Transcript Psychology and Economics

Making Better Decisions
Itzhak Gilboa
1
Judgment and Decision
Biases
2
Problems 2.1, 2.12
A 65-year old relative of yours suffers from a
serious disease. It makes her life miserable,
but does not pose an immediate risk to her
life. She can go through an operation that, if
successful, will cure her. However, the
operation is risky. (A: 30% of the patients
undergoing it die. B: 70% of the patients
undergoing it survive.) Would you
recommend that she undergoes it?
3
Framing Effects
• Representations matter
• Implicitly assumed away in economic
theory
• Cash discount
• Tax deductions
• Formal models
4
Problems 2.2, 2.13
A: You are given $1,000 for sure. Which of the following
two options would you prefer?
a. to get additional $500 for sure;
b. to get another $1,000 with probability 50%, and
otherwise – nothing more (and be left with the first
$1,000 ).
B: You are given $2,000 for sure. Which of the following
two options would you prefer?
a. to lose $500 for sure;
b. to lose $1,000 with probability 50%, and otherwise – to
lose nothing.
5
Problem 2.2, 2.13
In both versions the choice is between:
a.
$1,500 for sure;
b.
$1,000 with probability 50%,
and $2,000 with probability 50%.
Framing
6
Gain-Loss Asymmetry
• Loss aversion
• Relative to a reference point
• Risk aversion in the domain of gains, but
evidence of loss aversion in the domain of
losses
7
Is it rational to fear losses?
Three scenarios:
• The politician
• The spouse
• The self
The same mode of behavior may be rational
in some domains but not in others
8
Endowment Effect
What’s the worth of a coffee mug?
• How much would you pay to buy it?
• What gift would be equivalent?
• How much would you demand to sell it?
Should all three be the same?
9
Standard economic analysis
Suppose that you have m dollars and 0
mugs
• How much would you pay to buy it?
(m-p,1)~(m,0)
• What gift would be equivalent?
(m+q,0)~(m,1)
• How much would you demand to sell it?
(m,1)~(m+q,0)
10
Standard analysis of mug
experiment
• How much would you pay to buy it?
• What gift would be equivalent?
• How much would you demand to sell it?
The last two should be the same
11
Results of mug experiment
•
How much would you pay to buy it?
$2.87
•
What gift would be equivalent?
$3.12
•
How much would you demand to sell it?
$7.12
12
The Endowment Effect
• We tend to value what we have more than
what we still don’t have
• A special case of “status quo bias”
• Related to the “disposition effect”
13
Is it rational
… to value
a house
your grandfather’s pen
your car
equity
… just because it’s yours?
14
Rationalization of Endowment
Effect
• Information (used car)
• Stabilization of choice
• Habits and efficiency (word processor)
15
Problem 2.3, 2.14
• 2.3: You go to a movie. It was supposed to be
good, but it turns out to be boring. Would you
leave in the middle and do something else
instead?
• 2.14 reads: Your friend had a ticket to a movie.
She couldn’t make it, and gave you the ticket
“instead of just throwing it away”. The movie
was supposed to be good, but it turns out to be
boring. Would you leave in the middle and do
something else instead?
16
Sunk Cost
•
Cost that is “sunk” should be ignored
•
Often, it’s not
•
Is it rational?
17
Decision Tree – Problem 2.3
Stay Home
Buy Ticket
Boring
Leave
18
Interesting
Stay
Decision Tree – Problem 2.14
Stay Home
Accept Free Ticket
Boring
Leave
19
Interesting
Stay
Consequentialism
• Only the consequences, or the rest of the
tree, matter
• Helps ignore sunk costs if we so wish
• Does it mean we’ll be ungrateful to our old
teachers?
20
Problem 2.4, 2.15
Linda is 31 years old… etc.
How many ranked
f. Linda is a bank teller
Below
h. Linda is a bank teller who is active in a feminist
movement ?
21
Representativeness Heuristic
The conjunction fallacy
Many explanations:
– “a bank teller” – “a bank teller who is not
active”?
– Ranking propositions is not a very natural task
Still, the main point remains: heuristics can
be misleading
22
Problems 2.5, 2.16
A:
In four pages of a novel (about 2,000 words)
in English, do you expect to find more than ten
words that have the form _ _ _ _ _n _ (sevenletter words that have the letter n in the sixth
position)?
B:
In four pages of a novel (about 2,000 words)
in English, do you expect to find more than ten
words that have the form _ _ _ _ ing (sevenletter words that end with ing)?
23
Availability Heuristic
In the absence of a “scientific” database, we
use our memory
Typically, a great idea
Sometimes, results in a biased sample
24
Problems 2.6, 2.17
A: What is the probability that, in the next 2 years,
there will be a cure for AIDS?
B: What is the probability that, in the next 2 years,
there will be a new genetic discovery in the
study of apes, and a cure for AIDS?
Availability heuristic
25
Problems 2.7, 2.18
A: What is the probability that, during the next year, your car would be
a "total loss" due to an accident?
B: What is the probability that, during the next year, your car would be
a.
b.
c.
d.
e.
a "total loss" due to:
an accident in which the other driver is drunk?
an accident for which you are responsible?
an accident occurring while your car is parked on the street?
an accident occurring while your car is parked in a garage?
one of the above?
Availability heuristic
26
Problems 2.8, 2.19
Which of the following causes more deaths
each year:
a. Digestive diseases
b. Motor vehicle accidents?
Availability heuristic
27
Problems 2.9, 2.20
A newly hired engineer for a computer firm in
Melbourne has four years of experience
and good all-around qualifications.
Do you think that her annual salary is above
or below [A: $65,000; B: $135,000 ]?
______
What is your estimate?
28
Anchoring Heuristic
• In the absence of solid data, any number
can be used as an “anchor”
• Is it rational?
• Can be used strategically
29
Problems 2.10, 2.21
• A: You have a ticket to a concert, which cost you
$50. When you arrive at the concert hall, you
find out that you lost the ticket. Would you buy
another one (assuming you have enough money
in your wallet)?
• B: You are going to a concert. Tickets cost $50.
When you arrive at the concert hall, you find out
that you lost a $50 bill. Would you still buy the
ticket (assuming you have enough money in
your wallet)?
30
Mental Accounting
• Different expenses come from “different”
accounts.
• Other examples:
– Your spouse buys you the gift you didn’t
afford
– You spend more on special occasions
(moving; travelling)
– Driving to a game in the snow (wouldn’t do it if
the tickets had been free)
31
Is it rational?
Mental accounting can be useful:
– Helps solve a complicated budget problem
top-down
– Helps cope with self-control problems
– Uses external events as memory aids
… but it can sometimes lead to decisions we
won’t like (driving in the snow…)
32
Problems 2.11, 2.22
A: Which of the following two options do you
•
•
prefer?
Receiving $10 today.
Receiving $12 a week from today.
B: Which of the following two options do you
prefer?
• Receiving $10 50 weeks from today.
• Receiving $12 51 weeks from today.
33
Discounting
• Classical:
U(c1,c2,…) = [1/(1- δ )] * ∑i δi u(ci)
-- violated in the example above.
• Hyperbolic: more weight on period no. 1
34
Summary
• Many of the classical economic
assumptions are violated
• Some violations are more rational than
others
• Formal models may help us decide which
modes of behavior we would like to
change
35
A catalogue of pitfalls
•
•
•
•
•
•
•
•
36
Framing effects
Mental accounting
Endowment effect
Representativeness heuristic
Availability heuristic
Anchoring
Loss aversion and gain/loss asymmetry
Non-stationary discounting
Consuming Statistical Data
37
Problem 3.1
A newly developed test for a rare disease has the
following features: if you do not suffer from the
disease, the probability that you test positive
("false positive") is 5%. However, if you do have
the disease, the probability that the test fails to
show ("false negative") is 10%.
You took the test, and, unfortunately, you tested
positive. The probability that you have the
disease is:
38
The missing piece
The a-priori probability of the disease,
P (D) = p
Intuitively, assume that p=0 vs. p=1
39
Conditional probabilities
p
1- p
D
.90
T
.90 p
40
~D
.10
.05
~T
.10 p
T
.05(1- p)
.95
~T
.95(1- p)
The calculation
P(D|T)
=P(D∩T)/P(T)
= .90 p / [.90 p +.05(1- p) ]
with p=P(D)
… can be anywhere between 0 and 1 !
41
For example…
If, say, P(D)=.01,
P(D|T) = .01*.90 / [.01*.90 + .99*.05 ] =
= .009 / [ .009 + .0495 ] =.009 / .0585 = 15.3%
P(D|T)>P(D)
but
P ( D | T ) < 50%
42
The frequency story
100 sick
9,900 healthy
43
The frequency story cont.
495 healthy,
positive
90 sick,
positive
10 sick,
negative
44
9,405 healthy,
negative
Ignoring Base Probabilities
P ( A | B ) = [P(A)/P(B)] P ( B | A )
In general,
P(B|A) ≠ P(A|B)
45
Why do we get confused?
P ( A | B ) > P ( A | ~B )
is equivalent to
P ( B | A ) > P ( B | ~A )
Correlation is a symmetric relation; still:
P(B|A) ≠ P(A|B)
46
Social prejudice
It is possible that:
Most top squash players are Pakistani
but
Most Pakistanis are not top squash players
47
Problem 3.2
You are going to play the roulette. You first
sit there and observe, and you notice that
the last five times it came up "black."
Would you bet on "red" or on "black"?
48
The Gambler’s Fallacy
• If you believe that the roulette is fair, there
is independence
• By definition, you can learn nothing from
the past about the future
• Law of large numbers: errors do not get
corrected, they get diluted.
49
Errors are “diluted”
• Suppose we observe 1000 blacks
• The prediction for the next 1,000,000 will
still be
500,000 ; 500,000
• Resulting in
501,000 ; 500,000
50
Maybe the roulette is not fair?
• Indeed, we will have to conclude this after,
say, 1,000,000 blacks.
• But then we will expect black, not red
• We assume independence given the
parameter of the roulette wheel, but we
can learn from observations about this
parameter
51
Problem 3.3
• A study of students’ grades in the US
showed that immigrants had, on average,
a higher grade point average than did USborn students. The conclusion was that
Americans are not very smart, or at least
do not work very hard, as compared to
other nationalities.
52
Biased Samples
• The point: immigrants are not necessarily
representative of the home population
• The Literary Digest 1936 fiasco
• Students who participate in class
• Citizens who exercise the right to vote
53
Problem 3.4
• In order to estimate the average number of
children in a family, a researcher sampled
children in a school, and asked them how
many siblings they had. The answer, plus
one, was averaged over all children in the
sample to provide the desired estimate.
54
Inherently Biased Samples
Here the very sampling procedure
introduces a bias.
A family of 8 children has 8 times higher
chance of being sampled than a family of
1.
(8 * 8 + 1 * 1)/9 = 7.22 > (8 + 1) /2 = 4.5
55
Problem 3.5
• A contractor of small renovation projects
submits bids and competes for contracts.
He noticed that he tends to lose money on
the projects he runs. He started
wondering how he can be so
systematically wrong in his estimates.
56
The Winner’s Curse
• Firms that won auctions tended to lose
money
• Even if the estimate is unbiased ex-ante, it
is not unbiased ex-post, given that one
has one the auction.
• If you won the auction, it is more likely that
this was one of your over-estimates rather
than one of your under-estimates.
57
Problem 3.6
Ann: “Do you like your dish?”
Barbara: “Well, it isn’t bad. Maybe not as
good as last time, but…”
If the restaurant isn’t so new, how would you
explain it?
58
Regression to the Mean
Regressing X(t+1) on X(t)
59
Regression to the Mean – cont.
• We should expect an increasing line
• We should expect a slope < 1
• Students selected by grades
• Your friend’s must-see movie
• Selection brokers; electing politicians
60
Problem 3.7
Studies show a high correlation between
years of education and annual income.
Thus, argued your teacher, it’s good for
you to study: the more you do, the more
money you will make in the future.
61
Correlation and Causality
Possible reasons for correlation between X
and Y:
• X is a cause of Y
• Y is a cause of X
• Z is a common cause of both X and Y
• Coincidence (should be taken care of by statistical
significance)
62
Problem 3.8
• In a recent study, it was found that people
who did not smoke at all had more visits to
their doctors than people who smoked a
little bit. One researcher claimed,
“Apparently, smoking is just like
consuming red wine – too much of it is
dangerous, but a little bit is actually good
for your health!”
63
Correlation and Causality –
cont.
Other examples:
• Do hospitals make you sick?
• Will a larger hand improve the child’s
handwriting?
64
Problem 3.9
Daniel: “Fine, it’s your decision. But I tell
you, the effects that were found were
insignificant.”
Charles: “Insignificant? They were
significant at the 5% level!”
65
Statistical Significance
• Means that the null hypothesis can be
rejected, knowing that, if it were true, the
probability of being rejected is quite low
• Does not imply that the null hypothesis is
wrong
• Does not even imply that the probability of
the null hypothesis is low
66
The mindset of
hypotheses testing
• We wish to prove a claim
• We state as the null hypothesis, H0, its
negation
• By rejection the negation, we will “prove”
the claim
67
The mindset of
hypotheses testing – cont.
• A test is a rule, saying when to say “reject”
based on the sample
• Type I error: rejecting H0 when it is, in fact,
true
• Type II error: failing to reject H0 when it is,
in fact, false
68
The mindset of
hypotheses testing – cont.
• What is the probability of type I error?
– Zero if the null hypothesis is false
– Typically unknown if it is true
– Overall, never known.
• So what is the significance level, α ?
– The maximal probability possible (over all
values consistent with the null hypothesis)
69
The mindset of
hypotheses testing – cont.
• We never state the probability of the null
hypothesis being true
• Neither before nor after taking the sample
• This would depend on subjective judgment
that we try to avoid
70
Problem 3.10
Mary: “I don’t get it. It’s either or: if you’re so
sure it’s OK, why isn’t it approved? If it’s
not yet approved, it’s probably not yet OK.”
71
Classical and Bayesian
Statistics
• Bayesian:
– Quantify everything probabilistically
– Take a prior, observe data, update to a
posterior
– Can treat an unknown parameter, µ, and the
sample, X, on equal ground
– A priori beliefs about the unknown parameter
are updated by Bayes rule
72
Classical and Bayesian
Statistics – cont.
• Classical:
– Probability exists only given the unknown
parameter
– There are no probabilistic beliefs about it
– µ is a fixed number, though unknown
– X is a random variable (known after the
sample is taken)
– Uses “confidence” and “significance”, which
are not “probability”
73
Classical and Bayesian
Statistics – cont.
• Why isn’t “confidence” probability?
• Assume that X ~ N (µ,1)
Prob ( |X - µ| ≤ 2 ) = 95%
• Suppose
X=4
• What is
Prob ( 2 ≤ µ ≤ 6 ) = ?
74
Classical and Bayesian
Statistics – cont.
• The question is ill-defined, because µ is
not a random variable. Never has been,
never will.
• The statement
Prob ( |X - µ| ≤ 2 ) = 95%
is a probability statement about X, not
about µ
75
Classical and Bayesian
Statistics – cont.
• If Y is the outcome of a roll of a die,
Prob ( Y = 4 ) = 1/6
• But we can’t plug the value of Y into this,
whether Y=4 or not.
76
Why do we use
Classical Statistics?
• Suppose we were Bayesian
• Court analogy: the null hypothesis is that
the defendant is innocent
• Do you want to be judged based on the
judge’s hunch and intuition?
77
Different Methods for
Different Goals
Classical
Bayesian
Goal
To be objective
To express also
subjective biases and
intuition
For
Making statements in a
society
Making the best decision
for oneself
Analogous to
Rules of evidence
Self-help tool
To be used when you try
To make a point
To make a decision
78
Decision Under Risk
79
Problems 4.1 and 4.6
Problem 4.1
P=(1,500)
Q=(.5, 0 ; .5, 1000)
P
Q
Problem 4.6
.60
400
80
.40
P
.60
.40
400
Q
Problems 4.2 and 4.7
Problem 4.2
P=(1,500)
Q=(.2, 0 ; .8, 1000)
P
Q
Problem 4.7
.50
400
81
.50
P
.50
.50
400
Q
Problems 4.3 and 4.8
Problem 4.3
P=(.5, 2000 ; .5, 4000)
Q=(.5, 1000 ; .5, 5000)
P
Q
Problem 4.8
82
.60
.40
.60
6000
P
6000
.40
Q
Problems 4.4 and 4.9
Problem 4.4
P=(.5, 2000 ; .5, 4000)
Q=(.4, 1000 ; .6, 5000)
P
Q
Problem 4.9
83
.60
.40
.60
6000
P
6000
.40
Q
Problems 4.5 and 4.10
Problem 4.5
P=(.2, 0 ; .8, 4000)
Q=(1, 3000)
P
Q
Problem 4.10
84
.75
.25
0
P
.75
.25
0
Q
The Independence Axiom
The choice
P
Q
Should be the same as
1-α
R
85
α
P
1-α
α
R
Q
The Independence Axiom and
dynamic consistency
Compare
α
1-α
R
and
P
1-α
R
86
α
P
1-α
α
R
Q
Q
The Independence Axiom
as a formula
• The preference between two lotteries P
and Q is the same as between
( α, P ; (1- α), R )
and
( α, P ; (1- α), R )
87
von-Neumann Morgenstern’s
Theorem
• A preference order ≥ over lotteries (with
known probabilities) satisfies:
– Weak order (complete and transitive)
– Continuity
– Independence
IF AND ONLY IF
• It can be represented by the maximization
of the expectation of a “utility” function
88
Expected Utility
• Suggested by Daniel Bernoulli in the 18th
century
• A lottery
(p1,x1; … ; pn, xn)
is evaluated by the expectation of the
utility:
p1*u(x1) + … + pn*u(xn)
89
Implications of the Theorem
• Descriptively: it is perhaps reasonable that
maximization of expected utility is a good
model of people’s behavior
• Normatively: Maybe we would like to
maximize expected utility even if we don’t
do it anyway
90
Calibration of Utility
• If we believe that a decision maker is an
EU maximizer, we can calibrate her utility
function by asking, for which p is
( 1, $500 )
Equivalent to
( (1-p), $0 ; p, $1,000 )
91
Calibration of utility – cont.
• If, for instance,
u( $0 ) = 0
u( $1,000 ) = 1
Then
u( $500 ) = p
92
Problem 4.1
• Do you prefer $500 for sure or
( .50, $0 ; .50, $1,000 ) ?
• Preferring 500 for sure indicates risk
aversion.
• Defined as preferring E(X) to X for any
random variable X.
93
Risk aversion
• Is equivalent to the function u being
concave
94
Ample evidence that
• The independence axiom fails in examples
such as 4.5 and 4.10.
• Why?
• A version of Allais’ paradox
• Kahneman and Tversky: Certainty Effect
95
Do People Satisfy the
Independence Axiom?
• Compare your choices in problems 4.5
and 4.10.
• The Independence Axiom suggests that
you make the same choices in both.
96
Another example
• Do you prefer
$1,000,000 with probability 0.8
or
$2,000,000 with probability 0.4 ?
• How about
$1,000,000 with probability 0.0008
or
$2,000,000 with probability 0.0004 ?
97
Prospect Theory
Two main components:
• People exhibit gain-loss asymmetry
• People “distort” probabilities
98
Insurance and gambling
• How can we explain both?
• One idea:
99
Insurance and gambling – cont.
• How come all these people are around the
inflection point of their utility function?
• To be honest, it’s not clear that the
outcomes are properly defined
• Yet: maybe the inflection point moves
around with their wealth?
100
Gain-Loss Asymmetry
• Positive-negative asymmetry in
psychology
• A “reference point” relative to which
outcomes are defined
• “Prospects” as opposed to “lotteries”
101
Probability Distortion
• Small probabilities appear larger than they
are in terms of their “decision weights”
• State lotteries and insurance
• A probability distortion function
– (Different for losses and for gains)
102
The probability distortion
function
1
1
103
p
Decision Under Uncertainty
104
Problems 5.1 and 5.6
• A: It will snow on February 1st
B: A roulette wheel yields the outcome 3
vs.
• A: It will not snow on February 1st
B: A roulette wheel yields an outcome
different than 24
105
Calibration of probability
• To assess our subjective probabilities, we
can compare them to “objective” ones.
• Our willingness to bet is a concrete,
behavioral way to elicit our beliefs.
• This is similar to the calibration of utility.
106
Incompatible
with probabilistic beliefs
• (A) in 5.1:
Prob (snow) > Prob (outcome 3) = 1/37
or
Prob (no-snow) = 1 – Prob (snow) < 36/37
• But (A) in 5.6:
Prob (no-snow) >
Prob (outcome different from 24) = 36/37
107
Problems 5.2 and 5.7
• A: The student next to you gets A in this
class
B: Two consecutive tosses of a fair coin
come up Head
vs.
• A: The student next to you gets less than
A in this class
B: A roulette wheel yields an outcome less
than 24
108
Problems 5.3 and 5.8
• A: Your next flight is delayed by more than
an hour
B: A roulette wheel yields an outcome in
the range 0-5
vs.
• A: Your next flight is on time
B: In two consecutive tosses of a fair coin
there is at least one Head
109
Problems 5.4 and 5.9
• A: The CAC-40 ≥ current value at the end of the
year
B: A toss of a fair coin is Head
vs.
• A: The CAC-40 < current value at the end of the
year
B: A toss of a fair coin is Head
110
Problems 5.5 and 5.10
• A: The next president of the US is a Democrat
B: A toss of a fair coin is Head
vs.
• A: The next president of the US is a Republican
B: A toss of a fair coin is Head
111
Problem 5.11
Putting it all together
• A.
$1,000
$2,000
-$1,000
$0
if
if
if
if
1% < Δ
0
< Δ ≤ 1%
-0.5% < Δ ≤ 0
Δ ≤ -0.5%
or
• B. -$1,000
$1,000
$500
$2,000
if
if
if
if
0.8% < Δ
-0.1% < Δ ≤ 0.8%
-0.7% < Δ ≤ -0.1%
Δ ≤ -0.7%
112
Problem 5.12
• First, let’s convince ourselves that “switch”
wins with probability 2/3.
• Suppose we named door A.
• If the car really is behind A, we lost.
• If the car is behind B, Monty opens C.
“Switching” means “switching to B” and we
win.
• Similarly if the car is behind C.
113
Problem 5.12 – cont.
• We conclude that
114
Car is…
A
B
C
Stick
1
0
0
Switch
0
1
1
Problem 5.12 – cont.
• To be convinced:
– Assume that there were 1000 doors (and he
has to open 998)
– Realize that “switching” is not always to the
same door
– Monty Hall “helps” us by ruling out one of the
bad choices we could have made.
115
But…
• What’s wrong with the simple calculation:
State (car is
behind…)
116
A
B
C
Prior
1/3
1/3
1/3
Posterior
(after
opening B)
1/2
0
1/2
The problem
• The state space above {A,B,C} is not rich
enough
• Assumptions that are hidden in the
definition of the states will never be
challenged by Bayesian updating
• In particular, the way we get information,
and the fact that we know something, may
be informative in itself.
117
A correct Bayesian analysis
• Suppose we named door A
• Two sources of uncertainty
– Where the car really is
– Which door Monty Hall opens
• Thus, there are 9 states of the world, not
3:
118
The states of the world
MH opens/
car is
behind…
A
B
C
Marginal
119
A
B
C
Marginal
The states of the world
MH opens/
car is
behind…
A
0
B
0
C
0
Marginal
120
A
B
C
Marginal
The states of the world
MH opens/
car is
behind…
A
0
B
0
C
0
Marginal
121
A
B
C
0
0
Marginal
The states of the world
MH opens/
car is
behind…
A
0
B
0
C
0
Marginal
122
A
B
C
Marginal
1/3
0
1/3
0
1/3
1
The states of the world
MH opens/
car is
behind…
B
C
Marginal
A
0
B
0
0
1/3
1/3
C
0
1/3
0
1/3
Marginal
123
A
1/3
1
The states of the world
MH opens/
car is
behind…
A
B
C
Marginal
A
0
1/6
1/6
1/3
B
0
0
1/3
1/3
C
0
1/3
0
1/3
0
1/2
1/2
1
Marginal
124
Conditioning on “not B”
MH opens/
car is
behind…
A
B
C
Marginal
A
0
1/6
1/6
1/3
B
0
0
1/3
1/3
C
0
1/3
0
1/3
0
1/2
1/2
1
Marginal
125
Conditioning on “not B”
MH opens/
car is
behind…
A
B
C
Marginal
A
0
1/6
1/6
(1/3)/(2/3)
=1/2
B
0
0
1/3
1/3
C
0
1/3
0
(1/3)/(2/3)
=1/2
0
1/2
1/2
1
Marginal
126
Conditioning on “MH opened B”
MH opens/
car is
behind…
A
B
C
Marginal
A
0
1/6
1/6
1/3
B
0
0
1/3
1/3
C
0
1/3
0
1/3
0
1/2
1/2
1
Marginal
127
Conditioning on “MH opened B”
MH opens/
car is
behind…
A
B
C
Marginal
A
0
(1/6)/(1/2)
=1/3
1/6
1/3
B
0
0
1/3
1/3
C
0
(1/3)/(1/2)
=2/3
0
1/3
0
1/2
1/2
1
Marginal
128
And what if…
MH opens/
car is
behind…
A
B
C
Marginal
A
0
α
1/3- α
1/3
B
0
0
1/3
1/3
C
0
1/3
0
1/3
Marginal
129
0
1/3+ α 2/3- α
1
It is important, however…
MH opens/
car is
behind…
A
B
C
Marginal
A
0
1/6
1/6
1/3
B
1/6
0
1/6
1/3
C
1/6
1/6
0
1/3
1/3
1/3
1/3
1
Marginal
130
Problem 5.13
• It appears like cutting prices is dominant:
State of world/
act
Cut Prices
L,L
Competitor
doesn’t cut
prices
L,H
Not Cut Prices
H,L
H,H
131
Competitor
cuts prices
And yet…
• Would you do it for real money?
• If not, how would you justify it?
132
An alternative analysis
• The states of the world should reflect all
possible causal relationships
• In order to assume nothing a-priori, we
define states as functions from acts to
outcomes:
133
The states of the world
In reply to…
Cut
Not Cut
State 1
Cut
Cut
State 2
Cut
Not Cut
State 3
Not Cut
Cut
State 4
Not Cut
Not Cut
134
The new decision matrix
State 1
State 2
State 3
State 4
(C,C)
(C,NC)
(NC,C)
(NC,NC)
Cut
L,L
L,L
L,H
L,H
Not Cut
H,L
H,H
H,L
H,H
135
Important lessons
• The states of the world should specify all
the information that might be relevant,
including how information was given.
• The states of the world should allow for all
causal relationships.
• Other examples: political discussions
136
Problem 5.14
(Ellsberg’s Two-Urn Paradox)
• How many exhibit any preference between
betting on Urn A vs. Urn B?
• Of those – which did you prefer?
137
Evidence
• A significant proportion of decision makers
prefer to bet with known probabilities
rather than with unknown ones.
• Knight (1921) distinguished “risk” from
“uncertainty”
• Uncertainty/Ambiguity aversion
138
Problem 5.15
(Ellsberg’s Single-Urn Paradox)
• Many prefer betting on red to betting on
blue (or yellow)
• But also betting on not-red to betting on
not-blue (or not-yellow)
• In both cases, the bets defined by red
have known probabilities, as opposed to
those defined by blue (or yellow)
139
Is this rational?
Red
Blue
Yellow
Red
100
0
0
Blue
0
100
0
Not red
0
100
100
Not blue
100
0
100
140
Focus on Red and Blue
Red
Blue
Yellow
Red
100
0
0
Blue
0
100
0
Not red
0
100
100
Not blue
100
0
100
141
Why should preferences depend on
the outcome for Yellow?
Red
Blue
Yellow
Red
100
0
0
Blue
0
100
0
Not red
0
100
100
Not blue
100
0
100
142
Let us change the outcome for
Yellow
Red
Blue
Yellow
Red
100
0
100
Blue
0
100
100
Not red
0
100
100
Not blue
100
0
100
143
The Sure Thing Principle
(Savage)
• We compare acts, which are functions
from states of the world to outcomes
• If two acts are equal on a certain event, A,
then the preferences between them should
only depend on their values off A (on ~A).
• Or: if we change them on A, but keep
them equal to each other on A,
preferences between them shouldn’t
change.
144
Back to Problem 5.14
• In order to see that the Sure Thing
Principle is violated in this problem, we
first have to define the state space
• Recall that states assign outcomes to acts
• From each urn we may draw either a red
or a black ball
• Hence there are four states of the world:
145
The states of the world
State 1
Ball drawn out
of Urn A
red
Ball drawn out
of Urn B
red
State 2
red
black
State 3
black
red
State 4
black
black
146
The decision matrix
AR
State 1
(RR)
100
State 2
(RB)
100
State 3
(BR)
0
State 4
(BB)
0
AB
0
0
100
100
BR
100
0
100
0
BB
0
100
0
100
147
Compare AR and BB
AR
State 1
(RR)
100
State 2
(RB)
100
State 3
(BR)
0
State 4
(BB)
0
AB
0
0
100
100
BR
100
0
100
0
BB
0
100
0
100
148
Preference between AR and BB
should not change…
AR
State 1
(RR)
100
State 2
(RB)
0
State 3
(BR)
100
State 4
(BB)
0
AB
0
0
100
100
BR
100
0
100
0
BB
0
0
100
100
149
But then we get BR and AB…
AR
State 1
(RR)
100
State 2
(RB)
0
State 3
(BR)
100
State 4
(BB)
0
AB
0
0
100
100
BR
100
0
100
0
BB
0
0
100
100
150
Violations of EUT
• Ellsberg’s examples are violations of
expected utility theory
• Clearly, summation of utilities weighted by
probabilities will satisfy the Sure Thing
Principle
• But even other theories will
• Ellsberg shows problems with
summarizing information in probability
151
Realistic examples
•
•
•
•
152
… are even more complicated
Ellsberg’s urns have symmetries
Real life often doesn’t
The basic lesson: often we don’t have
enough probability to generate a
probabilistic belief
So what do we do?
• One alternative: have a set of priors
instead of one
• How do we make decisions with a set of
priors?
153
Decisions with sets of priors
• Simple possibility: take worst case
• C – a set of probability measures
• f – an act
U (f) = min pεC ∑ωεΩ p(ω)u(f(ω))
154
Comments
• The minimum reflects uncertainty aversion
• It might be extreme and you can use other
alternatives
• In particular, using a set C which is a
subset of the probability measures actually
possible (based on your information).
155
Comments
• The minimum reflects uncertainty aversion
• It might be extreme and you can use other
alternatives
• In particular, using a set C which is a
subset of the probability measures actually
possible (based on your information).
156
Problem 5.16
• How can my doctor figure out the
probability of success in my case?
• Aren’t we all different?
157
Logistic Regression
P(Y=1) = F ( β0 + β1 X1 + … + βn Xn )
The X’s – explanatory variables
F – a distribution function
158
Similarity-weighted
empirical frequency
• We want to rely on relative frequencies in the
past
• But to give more similar cases more weight
s( successes)

p
 s(all cases)
159
Extreme special cases
• All past cases equally similar – simple
empirical frequency
• Only identical cases are similar to degree
1 (others – 0) – conditional empirical
frequency
160
Problem 5.17
• But what about war?
• As in the medical example, each case is
different
• A new problem: cases are not causally
independent.
161
Well Being and Happiness
162
Problem 6.1 and 6.4
• Mary’s direct boss just quit, and you’re looking
for someone for the job. You don’t think that
Mary is perfect for it. By contrast, Jane seems a
great fit. But it may be awkward to promote
Jane and make Mary her subordinate. A
colleague suggested that you go ahead and do
this, but give both of them a nice raise to solve
the problem.
163
Money and Well-Being
• Money isn’t everything
– Low correlation between income and wellbeing
– The relative income hypothesis
– Higher correlation within a cohort than across
time
– Relative to aspiration level
164
Other Determinants of Well-Being
• Love, friendship
• Social status
• Self fulfillment
– So how should we measure “success”?
165
Problem 6.2
• As we’re nearing the end of this book, it is time to get
some feedback. Please answer the following questions:
• Did you find the explanations clear?
• Did you find the topics interesting?
• Did you find the graphs well done?
• On a scale of 0-10, your overall evaluation for the book
is:
166
Problem 6.5
• As we’re nearing the end of this book, it is time to get
some feedback. Please answer the following questions:
• On a scale of 0-10, your overall evaluation for the book
is:
• Did you find the explanations clear?
• Did you find the topics interesting?
• Did you find the graphs well done?
167
Subjective Well Being
• A common measure
• Prone to manipulations
– Dates and overall well-being
– The weather: “deducting” irrelevant effects
• Kahnemans’ Day Reconstruction Method
168
Problems 6.3 and 6.6
Robert is on a ski vacation with his wife, while
John is at home. He can’t even dream of a ski
vacation with the two children, to say nothing of
the expense. In fact, John would be quite happy
just to have a good night sleep.
Do you think that Robert is happier than John?
169
What’s Happiness?
• Both subjective well being and day
reconstruction would suggest that Robert is
happier
• And yet…
• What’s happiness for you?
• How should we measure happiness for social
policies?
170