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 , Pr(A) ≥ Pr (B)• Moreover:

• Intuitive reasoning– Not extensional– Heuristic

• Availability and Representativity.

BABBA )&( 1. _ _ _ _ ing

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

• Watching TV affect our probability assessment of violent crimes, divorce and heroic doctors. (O’Guinn and Schrum)

BA

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 ∑ iv(xi-r)

( Replaces Max ∑ piu(xi) )

Evidence; Decision weights

• Problem 3– A: (4 000, 0.80) or B: (3 000)– N=95 [20] [80]*

• Problem 4– C: (4 000, 0.20) or D: (3 000, 0.25)– N=95 [65]* [35]

• 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 functionReflection effect

• Problem 3– A: (4 000, 0.80) or B: (3 000)– N=95 [20] [80]*

• Problem 3’– A: (-4 000, 0.80) or B: (-3 000)– N=95 [92]* [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) or B: (500)– N=95 [16] [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) or B: (-500)– N=95 [69]* [31]

• Both equivalent according to EU, but the initial instruction affect the reference point.

Decision weights

• Suggested by Allais (1953).

• Originally a function of probabilityi = 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

• Decision weights for losses

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Cumulative prospect theory

• Value-function– Concave for gains– Convex for losses– Kink at 0

• Decision weights– Adjust cumulative

distribution from above and below

• Maximize

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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.