Touchdown?
• Field goal is worth 3 points.
• Touchdown is worth 7 points.
Which is better? Sure field goal or probability ½
of touchdown?
Finding the coach’s von Neumann
Morgenstern utilities
• Set utility of touchdown u(T)=1
• Set utility no score u(0)=0
The utility of a gamble in which you get a touchdown
with probability p and no score with probability 1-p is
pu(T)+(1-p)u(0).
What utility u(F) to assign to a sure field goal?
Let p* be the probability such that the coach is indifferent
between scoring a touchdown with probability p* (with
no score with prob 1-p*) and having a sure field goal.
Then u(F)=p*u(T)+(1-p*)u(0)=p*x1+(1-p*)x0=p*.
Advanced Hide and Seek
Hider’s Choice
q
p
Plains
Seeker’s
Choice
1-p
Forest
Plains
1-q
Forest
3,-3
-1,1
-1,1
1,-1
Guard and shoot game
Guard protects
q
Left
Shooter
uses
P
Right
1-p
Left
1-q
Right
D, -D
B, -B
A, -A
C, -C
Probabilities of scoring: A>C , B>D, B>C
Nash Demand Game
There are N dollars to be divided among 2
players. Each player gets to make a demand.
The demand must be a positive integer. If sum
of demands is not greater than N, each gets
what she demanded. If sum of demands exceeds
N, both get nothing.
Nash demand game, N=3
A Game of Chicken
Two teenagers have their fathers’ Buicks and
decide to play Chicken. There are two strategies,
Swerve and Don’t Swerve. If neither swerves,
disastrous consequence for both.
Like the hawk-dove game.
Alternative story. Two countries have mobilized
their armies with threats and counterthreats. Will
one of them yield? If not, there is a disastrous war.
A Chicken payoff matrix