1 st 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. 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)

Expected utility Preferences over lotteries Notation –(x 1,p 1 ;…;x n,p n )= x 1 with probability p 1 ; … and x n with probability p n –Null outcomes not listed: (x 1,p 1 ) means x 1 with probability p 1 and 0 with probability 1-p 1 –(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 ∑ p i u(x i )

Proof Start with (x 1,p 1 ;x 2,p 2 ) Now –x 1 ~ (b,f(x 1 );w,1-u(x 1 )) –x 2 ~ (b,f(x 2 );w,1-u(x 2 )) Replace x 1 and x 2 by the equally good lotteries and collect terms (x 1,p 1 ;x 2,p 2 ) ~ (b,p 1 u(x 1 )+p 2 u(x 2 ); w,1-p 1 u(x 1 )+p 2 u(x 2 )) 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 ∑  i v(x i -r) ( Replaces Max ∑ p i u(x i ) )

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 function Reflection 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 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 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 probability  i = f(p i ) This formulation violates stochastic dominance and are difficult to generalize to lotteries with many outcomes (p i →0) The standard is thus to use cumulative prospect theory

Rank dependent weights Order the outcome such that x 1 >x 2 >…>x k >0>x k+1 >…>x n Decision weights for gains Decision weights for losses

Cumulative prospect theory Value-function –Concave for gains –Convex for losses –Kink at 0 Decision weights –Adjust cumulative distribution from above and below Maximize

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.