Midterm Scores
• 181 points total
Grade Range
Score
Number of students in
this range
A
160-181
16
B
120-159
16
C
90-119
16
D
70-89
9
F
<40
4
Median Score: 120
Top Score 181
For those who did well…
f0rmty
And keep climbing…
For those who did poorly…
Fair warning…
• Course builds on what you have learned.
• Material becomes harder.
• If you are lost now, your chances of catching
up are almost zero.
• Last day to drop class is Feb 1.
What is a strategy?
• Most of you showed that you know.
• A strategy is NOT a possible “history of play.”
(i.e. Not, he did this, then he did that)
• It is a contingency plan. What will player do
at each information set that he might reach.
• If you didn’t list strategies correctly wherever
required, Study the textbook! Read pages pp
36-38 carefully and thoughtfully.
Mixed Strategies
Mixed strategies
• If a player’s strategy is to do something for
sure at each information set, we say it is a
pure strategy.
• A player who ``randomizes’’ between ``pure
strategies’’, assigning a specific probability to
taking each possible pure strategy is said to
use a mixed strategy.
• Why might you do that?
Simple hide and seek
Player 2 (Seeker)
Look Upstairs
p upstairs
Hide
Player 1
(Hider) (
Hide
downstairs
0,1
1,0
Look Downstairs
1,0
0, 1
The game of hide and seek has
A) Two pure strategy Nash equilibria
B) One pure strategy Nash equilibrium
C) No pure strategy Nash equilibrium
Look upstairs
0,1
H
i
d
e
Hide
u downstairs
p
s
t
1,0
Look downstairs
1,0
0, 1
Best response
Suppose Hider randomizes and hides upstairs
with probability 2/3. What is Seeker’s best
response?
A) Look upstairs for sure
B) Look downstairs for sure
C) Randomize with Probability of upstairs 2/3.
D) Randomize with Probability of downstairs 2/3
Could there be a Nash equilibrium
strategy profile in which Hider is more
likely to hide upstairs than downstairs?
A) Yes
B) No
C) Maybe
In Hide and Seek, is there a Nash
equilibrium in which one of the two
players uses a pure strategy?
A) Yes
B) No
C) Maybe
Mixed strategy as best response
• In a two-player, two-strategy game playing a
mixed strategy is a best response to what the
other player is doing only if your payoffs from
the two pure strategies are equal.
• A) True
• B) False
Mixed strategies for Hide-and-Seek
• Let p be the probability that hider hides
upstairs. For what value of p would seeker be
willing to use a mixed strategy?
• Let q be the probability that seeker looks
upstairs. For what value of q would hider be
willing to use a mixed strategy?
A Fundamental Theorem
• Some games have no equilibrium in pure
strategies: Examples: matching pennies; rock,
paper scissors
• Every game in which there is a finite number
of pure strategies has at least one mixed
strategy equilibrium.
Advanced Hide and Seek
Large Attacking Army
Plains
Plains
Smaller
Retreating
Army
Forest
Forest
-3,3
1,-1
1,-1
-1, 1
Mixed strategy equilibrium
• In a mixed strategy equilibrium, all strategies
that are assigned positive probability have
equal expected value.
• You can use this fact to find mixed strategy
Nash equilibria.
Example: Advanced Hide and Seek
• When does Attacker have a mixed strategy best
response. The payoffs to Attacker from looking in
the plains and looking in the forest must be the
same.
• Where p is probability Retreater is in the plains
• For the attacker
– Payoff to Plains is p3+(1-p)(-1)=4p-1.
– Payoff to Forest is -1p +(1-p)1=1-2p
– 4p-1=1-2p if and only if 6p=2, p=1/3.
How about Retreater?
• When does Retreater have a mixed strategy
best response?
• Where q is the probability that Attacker comes
through the plains
• For the retreater, the expected payoffs are
– go through plains -3q+1x(1-q) =1-4q
– go through forest q-1x(1-q)=2q-1
– Payoffs are same if 1-4q =2q-1 which implies that
q=1/3.
The mixed strategy Nash equilibrium
• Retreating army goes to the forest with
probability 2/3 and plains with probability 1/3.
• Attacking army goes to the forest with probability
2/3, plains with probability 1/3.
• Probability both go to plains =1/9
• Probability both go to forest=4/9.
• Probability they go to different places 4/9.
• Expected payoff
– For Attacking army is 4/9x1+1/9x3+4/9x(-1)=1/3
– For Retreating army is =1/3.
Best response Mapping
1
q=probability
Player 1’s
2 chooses Plains Reaction
Player 2’s Reaction Function (in Red)
Function
(in Green)
1/3
0
1/3
1
p= Probability 1 chooses Plains
Chicken Game
Player 2
q
Swerve
P
Swerve
1-q
Don’t Swerve
0, 0
0, 1
1, 0
-10, -10
Player 1
Don’t Swerve
1-p
Two Pure Strategy Nash equilibria
Mixed Strategy
When is Player 1 indifferent between the two
strategies, Swerve and Don’t Swerve?
Expected payoff from Swerve is 0.
Expected payoff from Don’t Swerve is
q-10(1-q).
So Player 1 will use a mixed strategy best response
only if 0=11q-10 or q=10/11.
Similar reasoning inplies that in Nash equilibrium
p=10/11.
Crash occurs with probability 1/121.
A Dark Tale from the Big Apple
The Kitty Genovese Case
• In 1964, as she returned home from work late
at night, Kitty Genovese was assaulted and
murdered, near her apartment in Queens,
New York City.
• According to a story in the New York Times
– For more than half an hour, 38 respectable, lawabiding citzens in Queens watched a killer stalk
and stab a woman in three separate attacks. . . .
Not one person telephoned the police during the
assault”
Pundits’ Reactions
Pundits found this “emblematic of the callousness or
apathy of life in big cities, particularly New York.”
The incident was taken as evidence of ``moral decay’’
and of “dehumanization caused by the urban
environment.”
In Defense of New Yorkers?
• Sociologists, John Darley and Bibb Latane suggested an
alternative theory
– City dwellers might not be “callous” or “dehumanized.” They
know many are present and believe that it is likely that
someone else will act.
• Darley and Latane called this the “bystander-effect.”
They found this effect in lab experiments: Someone
pretended to be in trouble,
– When subjects believed nobody else could help, they did so
with probability .8.
– When they believed that 4 others observed the same events,
they helped with probability .34.
Volunteer’s Dilemma Game
• Andreas Diekmann, a sociologist, created a
game theoretic model, the “Volunteer’s
Dilemma”
• N-player simultaneous move game: Strategies
Act or Not.
– All who act pay C. If at least one acts, those who
acted get B-C.. Those who didn’t act get B. If
nobody acts, all get 0.
• In symmetric mixed strategy Nash equilibrium,
as N increases, it less likely that any one person
calls. In fact, it is more likely that nobody calls.
Volunteers’ Dilemma
N people observe a mugging. Someone needs to
call the police. Only one call is needed. Cost of
calling is c. Cost of knowing that the person is
not helped is T. Should you call or not call?
T>c>0. Many asymmetric pure strategy
equilibria.
Also one symmetric mixed strategy equilibrium.
Mixed strategy equilibrium
• Suppose everybody uses a mixed strategy with probability
p of calling.
• In equilibrium, everyone is indifferent about calling or not
calling if expected cost from not calling equals cost from
calling.
Expected Cost of of not calling is
T(1-p)N-1
• Expected cost of calling is c.
• Equilibrium has c= T(1-p)N-1 so 1-p=(c/T)1/N-1
• Then (1-p)N=(c/T)N/N-1 is the probability that nobody calls.
This is an increasing function of N. So the more
People who observe, the less likely that someone calls.
Defense of New Yorkers
• In what sense does the Volunteer’s dilemma
game suggest that New Yorkers may not be
“callous and dehumanized”?
Further defense of New Yorkers
• Less interesting for theory, but facts deserve
respect.
• Fact-checkers later found the journalists’ story
partly fabricated (albeit by NYC-based
journalists).
– No evidence that 38 people knew what was going
on. It was 3 am on a cold night. Windows were
closed. One person tried to help.
Battle of Sexes
Bob
Movie A
Movie A
Alice
Movie B
BRA(A)=A
BRA(B)=B
Movie B
3,2
1,1
0,0
2,3
BRB(A)=A
BRB(B)=B
Does this game have pure strategy
Nash equilibria?
A) Yes, there is exactly one
B) Yes, there are two
C) No there are no pure strategy Nash equilibria
Mixed Strategy Equilibrium
• Let p be probability Alice goes to movie A and
q the probability that Bob goes to movie B.
• When is there a mixed best response for
Alice?
– Expected payoff for Movie A for Alice is
3(1-q)+ q1=3-2q.
– Expected payoff to Movie B for Alice is
2q+(1-q)0=2q
• Payoffs are the same if 3-2q= 2q, so q=3/4.
Similar for Bob
• From the symmetry of the game, we see that a
mixed strategy is a best response for Bob if
p=3/4.
• In a symmetric mixed strategy, each goes to his or
her favorite movie with probability ¾.
• Probability that they get together at Movie A is
3/4x1/4=3/16. Probability that they get together
at Movie B is also 3/16. Probability that they miss
each other is 5/8. Probability that each goes to
favorite movie is 9/16. Probability that they each
go to less preferred movie is 1/16.
Have a nice weekend!