8/25/2014
ECON4260 Behavioral
Economics
2nd lecture
Cumulative Prospect Theory
Which lottery do you prefer?
NB: No right or wrong answer
Lottery A
4000 4000 4000 4000 4000
0
Lottery B
3000 3000 3000 3000 3000 3000
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Which lottery do you prefer?
We first flip a coin,
if head the lotteries are as on last page
if tail you get nothing
Lottery C
4000 4000 4000 4000 4000
0
0
0
0
0
0
0
Lottery D
3000 3000 3000 3000 3000 3000
0
0
0
0
0
0
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Do you think it makes an essential
difference for your preference if:
Base alternative: The coin is flipped, the face announced and
then you choose between C and D and the dice is rolled - or
1. You choose between C and D, then the coin is flipped, the
face announced and finally the dice is rolled
2. You choose between C and D and then the coin is flipped
and the dice is rolled at the same time.
3. You choose C or D, then the dice is rolled, the number of
eyes announced and the coin is flipped
4. None of them makes any difference
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The independence axiom
If you prefer A to B
It is consistent to prefer C to D:
Similarly with B preferred to A or indifference
Lottery C:
Lottery A
0
Lottery D:
Lottery B
0
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Standard notation
Lottery A
Lottery B
4000 4000 4000 4000 4000
0
3000 3000 3000 3000 3000 3000
Lottery A
4000 with probability 5/6
0 with probability 1/6
3000 with probability 1
Lottery B
Notation in Kahneman and Tversky (1979):
Lottery A: (0,1/6; 4000,5/6) that is (4000,5/6)
Lottery B: (3000, 1) thatØkonomisk
is Institutt (3000)
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Deriving Expected utility
from the independence axiom
• To simplify we only look at lotteries with prices between 0 and
100.
• Consider the lottery A: (100,p; x,q; 0,1-p-q)
• We want to find a lottery of the from
A’: (100,P; 0;1-P) (only best/worst outcome)
such that A~A’
• Additional assumption:
– For any x there is a probability u such that (x) ~ (100,u; 0,1-u)
– This probability depend on x, write it u(x).
• Using the independence axiom
– Replace x with (100,u(x); 0,1-u(x)) innside lottery A
– Yields lottery A’=(100,p+qu(x); 0, 1-p-qu(x))
– Probability of winning is P = p + qu(x) = pu(100) + qu(x) + (1-p-q)u(0) = EU
• Do the same for any lottery B to find B’
• A preferred to B if A’ preferred to B’
– If A’ has higher winning probability than B’
– Expected utility of A higher than Expected utility of B.
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Positive linear transforms
• Note that if
,
0
is equivalent to maximizing
• Then maximizing
• Useful in the following
– Can choose
– Use
0
0
, with
0)
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Applying EU in Economics
- no lottery is an isolated event
• Suppose you win 100 kroner in a lottery
– You can spend them today
– Keep them until tomorrow or some later day
• Insurance is like a lottery
– You get 1 million NOK if your house burns
– Nothing otherwise
– Insurance only make sense if you see this lottery and the value of
the house together.
• Let
denote the outcome of the lottery
– May be positive or negative
– Utility would be
where
is wealth
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Prospect theory
Lotteries are valued in isolation
• Let denote the outcome of the lottery
– While
will never be negative, can be negative – a loss!
– In general: evaluated relative to a reference point :
Preference may violate the independence axiom
• Decisions weights replace probabilities
– While
EU   pi u( xi )
– Prospect theory uses
 u( x )
i
i
where  i are decision weights
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Evidence; Decision weights
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]*
or
D: (3 000, 0.25)
[35]
• Problem 4
– C: (4 000, 0.20)
– N=95 [65]*
• 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%
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Value function
Reflection effect
• Problem 3
– A: (4 000, 0.80)
– N=95 [20]
or
B: (3 000)
[80]*
• Problem 3’
– A: (-4 000, 0.80)
– N=95 [92]*
or
B: (-3 000)
[8]
• Ranking reverses with different sign (Table 1)
• Concave (risk aversion) for gains and
• Convex (risk lover) for losses
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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)
– N=95 [16]
or
B: (500)
[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)
– N=95 [69]*
or
B: (-500)
[31]
• Both equivalent according to EU, but the
initial instruction affect the reference point.
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Decision weights
• Suggested by Allais (1953).
• Originally a function of probability
i = 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
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Isolation Effect
(recall the independence axiom)
”In order to simplify the choice between alternatives,
people often disregard components that the
alternatives share and focus on the components
that distinguishes them”
•
Problem 10:
Consider the two-stage game. The first stage is (2. stage,
0.25; 0, 0.75) (proceed to stage to with 25% probability. If
you reach the second stage you have the choice between
A: (4000, 0.80) and B (3000) [78%]. Your choice must be
made before the game starts.
•
The choice in 10 is equivalent to.
A’: (4000, 0.20) [65%] and B’: (3000,0.25)
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The editing phase
Finding the reference point
• Combination
(200,0.25,200,0.25) =(200,0.5)
• Segregation
(300,0.8;200,0.2)=200 + (100,0.8)
• Cancellation
(200,0.2;100,0.5;-50,0.3) vv (200,0.2;150,0.5;-100,0.3)
Can be seen as a choice between
(100,0.5;-50,0.3) vv (150,0.5;-100,0.3)
• Simplifications
– (500,0.2 ; 99,0.49) dominates
(500,0.15; 101,0.51) if the last part is simplified to
(…
; 100,0.50) Økonomisk Institutt
Stochastic dominance
Lottery A (58%)
White
red
green
yellow
Probability %
90
6
1
3
Price
0
45
30
-15
Lottery B (42%)
White
red
green
yellow
Probability %
90
7
1
2
Price
0
45
-10
-15
White
red
green
blue
Prob. %
90
6
1
1
yellow
2
Lottery C (A)
0
45
30
-15
-15
Lottery D (B)
0
45
45
-10
-15
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Lotto
(Think about this until next week)
• 50% of the money that people spend on
Lotto is paid out as winning prices
• Stylized:
– Spend 10 Kroner
– Win 1 million kroner with probability 1 to 200 000
• Would a risk avers expected utility maximizer
play Lotto?
– Is Lotto participation a challenge to expected
utility?
• Can prospect theory explain why people
participate in Lotto?
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Rank dependent weights
• Order the outcome such that
x1>x2>…>xk>0>xk+1>…>xn
• Decision weights for gains

j


j 1


 i 1

 j  w   pi   w    pi  for all j  k
 i 1
• Decision weights for losses

n



 i  j 1 
n

 j  w    pi   w    pi  for all j  k
 i j
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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
n
  v( x )
i 1
i
i
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Suggested approximation
(See Benartzi and Thaler, 1995)
w( p ) 
p
p

 (1  p )
1

1/ 
0,9
0,8
Gains
0,7
Losses
0,6
0,5
  0.61 for gains
0,4
  0.69 for losses
0,2
0,3
0,1
0
0
0,2
0,4
0,6
0,8
1
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The value function
(see Benartzi and Thaler, 1995)
 x
v( x)  

  (  x )
10
if x  0
if x  0
5
0
-10
-5
• =  = 0.88
•  = 2.25
0
5
10
-5
-10
-15
-20
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The status of cumulative
prospect theory
(See Starmer 2000)
• Explains data much better than alternative theories.
– Rank dependent utility, does a fair job but not as good
• Starmer claim a limited impact on economic theory
– But loss aversion is increasingly referred to
• But, some very interesting application
– We will use it to understand equity return
• See Camerer 2000 for other examples.
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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.
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A critique of Prospect theory
• The theory is open-ended on the reference point.
• The reference point is sometimes zero but other
times it may be something else like the outcome of a
safe alternative.
• K&T are free to choose the reference point to make
the theory fit the data
– No wonder they can explain a lot
• A theory determining the reference point wanted
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Köszegi and Rabin
Utility u( c | r )  m( c )  n( c | r ) with
m( c ) being "consumption utility"
n( c | r ) is gain-loss utility with r as reference point
Both stochastic reference point (distribution g ( r )  G '( r ))
and stochastic consumption (distribution f ( r )  F '( r ))
U ( F | G )   u( c | r ) dG( r )dF (c )   u (c | r ) g ( r ) f (c ) drdc
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K&R Med additive nyttefunksjon
Nytten for hver vare er uavhengig
av konsumet av andre varer:
m(c )  m1 (c1 )  m2 (c2 )  ...  mK ( cK )
der mk ( ck ) er nytten av vare k
nk ( ck | kk )   ( mk ( ck )  mk ( rk ))
der  har knekkpunkt i 0
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K&R. Theory of the reference point
• The reference point r is a personal equilibrium (PE)
if a person would choose c=r if r where the reference
point.
– Generally: the person would lottery F if F where his reference
lottery
• There may be many equilibrium:
– A Preferred Personal Equlibrium is the best PE
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Two lotteries. A and B
• A: (100,50%; 0,50%) and B: (300,50%; -100,50%)
• Utility u(c|r)=2c+(r-c)+
• With A as reference:
– U(B|A)= ¼ (600 + 600+(-200-200) + (-200 -100))=500/4 =125
– U(A|A)= ½ (200 + 0)= 100 > U(B|A)
• With B as reference
– U(A|B) = ¼ ((200 -200) + 200 + (0-300)+0) = -100/4 =-25
– U(B|B) = ½ (600 – 200) = 200
• B is a personal equilibrium
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Shopping
• Two “goods”: Shoes and money
• If you enter the shop expecting to buy
– Leaving without the shoes is a loss
– Leaving with the money is a gain
– Losses weight more than gains so you are inclined to buy
• If you enter the shop expecting not to buy
– Leaving the shop with the shoes is a gain
– Leaving without the money is a loss
– As losses weigh more you are inclined to leave without shoes
• You are more likely to buy if buying is you reference point
– Make sure to leave home with you PPE as reference!!
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Raise your hand if you disagree
Suppose that you prefer lottery A to lottery B
You are then offered lotteries C and D
C: Flip a coin and you get A if head (otherwise 0)
D: Flip a coin and you get B if head (otherwise 0)
It is rational to prefer C to D, as the lottery you get if
head is then preferable (A better than B)
Claim: “I wish my preferences where so consistent!”
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Rabin’s Theorem
A teaser
• How many of you would participate in the following
lotteries (the alternative is (0)).
– A: (-100, 33 %, +100, 67 %)
– B: (-100, 45 %, +100, 55 %)
• Would changes in wealth (±10 000 kroner) affect
your preferences?
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Are you consistent?
• If you decline A (B), or were indifferent, then the
logical implication is to decline C (D):
– C: (-100, 50 %, +10 billions, 50%)
– D: (-100, 85 %, +10 billions, 15%)
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Lotto
• 50% of the money that people spend on Lotto is paid
out as winning prices
• Stylized:
– Spend 10 kroner
– Win 1 million kroner with probability 1 to 200 000
• Would a risk avers expected utility maximizer play
Lotto?
– Is Lotto participation a challenge to expected utility?
• Can prospect theory explain why people participate
in Lotto?
• What is maximum willingness to pay for this winning
prospect, for an
– Risk avers expected utility maximizer?
– A person acting acording to prospect theory?
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Suggested answer
• A risk neutral expected utility maximizer will value the
winning prospect to the expected value
– 1 million kroner* (1/200 000) = 5 kroner
– WTP for a risk avers person < 5 kroner
• Prospect theory
–
–
–
–
–
= w(1/200 000) ≈ 0.0002
v(1 million) = (1 000 000)^0.88 ≈ 190 000
WTP = x where: 2.25(x)^0.88 ≈ 190 000 * 0,0002
Solution: WTP ≈ 27.5 kroner
Would buy Lotto even with only 2 kroner expected value for
each 10 kroner spent. (5 kroner/2.7)
• Some people do NOT buy Lotto tickets
– Is that a challenge to CPT?
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