paradoxes in decision making with a solution. lottery 1 $3000 s1 $4000 $0 80% 20% r1 80%20%

Paradoxes in Decision Making

With a Solution

Lottery 1

$3000

S1

$4000 $0

80% 20%

R1

80% 20%

Lottery 2

$3000 $0

25% 75%

S2

$4000 $0

20% 80%

R2

Lottery 2

$3000 $0

25% 75%

S2

$4000 $0

20% 80%

R2

35% 65%

Lottery 3

$1,000,000

S3

$5,000,000 $1,000,000 $0

10% 89% 1%

R3

Lottery 4

$1,000,000 $0

11% 89%

S4

$5,000,000 $0

10% 90%

R4

Lotteries 3 and 4

60% migration from S3 to R4

Is this a problem???

Allais Paradox (1953)

Violates “Independence of Irrelevant Alternatives” Hypothesis

(or possibly reduction of compound lotteries)

Example: Offered in restaurant Chicken or Beef

order Chicken.Given additional option of Fish

order Beef

Restatement - Lottery 1

S1

oooo o

$3000

R1

oooo o

$4000 $0

Restatement - Lottery 2

S2

oooo o

$3000

oooo ooooo ooooo o

$0

R2

oooo o

$4000 $0 (80%) (20%)

oooo ooooo ooooo o

$0

Restatement - Lottery 3S4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooo$1,000,000

o$1,000,000

oooooooooo$1,000,000

R4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooo$1,000,000

o$0

oooooooooo$5,000,000

Restatement - Lottery 4S4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooo$0

o$1,000,000

oooooooooo$1,000,000

R4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooo$0

o$0

oooooooooo$5,000,000

p3

p1

p2

Marschak-Machina Triangle3 outcomes: Probabilities:

123 xxx 1123 ppp

1,,0 321 ppp

4000

0

p2

p3

p1

3000

R1 (0.2, 0, 0.8)

S1

R2 (0.8, 0, 0.2)

S2 (0.75, 0.25, 0)

p3

p1

P2=0

Reduce to two dimensions

p3

p1

Subjective Expected Utility Theory (SEUT)

Betweenness Axiom:

If G1~G2 then [G1, G2; q, 1-q]~G1 ~G2

So, indifference curves linear!

Independence Axiom:

If G1~G2 then

[G1, G3; q, 1-q]~ [G2, G3; q, 1-q]

So, indifference curves are parallel!!

Risk Neutrality:

Along indifference curve p1x1+p2x2+p3x3=c

p1x1+(1-p1-p3)x2+p3x3=c

123

12

23

23 p

xx

xx

xx

xcp

Linear and parallel

Risk Averse:

Along indifference curve p1u(x1)+p2u(x2)+p3u(x3)=c

p1u(x1)+(1-p1-p3) u(x2)+p3u(x3)=c

123

12

23

23 )()(

)()(

)()(

)(p

xuxu

xuxu

xuxu

xucp

Linear and parallel

p3

p1

R1

S2S1

R2

Common Ratio Problem

p3

p1

R3

S4S3

R4

Common Consequence Problem

Prospect TheoryKahneman and Tversky

(Econometrica 1979)

Certainty EffectReflection EffectIsolation Effect

Certainty Effect

People place too much weight on certain events

This can explain choices above

Ellsberg Paradox

Certainty Effect

G1 $1000 if red

G2 $1000 if black

G3 $1000 if red or yellow

G4 $1000 if black or yellow

33

67

Ellsberg Paradox

Most people choose G1 and G4.

BUT: Yellow shouldn’t matterRed Black Yellow

G1 $1000 $0 $0

G2 $0 $1000 $0

G3 $1000 $0 $1000

G4 $0 $1000 $1000

Reflection Effect

All Results get turned around when discussing Losses instead of Gains

Isolation Effect

Manner of decomposition of a problem can have an effect.

Example: 2-stage game

Stage 1: Toss two coins. If both heads, go to stage 2. If not, get $0.

Stage 2: Can choose between $3000 with certainty, or 80% chance of $4000, and 20% chance of $0.

This is identical to Game 2, yet people choose like in Game 1 (certainty), even if they must choose ahead of time!

Example

We give you $1000. Choose between:

a) Toss coin. If heads get additional $1000, if tails gets $0.

b) Get $500 with certainty.

Example

We give you $2000. Choose between:

a) Toss coin. If heads lose $0, if tails lose $1000.

b) Lose $500 with certainty.

84% choose +500, and 69% choose [-1000,0]

Very problematic, since outcomes identical! 50% of $1,000 and 50% chance of $2,000

or $1,500 with certainty

Prospect Theory explanation: isolation effect - isolate initial receipt of money from

lottery reflection effect - treat gains differently from losses

Preference Reversals(Grether and Plott)

Choose between two lotteries:($4, 35/36; $-1 1/36) or ($16, 11/36; $-1.50, 25/36)Also, ask price willing to sell lottery for.Typically – choose more certain lottery (first one)

but place higher price on risky bet.Problem – prices meant to indicate value, and

consumer should choose lottery with higher value.

Wealth Effects

Problem: Subjects become richer as game proceeds, which may affect behavior

Solutions: Ex-post analysis – analyze choices to see if changed Induced preferences – lottery tickets Between group design – pre-test Random selection – one result selected for payment

Measuring Preferences

Administer a series of questions and then apply results.

However, sometimes people contradict themselves – change their answers to identical questions