Download January 20 Lecture Econ 171

Midterm Scores
• 20 points per question
Grade Range
Score
Number of students in
this range
A
100-120
22
B
80-99
13
C
60-79
12
D
40-59
3
F
<40
4
Median Score: 90
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 strategy Nash Equilibrium
• A player using a mixed strategy chooses to
``randomizes’’ between ``pure strategies’’,
assigning a specific probability to taking each
possible pure strategy.
• Assume that if the other player is using a
mixed strategy, your best response is to
choose a strategy that maximizes your
expected payoff.
Matching Pennies:
or Simple hide and seek_
Player 2 (Seeker)
q
Heads
p
Player 1
(Hider)
1-p
Tails
Heads
1-q
Tails
-1,1
1,-1
1,-1
-1, 1
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.
Nash equilibrium in Matching Pennies
Suppose Player 1 randomizes and plays Heads
2/3 of the time, what is Player 2’s best
response?
A) Heads for sure
B) Tails for sure
C) Randomize with Probability of Heads 2/3.
D) Randomize with Probability of Tails 2/3
Mixed strategy as best response
• In a two-player, two-strategy game your best
response is a mixed strategy with positive
probabilities of playing both pure strategies
only if your payoffs from the two pure
strategies are equal.
Mixed strategies for Hide-and-Seek
• Let pH be the probability that hider plays
Heads and 1-pH the probability that hider
plays Tails.
• When would seeker get the same payoff from
playing Heads or Tails?
• Expected payoff to seeker from Heads is
pH×1 +(1- pH) ×( -1)=2 pH -1.
Best response Mapping
Player 2’s Reaction Function (in Red)
1
q=probability
2 chooses H
Player 1’s
Reaction
Function
(in Green)
1/2
0
1/2
1
p= Probability 1 chooses H
Nash equilibrium in Mixed Strategies
• Intersection of Reaction Functions
• Each is doing best response to other’s 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
Seeker’s Choice
q
p
Plains
Hider’s
Choice
1-p
Forest
Plains
1-q
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 Seeker have a mixed strategy best
response. The payoffs to looking in the plains
and looking in the forest must be the same.
• Where p is probability Hider is in the plains,
• 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.
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
Expected Utility Theory of Choice
Under Uncertainty
• Suppose that you face random outcomes. You
assign a “utility” to each possible outcome in
such a way that your choices among uncertain
prospects are those that maximize “expected
utility”.
Expected utility Example:
Utility of money
• Suppose you have a lottery that will with
probability 1/4 win 10 million dollars and
with probability ¾ will be worthless. You get
just one chance to sell your ticket.
Would you sell it for 2.5 million dollars?
A) Yes
B) No
Expected utility Example:
Utility of money
• Suppose you have a lottery that will with
probability 1/4 win 10 million dollars and
with probability ¾ will be worthless. You get
just one chance to sell your ticket.
Would you sell it for 1 million dollars?
A) Yes
B) No
Expected utility Example:
Utility of money
• Suppose you have a lottery that will with
probability 1/4 win 100 million dollars and
with probability ¾ will be worthless. You get
just one chance to sell your ticket.
Would you sell it for 500 thousand dollars?
A) Yes
B) No
Construct a utility scale
• Let u(10 million)=1 Let u(0)=0.
• Then ask question. How much money X for sure
would be just as good as having a ¼ chance of
winning 10 million and ¾ chance of 0?
• Then assign u(X)=(3/4)u(0)+(1/4)u(10,000,000)=
(3/4)0+(1/4)1=1/4.
Assigning utility to any income
• Lets choose a scale where u(0)=0 and u(10
million)=1.
• Take any number X. Find a probability p(X) so
that you would just be willing to pay $X for a
lottery ticket that pays 10 million with
probability p(X) and 0 with probability 1-p(x).
• Assign utility p(X) to having $X.
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*.
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.
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.
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
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!