8/22/2011
ECON4260 Behavioral
Economics
1st lecture
Probabilities, Expected utility
and
Prospect Theory
Probabilities
•
In a text over 10 standard novel-pages, how many
7-letter words are of the form:
1.
2.
3.
_ _ _ _ _n_
_ _ _ _ _ ly
_ _ _ _ ing
Practical Issues
• Dates for seminar on web-page
– Plan for the course and the lectures
– Delayed start due to 200 years anniversary
• Most paper available online
– The rest in a compendium
• For my topic:
–
–
–
–
Relevant lecture indicated in the reading list.
The fist handout will cover more than 1st lecture
This lecture will be mostly PowerPoint
In the three next I’ll mainly use the
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Three main topics
• Decision theory (Lectures 1-4)
• Time preferences (Lectures 5-8)
– 10$ today versus 11$ tomorrow
– 10$ ten days from versus 11$ after 11 days
• Justice / Non-selfish behavior (L 9-13)
– Share 100 kroner with a recipient/responder
– Dictators share
– Responders reject unfair offers
Time preferences / Self control
• It is a good idea to read the papers before the lectures,
and to allocate work evenly over the semester
– Most students know
– Some lack the self control to do it.
• But then:
– Who is the ’self’ if not the student?
– If it is the student, who is the ’self’ controlling?
• Theories of multiple selves
• Enjoyable read (not curriculum): Robert Kurzban (2011)
”Why everyone (else) is a hypocrite” Princeton UP
Study pre-commitment technique
• Suppose at the start of the semester you decide to
– Solve all seminar exercises in advance
– Read all relevant papers on the reading list before each lecture
– Attend all lectures and seminars
• But you know that you (maybe) will not follow through
– And that you will regret as exams are approaching
• Make a contract with another student
– Attend at least 90% of lectures and seminars – have someone to sign.
– Have written answers to 80% of all seminar problem (signed)
– If the contract is not met – give 5000 kroner to an organization that
you disagree strongly with. (FrP? SV? )
• Homo oeconomicus would not need this contract
– Why do we need it?
Institutt for statsvitenskap
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Social preferences
• When you watch someone in pain and when you
yourself is in pain, some of the same neurons
light up in your brain.
• Old wisdom: We share others pain, sorrow,
happiness.
– But may enjoy their pain if they have done us wrong
• Is it then reasonable to assume my utility only
depend on my own consumption?
• Enjoyable read (no curriculum): Franz de Waal
(2009) ”The age of empathy.” Three river press
– Morality evolved from the ability to mimic.
The economic problem
• What is important when we want to understand
the economy?
• Example of social preferences:
– My utility depend on my income and yours (Fehr and Smith)
– Reflect the mirror neurons?
– But yields an even more complex optimization
• What is the ideal model?
– The simplest possible structure that fits data (Gul and Pesendorfer)
– Representing what we know about the brain (Camerer m.fl.)
– Representing the decision making process (Gigerenzer)
References
• Berg, Nathan and Gigerenzer, Gerd (2010). As-If
Behavioral Economics: Neoclassical Economics in
Disguise? History of Economic Ideas, Vol. 18, No. 1,
pp. 133-166, 2010. Available at SSRN:
http://ssrn.com/abstract 1677168
http://ssrn.com/abstract=1677168
• Camerer, Colin F., George Loewenstein and Drazen
Prelec (2004) Neuroeconomics: Why Economics
Needs Brains, Scandinavian Journal of Economics,
Vol 105, pp 555-579.
• Gul, Farok and Wolfgang Pesendorfer (2005): The
Case for Mindless Economics, Princeton
Institutt for statsvitenskap
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Gerd Gigerenzer: Fetching a ball
• What is the best strategy to fetch a ball in the air (e.g.
baseball)
• Optimize
– Estimate where it will land, run as fast as possible, update estimate
with new information
• Use a rule of thumb
– Run to keep the ball at a constant angle.
• Gigerenzer: We use the rule of thumb, and
optimization is in fact an inferior strategy.
Back to decision theory
How to make the optimal decision in
theory
• For each alternative action:
– Make an assessment of the probability distribution of outcomes
– Compute the expected utility associated with each such probability
distribution
– Choose the action that maximize expected utility
• How do people make probability assessment?
Fundamental law of statistics
• If the event A is contained in B then
Pr(A) ≤ Pr(B)
• Example: An urn contains Red,
Red Blue and
Green balls. A ball is drawn at random
Pr(Red OR Blue) ≥ Pr(Red)
• Conjunctions: A&B is contained in B
Pr(A&B) ≤ Pr(B)
• Applies to all alternatives to probability, like
Belief functions and non-additive measures
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Linda
• “Linda is 31 years old, single, outspoken and very bright.
She majored in philosophy. As a student, she was deeply
concerned with issues of discrimination and social justice,
and also participated in anti-nuclear demonstrations.”
–
–
–
–
–
Linda is a teacher in elementary school
Linda is active in the feminist movement (F)
Linda is a bank teller (T)
Linda is an insurance sales person
Linda is a bank teller and is active in the feminist movement (T&F)
• Probability rank (1=most probable):
– Naïve: T&F : 3,3; T : 4,4
– Sophisticated: T&F : 3,2; T : 4,3.
• Conjunction rule implies
– Rank T&F should be lower than T (Less probable)
Bill
• Bill is 34 years old. He is intelligent but unimaginative,
compulsive, and generally lifeless. In school he was
strong in mathematics but weak in social studies and
humanities.
–
–
–
–
–
–
–
–
Bill is a p
physician
y
who p
play
yp
poker for a hobby
y
Bill is an architect
Bill is an accountant (A)
Bill plays jazz for a hobby (J) [Rank 4.5]
Bill surfs for a hobby
Bill is a reporter
Bill is an accountant who play jazz for a hobby (A & J) [Rank 2.5]
Bill climbs mountains for a hobby.
Indirect and Direct tests
• Indirect versus direct
– Are both A&B and A in same questionnaire?
– Paper show that direct and indirect tests yield
roughly the same result.
• Transparent (“increasingly
( increasingly desperate”)
desperate )
– Argument 1: Linda is more likely to be a bank teller than she
is to be a feminist bank teller, because every feminist bank
teller is a bank teller, but some bank tellers are not feminists
and Linda could be one of them (35%)
– Argument 2: Linda is more likely to be a feminist bank teller
than she is likely to be a bank teller, because she
resembles an active feminist more than she resembles a
bank teller (65%)
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Sophistication
• Graduate student social sciences at UCB and
Stanford
• Credit for several statistics courses
– ”Only 36% committed the fallacy”
– Likelihood rank T&F (3.5) < T (3.8) ”for the first time?”
• But:
– Report sophisticated in Table 1.1, no effect
As a lottery
• ”If you could win $10 by betting on an event, which of
the following would you choose to bet on? (check
one)”
– ”Only” 56 % choose T&F over F
Extensional versus intuitive
• Extensional reasoning
– Lists, inclusions, exclusions. Events
– Formal statistics.
• If A  B , Pr(A) ≥ Pr (B)
• Moreover: ( A & B)  B
• Intuitive reasoning
– Not extensional
– Heuristic
• Availability
• Representativity.
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Representative versus probable
• ”It is more representative for a Hollywood actress to
be divorced 4 times than to vote Democratic.” (65%)
• But
• ”Among Hollywood actresses there are more women
who vote Democratic than women who are divorced
4 times.” (83%)
Representative heuristic
• While people know the difference between
representative and probable they are often
correlated
• More probable that a Hollywood actress is divorced
4 times than a the probability that an average
woman is divorced 4 times.
• Thus representativity works as a heuristic for
probability.
Availability Heuristics
•
We assess the probability of an event by the
ease with witch we can create a mental
picture of it.
–
•
Works good most of the time.
Frequency of words
–
–
–
–
A: _ _ _ _ ing
(13.4%)
B: _ _ _ _ _ n _
( 4.7%)
Now,
and hence Pr(B)≥Pr(A)
But ….ing words are easier to imagine
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Predicting Wimbledon.
• Provided Bjørn Borg makes it to the final:
– He had won 5 times in a row, and was perceived
as very strong.
• What is the probability that he will (1
(1=most
most
probable)
– Lose the first set (2.7)
– Lose the first set but win the match (2.2)
• It was easier to make a mental image of
Bjørn Borg winning at Wimbledon, than
losing.
We like small samples to be
representative
•
•
•
Dice with 4 green (G) and two red (R) faces
Rolled 20 times, and sequence recorded
B t on a sequence, and
Bet
d win
i $25 if it appear
1. RGRRR
2. GRGRRR
3. GRRRRR
•
33%
65%
2%
Now most subject avoid the fallacy with the
transparent design
For the seminar
A dice has four Green (G) faces and two Red (R) faces. The dice will be
rolled 20 times, and the result (R or G) will be written down. This will
produce a sequence of 20 letters.
You can choose one of the three short sequences below:
1. RGRRR
2. GRGRRR
3 GRRRRR,
3.
GRRRRR
Suppose that if your chosen sequence appears in the sequence of 20
letters, you would win 500 kroner. Which one of the sequences 1.-3.
would you prefer? “
•
•
•
Ask 4 students each, two sophisticated and two non-sophisticated.
You may collaborate and attend lectures for first year students and
students at intermediate/advanced courses in statistics at the math
department.
Send me the results prior to 3rd lecture.
Økonomisk Institutt
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More varieties
• Doctors commit the conjunction fallacy in medical
judgments
• Adding reasons
– NN h
had
dah
heartt attack
tt k
– NN had a heart attack and is more than 55 years old
• Watching TV affect our probability assessment of
violent crimes, divorce and heroic doctors. (O’Guinn
and Schrum)
The critique from Gigerenser et.al
• The Linda-case provide lots of irrelevant information
• The word ’probability’ has many meanings
– Only some corresponds to the meaning in mathematical statistics.
• We are good at estimating probabilities
– But only in concrete numbers
– Not in abstract contingent probabilities.
Given a positive HIV-test
• Suppose HIV-test has the following quality
– Non-infected have 99.9% probability of negative
– Infected always test positive
– 1 out of 1000 who are tested, are infected.
• If y
you did a HIV-test and g
got a p
positive. What is the
probability that you are in fact infected?
– Write down your answer.
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Suppose we test 1001 persons
• Statistically 1 will be infected and test positive
• Of the 1000 remaining, 99,9% will test negative, and one
will test positive. (on average)
• If you did a HIV-test and got a positive. What is the
probability that you are in fact infected?
– Write down your answer.
An advise
• If you want to learn statistical theory, especially
understand contingent probabilities and Bayesian
updating:
– Translate into concrete numbers
• This will enhance
– Your understanding when you study it, and
– Your ability retain what you have learned 10 years from now.
Expected utility
• Preferences over lotteries
• Notation
– (x1,p1;…;xn,pn)= x1 with probability p1; … and xn with probability pn
– Null outcomes not listed:
• (x1,p1) means x1 with probability p1 and 0 with probability 1-p1
– (x) means x with certainty.
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Independence Axiom
• If A ~ B, then (A,p;L,1-p) ~ (B,p; L,1-p)
• Add continuity:
if b(est) > x > w(orst) then there is a p(=u(x))
p( u(x)) such
that (b,p;w,1-p) ~ (x)
• It follows that lotteries should be ranked according to
Expected utility
Max ∑ piu(xi)
Proof
• Start with any lottery (x1,p1;x2,p2 )
• Now, by continuity:
– x1~ (b,u(x1);w,1-u(x1))
– x2~ (b,u(x2);w,1-u(x2))
• Replace x1 and x2 by the equally good lotteries (IA)
• note that b and w are the only possible outcomes,
– Probability of outcome: Pr(b) = p1u(x1)+p2u(x2)
– (x1,p1;x2,p2 ) ~ (b,p1u(x1)+p2u(x2); w,1-p1u(x1)+p2u(x2))
• Note that Pr(b)=Eu(x)
• Maximizing Eu = max probability of winning b.
Positive linear transforms
• Note that if v(x)=au(x)+b, a>0
• Then
Ev= ∑ piv(xi) = ∑ piau(xi)+∑ pib =a Eu + b
• Maximizing Ev equivalent to maximizing Eu
• Useful in the following
– Can choose u(0)=0
– Use v(x)=u(x)+b, with b=-u(0)
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Prospect theory
• Loss and gains
– Value v(x-r) rather than utility u(x) where r is a reference point.
• Decisions weights replace probabilities
Max ∑ iv(xi-r)
( Replaces Max ∑ piu(xi) )
Evidence; Decision weights
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]*
or
D: (3 000, 0.25)
[35]
• Problem 4
– C: (4 000, 0.20)
– N=95 [65]*
• Violates expected utility
– B better than A :
u(3000) > 0.8 u(4000)
– C better than D: 0.25u(3000) > 0.20 u(4000)
• Perception is relative:
– 100% is more different from 95% than 25% is from 20%
Value function
Reflection effect
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]
[80]*
• Problem 3’
– A: (-4 000, 0.80)
– N=95 [92]*
or
B: (-3 000)
[8]
• Ranking reverses with different sign (Table 1)
• Concave (risk aversion) for gains and
• Convex (risk lover) for losses
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The reference point
• Problem 11: In addition to whatever you own,
you have been given 1 000. You are now
asked to choose between:
– A: (1 000,
000 0
0.50)
50)
– N=95 [16]
or
B: (500)
[84]*
• Problem 12: In addition to whatever you own,
you have been given 2 000. You are now
asked to choose between:
– A: (-1 000, 0.50)
– N=95 [69]*
or
B: (-500)
[31]
• Both equivalent according to EU, but the
initial instruction affect the reference point.
Decision weights
• Suggested by Allais (1953).
• Originally a function of probability
i = f(pi)
• This formulation violates stochastic dominance and
are difficult to generalize to lotteries with many
outcomes (pi→0)
• The standard is thus to use cumulative prospect
theory
Lotto
(Think about this until next week)
• 50% of the money that people spend on
Lotto is paid out as winning prices
• Stylized:
y
– Spend 10 Kroner
– Win 1 million kroner with probability 1 to 200 000
• Would a risk avers expected utility maximizer
play Lotto?
– Is Lotto participation a challenge to expected
utility?
• Can prospect theory explain why people
participate in Lotto?
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Rank dependent weights
• Order the outcome such that
x1>x2>…>xk>0>xk+1>…>xn
• Decision weights for gains

j


j 1


 i 1

 j  w   pi   w    pi  for all j  k
 i 1
• Decision weights for losses

n



 i  j 1 
n

 j  w    pi   w    pi  for all j  k
 i j
Cumulative prospect theory
• Value-function
– Concave for gains
– Convex for losses
– Kink at 0
• Decision weights
– Adjust cumulative
distribution from above
and below
• Maximize
n
  v( x )
i 1
i
i
Main difference between CPT and
EU
• Loss aversion
– Marginal utility twice as large for losses compared
to gains
• Certainty effects
– 100% is distinctively different from 99%
– 49% is about the same as 50%
• Reflection
– Risk seeking for losses
– Risk aversion form gains.
– Most risk avers when both losses and gains.
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