Critiques of Expected Utility Theory

14.123 Microeconomic Theory III Muha met Yildiz

Allais Paradox

 Choose A or B, then C or D.

(A) Win $1 million f o r s u r e .

(B) Win $ 5 M wit h 10% ch anc e, $1 M wit h 89%, nothing with 1%.

(C) Win $1 M wit h 11% ch anc e, no thing wit h 8 9 %.

(D) Win $5 M wit h 10% ch anc e, no thing wit h 9 0 %.

 Choi ce of A and D violates expected utility:

Resolutions

Allais Paradox, Graphically

Pr($ 5) 1

I n d i ff er e n ce cur v e s

∙ B’

A ∙ B

D ∙

0

∙

C

∙

1 Pr($ 0)

“Common consequence” paradox: A  B but D  C .

“Common ratio” paradox: A  B’ but D  C .

Probability Weighting Function

w

1

0

1

p

Ellsberg Paradox

 An urn contains 99 balls, colored, Red , Black and Green

 There are 33 Red balls;

 the combi nation of the other colors i s not known.

 You choose a color and we dr aw a ball.

 If the ball is of the color chosen, you win $1. What color woul d you cho o se?

 If the ball is not of the color chosen, you win $1. What color would you choose?

Resolution: Ambiguity Aversion

 Compounded lotteries are not reduced to simp le lo tteries

 Ambiguity aversion:

max a mi n p E p [ u ( a )]

 Smooth ambiguity aversion:

max a E [ v ( E p [ u ( a )])]

Framing

 “Outbreak of disease is about to kill 600 people. Choose treatment program A or B; then C or D.”

(A) 400 p eop le di e .

(B) No bo dy dies with 1/ 3 c h a n c e , 6 0 0 p e o p l e d i e with 2/3 c h a n c e .

(C ) 200 peop le sa ve d .

(D ) All sa ve d w i t h 1 / 3 cha n c e , n obod y sa ved w i t h 2 / 3 chance.

 78% of subjects pi ck B, 28% of subjects (in different group) pick D. But A is equi valent to C, B is equi v al ent to D (apart from wording).

Prospect Theory

 “Edit the decision problem”

 Distort the probabilities using inverted S shape

 Apply a reference-dependent S shaped utility function

 Ri sk aver sion towards gains

 Ri sk takin g towards losses

 “Loss aver sion”

Prospect Theory

Reference-dependent Utility Function

u

x 0

x

Prospect Theory Formula

 U ( x | w , x ₀ ) = ∫ u ( x | x ₀ ) dw ( F ( x ))

 Properties & Problems:

 What is reference point?

 Fra m ing

 Dynamic Programming