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1st lecture
Probabilities and
Prospect Theory
Probabilities
•
In a text over 10 standard novel-pages,
how many 7-letter words are of the form:
1. _ _ _ _ ing
2. _ _ _ _ _ ly
3. _ _ _ _ _n_
Linda and Bill
• “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 (B)
Linda is an insurance sales person
Linda is a bank teller and is active in the feminist movement (B&F)
• Probability rank:
– Naïve: B&F – 3,3; B – 4,4
– Sophisticated: B&F – 3,2; B – 4,3.
Indirect and Direct tests
• Indirect versus direct
– Are both A&B and A in same questionnaire?
• Transparent
– 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%)
Extensional versus intuitive
• Extensional reasoning
– Lists, inclusions, exclusions. Events
– Formal statistics.
• If A  B , Pr(A) ≥ Pr (B)
• Moreover: ( A & B)  B1. _ _ _ _ ing
• Intuitive reasoning
– Not extensional
– Heuristic
• Availability and Representativity.
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, A  B and hence Pr(B)≥Pr(A)
– But ….ing words are easier to imagine
Watching TV affect our probability assessment of
violent crimes, divorce and heroic doctors. (O’Guinn
and Schrum)
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.
Independence Axiom
• If A ~ B, then (A,p;…) ~ (B,p;…)
• Add continuity: if b(est) > x > w(orst) then
there is a 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 (x1,p1;x2,p2 )
• Now
– x1~ (b,f(x1);w,1-u(x1))
– x2~ (b,f(x2);w,1-u(x2))
• Replace x1 and x2 by the equally good
lotteries and collect terms
• (x1,p1;x2,p2 ) ~ (b,p1u(x1)+p2u(x2); w,1-p1u(x1)+p2u(x2))
• The latter is (b,Eu(x);w,1-Eu(x))
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 ∑ piv(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]*
• 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
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, 0.50)
– 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
pi = 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
Rank dependent weights
• Order the outcome such that
x1>x2>…>xk>0>xk+1>…>xn
• Decision weights for gains
j 1
 j



p j  w   pi   w   pi  for all j  k
 i 1 
 i 1 

• Decision weights for losses
n
n






p j  w   pi   w   pi  for all j  k
 i j 
 i  j 1 
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
 p 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.