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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