Individual decision-making I
Marta Serra-Garcia, University of Munich
Behavioral and Experimental Economics, SS 2011 – June 22nd, 2011
Outline
1.
The ‘Classical’ Framework:
•
2.
A famous criticism (of a particular case)
•
3.
Prospect Theory (Kahneman and Tversky, 1979)
Explaining ‘phenomena’ with Prospect Theory
•
5.
Rabin’s Calibration Theorem (Rabin, 2000)
A ‘Behavioral’ Framework:
•
4.
Expected Utility Theory – A very short refresher
Equity premium puzzle (Benartzi and Thaler, 1995)
Testing Prospect Theory
•
In the lab (Thaler, Tversky and Kahneman, 1997) and (Gneezy
and Potters, 1997)
2
1. EUT – A refresher
• A preference relation is rational iff:
– Complete
• for all x,y є X, either x is preferred to y, y to x or both
– Transitive
• for all x,y and z є X, if x is preferred to y and y to z, then x is
preferred to z
• Axioms:
– Continuity: no reversals in preference relations
• For pairs (xn, yn), if xn is preferred to yn for all n, in the limit
x is still preferred to y.
3
1. EUT – A refresher
– Independence: the preference ordering between two lotteries is
independent of a third lottery
• If L is preferred to L‘,
• then αL+(1- α)L‘‘ is preferred to αL‘+(1- α)L‘‘
• Suppose a rational preference relation satisfies the continuity
and independence axioms
• Then, it can be represented with the expected utility form:
– Take two lotteries L and L‘,
– If L is preferred to L‘, then
N
∑u
n =1
N
n
pn ≥ ∑ u n p 'n
n =1
4
1. EUT - Axiom Violations
• Allais Paradox (Allais, 1953)
– the independence axiom
• Certainty effect or common consequence effect
• Prospect 1:
-
s1 = (1 Mill., 1) or
-
r1 = (5 Mill., 0.1; 1 Mill., 0.89; 0; 0.01)
5
1. EUT - Axiom Violations
•
•
Prospect 2:
-
s2 = (1 Mill., 0.11; 0, 0.89) or
-
r2 = (5 Mill., 0.1; 0; 0.9)
EUT predicts choosing either s1 and s2 or r1 and r2, but many
people choose s1 and r2.
6
1. EUT - Axiom Violations
•
Reactions to the Allais Paradox:
–
Normative approach
–
Regret theory
–
Prospect theory
7
1. EUT – Risk Aversion
•
Take a continuous variable x, with utility values u(x) and
probability distribution F(x)
•
EUT allows us to use a v. N-Mn utility function U(.) to
represent the expected utility of the lottery F:
U ( F ) = ∫ u ( x)dF ( x)
8
1. EUT – Risk Aversion
•
Risk aversion is defined by concavity of u(x)
∫ u ( x)dF ( x) ≤ u (∫ u ( x)dF ( x))
•
for all F (.)
Consider an individual who is offered a lottery:
–
With probability 0.5, he wins $11
–
With probability 0.5, he loses $10
–
May accept it or reject it
Reject if: 0.5u(w+11)+0.5u(w-10)<u(w)
9
2. A Criticism - Rabin (2000)
• There is plenty of evidence showing that individuals would
reject the lottery:
– (0.5, 11; 0.5,-10)
• But these are relatively small stakes, what does it then imply
for larger stakes?
10
2. A Criticism - Rabin (2000)
• Suppose an individual rejects (0.5, $11; 0.5,-$10)
• What does this imply?
• Utility of 11th $ is at most 90% of that of -10%th $
• If the individual’s wealth increases, the decline in marginal
utility becomes ridiculously large
– Rejecting (0.5, $11; 0.5,-$10), implies a 10% decrease in marginal
utility for every increase of $21 in lifetime wealth
11
2. A Criticism - Rabin (2000)
12
2. A Criticism - Rabin (2000)
• Thus, if an individual rejects (0.5, 110; 0.5,-100)
• We can show that she will always reject a gamble with
(0.5,550;0.5,-400) or (0.5,∞;0.5,-1000)
• Intuition:
– Marginal utility declines very fast
– As increases become larger, the increase in utility will become
smaller and smaller
– On the other hand, decreases will imply larger and larger
decreases in utility
13
2. A Criticism
• EUT‘s assumptions or often-made assumptions:
– Continuity
Allais Paradox
– Independence
– Utility over final outcomes
– Risk aversion
Rabin‘s critique
14
2. A Criticism - Gains vs. Losses
• Kahneman and Tversky (1979)
• The Reflection effect
Positive prospects
3
(4000, 0.8)
(3000,1)
15
2. A Criticism - Gains vs. Losses
• Kahneman and Tversky (1979)
• The Reflection effect
Positive prospects
3
(4000, 0.8)
(3000,1)
4
(4000, 0.2) (3000, 0.25)
16
2. A Criticism - Gains vs. Losses
• Kahneman and Tversky (1979)
• The Reflection effect
Positive prospects
3
(4000, 0.8)
(3000,1)
4
(4000, 0.2) (3000, 0.25)
Negative prospects
3‘
(-4000, 0.8)
(-3000,1)
17
2. A Criticism - Gains vs. Losses
• Kahneman and Tversky (1979)
• The Reflection effect
Positive prospects
3
(4000, 0.8)
(3000,1)
4
(4000, 0.2) (3000, 0.25)
Negative prospects
3‘
(-4000, 0.8)
(-3000,1)
4‘
(-4000, 0.2)
(-3000, 0.25)
18
2. A Criticism - Isolation
• Suppose Steven plays the following game:
• Stage 1
– With prob. 0.75, the game ends: $0
– With prob. 0.25, the game enters stage 2
• Stage 2, one of two gambles is played:
– (4000, 0.8)
– (3000, 1)
• What gamble would you guess Steven is most likely to
choose, before the game starts?
19
2. A Criticism - Isolation
Consider Steven faces the following problems:
1. He has been given 1000, and asked to choose:
–
–
(1000,0.5)
(500, 1)
2. He has been given 2000, and asked to choose:
–
–
(-1000,0.5)
(-500, 1)
20
2. A Criticism – the Endowment Effect
• Kahneman, Knetsch and Thaler (1991)
• Mugs and pens of Cornell University
1.
Half the students were given a Cornell mug,
worth $6.00
• They acted as sellers, and their WTA was
elicited
• The rest acted as buyers, and their WTP was
elicited
2.
Half the students were given pens with a price
tag $3.98
• What happened?
21
2. A Criticism – the Endowment Effect
• Mugs/pens in the market: 22
• Expected number of trades: 11
• Actual number of trades:
– Mugs: 4,1,2 and 2
– Pens: 4 or 5
22
2. A Criticism – the Endowment Effect
• Another experiment...Kahneman and Loewenstein (1991)
• Half the subjects in a group of 63 were given pens
• Others a token redeemable for an unspecified gift
• In the experiment:
1.
They were asked to rank the attractiveness of 6 gifts
• Which could potentially be used as their remuneration
23
2. A Criticism – the Endowment Effect
2. Subjects were then given the choice between a pen and two
chocolate bars:
•
•
•
•
•
Those who owned a pen, chose it 56% of the time
Those who didn‘t own a pen, chose it 24% of the time
...although in previous stage subjects endowed with a pen
did not rate them as more attractive!!
These results suggest that once an individual has been
endowed with an object...
It is painful to give it up, though it is not valued more
24
2. A Criticism – the Endowment Effect
Does the ‚Endowment Effect‘ disappear with experience?
List, 2003, QJE
25
2. A Criticism – the Endowment Effect
• List, 2003, QJE
• WTP and WTA elicited from two different sets of
traders:
– Sportscards collectors
– Pin collectors
• At the market, where they go to sell their own goods or buy
new items
• Are the results due to experience or something else?
– It could be selection...
– List goes back to the sportscards market one year later
– Can examine effect of experience for subjects who participated 1
year before (eliminating selection effect)
– Results remain robust: experience decreases the endowment
effect
26
2. A Criticism
• Individuals often violate assumptions made within the
framework of EUT:
– Overweigh certain outcomes (Allais paradox)
– Do not value prospects purely in terms of the final outcome
– Are risk averse for gains, but risk neutral for losses
27
3. Prospect Theory
• Kahneman and Tversky (1979)
• Choices are made in two phases:
1.
Editing phase
• Simplifying the prospects
2. Evaluation phase
• Comparing the edited prospects
28
3. Prospect Theory
• The value of prospects V depends on:
– Decision weights
– Subjective values
V ( x, p; y, q) = π ( p )v( x) + π (q )v( y )
29
3. Prospect Theory – value function
Concavity for gains +
convexity for losses
Reference point
Steeper slope for losses
than gains
30
3. Prospect Theory – decision weights
•
•
π(0)=0
π(1)=1
•
Low probabilites are often overweighed
–
–
People often prefer (5000, 0.01) to (5,1)
But (-5,1) to (-5000,0.01)
• Large probabilities are underweighed (compared to 1)
– Certainty effect
31
3. Prospect Theory – decision weights
32
Kahneman and Tversky (1992)
3. Prospect Theory – decision weights
• How can decision weights be elicited?
• Booij, van Praag and van der Kuilen (2009)
33
3. Prospect Theory – decision weights
• Booij, van Praag and van de Kuilen (2009)
34
3. Prospect Theory
• Kahneman and Tversky‘s theory is now widely used:
– Loss aversion
– Diminishing sensitivity
– Over- and underweighing probabilities
• What kind of phenomena does it explain?
35
4. Explaining phenomena with PT
• An Example
– Consider the prospect (0.5, -100; 0.5,200)
– Suppose that you reject it
– Then, you have 0.5u(w-100)+0.5u(w+200)<u(w)
– Consider the repetition of the lottery
• 0.25, -200
• 0.5, 100
• 0.25, 400
– Would you accept it now?
• 0.25u(w-200)+0.5u(w+100)+0.25u(w+400)>u(w)
• But such a choice is inconsistent with EUT! (Samuelson‘s
theorem)
36
4. Explaining phenomena with PT
• Instead suppose you are loss averse
• In particular
– u(x)=
x if x≥0
2.5x if x<0
• Then, the single lottery yields
– U=0.5(200)+0.5(2.5(-100))<0!
• But the repeated lottery yields
– U=0.25(400)+0.5(100)+0.25(-2.5(-200))=150-125>0!
37
4. Explaining phenomena with PT
• Benartzi and Thaler (1997)
• Equity returns vs bond returns
Period
1802-1870
1871-1925
1926-1990
Equity
5.7
6.6
6.4
Bond
5.1
3.1
0.5
• Risk aversion?
• Habit formation?
38
4. Explaining phenomena with PT
• Loss aversion and narrow bracketing!
• Suppose Sally is loss averse
– If she brackets over 30 years, then equity >> bonds
– But if she brackets in 1 year periods:
• Equity is more likely to exhibit losses – this hurts
• Bonds are less likely...
39
4. Explaining phenomena with PT
• How often need the evaluation period be?
• Simulate the returns of stocks and bonds for different
bracketing periods
• For which ‚brackets‘are bonds preferred?
40
4. Explaining phenomena with PT
• If someone is loss averse and brackets for 1 year
• What is the optimal mixture of stocks and bonds?
41
4. Explaining phenomena with PT
• Conclusion:
– Loss aversion + narrow bracketing
– ...seem to provide an explanation for the equity premium puzzle
• Caveat?
– Results are consistent with the puzzle...
– But it is not direct evidence of loss aversion + bracketing affecting
the investment choices of individuals
• Solution: lab experiment
42
5. Testing PT
• Gneezy and Potters (1997)
• Research question: does the evaluation period actually affect
investment in risky assets?
• Design:
– Part 1:
• 9 rounds
• 200 tokens endowment in each round
• Subjects had to choose how much to bet in lottery (x):
– With prob. 1/3, lose x
– With prob. 2/3, gain 2.5*x
• End of part 1: earnings totaled
43
5. Testing PT
– Part 2:
• 3 rounds
• Subjects had to choose how much to bet in same lottery out of
their previous earnings
– Two treatments:
– High frequency: choose x in each round
– Low frequency: choose x for 3 rounds in round 1
(same x in 3 rounds, feedback after 3)
• What would we expect under EUT? And under PT?
– Note: information of subjects differs across treatments
– EUT ambiguous effect of treatment
• Small stakes, no effect
– PT: more investment in Low frequency
44
5. Testing PT
Treatment H
Treatment L
Rounds 1-3
52
66.7**
Rounds 4-6
44.8
63.7***
Rounds 7-9
54.7
71.9***
45
5. Testing PT
• Thaler, Tversky and Kahneman (1997)
• Ran a similar experiment
• Found similar results
• Conclusion:
– Investment choices in the lab are affected by the evaluation period
– This suggests that narrow bracketing + loss aversion lead to less
investment in risky assets
46
5. Summary
• EUT: widely used but suffers from limitations:
– Allais‘ Paradox
– Other ‚assumptions‘ that are usually unrealistic:
• Evaluation of final outcomes
• Risk aversion for small stakes
• Risk preferences over gains and losses
• Prospect theory: introduces
–
–
–
–
Probability weighting (addressing Allais Paradox)
Evaluations of outcomes in terms of losses and gains
Loss aversion (explaining the Endowment Effect)
Diminishing sensitivity
• It can be used to explain the equity premium puzzle
– And experimental results support this explanation
47