Modeling self-control problems I:
Multi-self vs. temptation
Lectures in Behavioral economics
Spring 2016, Part 2
People have demand for commitment
Ariely & Wertenbroch (2002)
Experiment where the students in a course had to hand in three
compulsory assignments before the final exam, they could choose
deadlines, and were punished if the deadlines were not observed.
Result: Many choose deadlines before the end of the semester,
& among these, students with evenly spread deadlines did better.
„
„
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G.B. Asheim, ECON4260, #2
1
Present-biased preferences: (EG)-pref.
U t (ct ,! , cT ) u (ct ) E ¦W
T
„
t 1
3
G.B. Asheim, ECON4260, #2
Hyperbolic discounting leads to time-inconsistent
preferences (Strotz, 1956), procrastination of tasks
with immediate cost, and makes commitment desirable, given awareness of the self-contr. problems.
Other ways to model the demand for commitment?
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„
G W t u (cW )
Yields time-inconsistent preferences.
Behavior with time-inconsistent preferences
„
„
„
Naive behavior: Choosing the best plan under the
presumption that it will be followed.
Sophisticated behavior: Choosing the best plan
among those that will actually be followed.
Multi-self model of sophisticated behavior: Let
every decision node correspond to a different “self ”.
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„
„
„
G.B. Asheim, ECON4260, #2
2
Outline
”Do it now or later”-article
O’Donoghue & Rabin (1999).
Interesting application of the multi-self approach
showing that sophisticates need not
realize better outcomes than naifs.
Problems with the multi-self approach
Alternative to the multi-self approach
Gul & Pesendorfer (2001, 2004).
Direct modeling of temptation.
Soft paternalism (”Nudge”)
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G.B. Asheim, ECON4260, #2
“Do it now or later” O’Donoghue & Rabin (1999)
„
Model: ʀ Must perform an activity exactly once.
“Do it now or later” (2)
„
ʀ T f periods in which to perform it.
Two
cases:
ʀ Each period, choose to “do it” or “wait”.
ʀ If wait until period T, must do it then.
„
„
If activity is done in period t, incur cost ct t 0 and
receive reward t 0.
ʀ Immediate costs: incur cost when you do
it, receive reward after some delay.
ʀ Immediate rewards: receive reward when
you do it, incur cost after some delay.
Assume (EG)-preferences with G (for simplicity):
Period-t utility for “do it” in period W t t:
For immedi ate costs : U t
vt
Reward schedule : v { (v1 , ! , vT )
For immedi ate rewards : U t
Cost schedule : c { (c1 , ! , cT )
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G.B. Asheim, ECON4260, #2
5
4
T
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„
Reward schedule : v { (0, 0, 0, 0)
Cost schedule : c { (3, 5, 8, 13)
„
Period-t utility
for “do it” in
period W t t:
„
W
t
3
0 (12)
1
W
2
5
2
W 3
4
( 2)
t 1 (23)
t
2
W
(3)
(3)
(1)
4
5
2
4
„
13
2
( 4)
13
2
( 4)
(3)
(1)
( 2)
132
4
5
(1)
( 2)
13
2
8
„
„
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G.B. Asheim, ECON4260, #2
­E vW cW if W t
®
¯E vW E cW if W ! t
­vW E cW if W t
®
¯E vW E cW if W ! t
G.B. Asheim, ECON4260, #2
6
“Do it now or later”: Ex. with immediate costs (2)
“Do it now or later”: Ex. with immediate costs
1
2
E
Naifs do it in
period 4.
„
Sophisticates do
t 3
it in period 2.
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Welfare comparisons of naive & sophistic. behavior:
W 2 is better than W 4 at both t 0, t 1 and t 2.
W 2 cannot be compared with W 4 at t 3.
Period-t utility
for “do it” in
period W t t:
Naifs do it in
period 4.
t
W
3
0 (12)
W
1
2
5
2
( 2)
t 1 (23)
t
5
2
(1)
5
2
( 2)
Sophisticates do
t 3
it in period 2.
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W 3
4
W 4
132
(3)
4
( 4)
132
(3)
4
( 4)
132
(1)
8
(3)
132
( 2)
(1)
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G.B. Asheim, ECON4260, #2
“Do it now or later”: Ex. with immediate rewards
1
2
E
4
T
Naifs do it in
period 3.
„
Period-t utility
for “do it” in
period W t t:
„
„
Reward schedule : v { (3, 5, 8, 13)
“Do it now or later”: Ex. with immediate rewards (2)
„
Cost schedule : c { (0, 0, 0, 0)
W
t
0
t 1
t
1
W
2
W
3
W
4
„
13
2
( 2)
( 4)
(3)
4
5
2
3
(3)
( 4)
5
2
3
2
2
Sophisticates do
t 3
it in period 1.
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( 2)
(1)
13
2
(1)
(1)
(3)
( 2)
13
2
4
5
( 2)
(1)
13
2
8
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G.B. Asheim, ECON4260, #2
General lesson: Sophistication about future self-control problems can mitigate or exacerbate misbehavior
„
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G.B. Asheim, ECON4260, #2
„
„
Welfare comparisons of naive & sophistic. behavior:
W 3 is better than W 1 at both t 0 and t 1.
W 3 cannot be compared with W 1 at t 2 and t 3.
Period-t utility
for “do it” in
period W t t:
Naifs do it in
period 3.
t
W
1
W
3
2
0
2
W
5
2
( 4)
(3)
3
W
( 2)
( 2)
(1)
13
2
4
( 2)
4
(1)
13
2
4
( 4)
5
2
13
2
4
(3)
5
2
3
t 1
t
Sophisticates do
t 3
it in period 1.
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„
Proposition 2. For both immediate costs and
immediate rewards, Ws d Wn.
ʀ Why? The future is always more promising from the
point of view of a naif, since a sophisticate removes some
future possibilities as unattainable without commitment.
Hence, for a sophisticate the present is relatively more
attractive, leading to the task being performed earlier.
ʀ With im. costs: Naifs may procrastinate due to presentbiased preferences. With im. rewards: Sophisticates may
preproperate since they realize their future misbehavior.
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(3)
(1)
13
2
8
(1)
( 2)
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G.B. Asheim, ECON4260, #2
Propositions 3 and 4. When evaluated from a prior
period 0, the following holds. For immediate costs, a
small self-control problem can cause servere welfare
losses if and only if you are naive. For immediate
rewards, a small self-control problem can cause servere
welfare losses if and only if you are sophisticated.
ʀ Why? Naifs with immediate costs may procrastinate
repeatedly even if E is close to 1.
Sophisticates with immediate rewards may preproperate
repeatedly even if E is close to 1.
General lesson: Even “small” self-control problems
can cause servere welfare losses.
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G.B. Asheim, ECON4260, #2
Procrastination example revisited
„
A task to be performed at time 0, 1, 2, …, or not at all.
Immediate cost: 25. Benefits at the next stage: 125.
„
(E, G)-preferences with E
50
40
1/2 and G 4/5.
But not
Better worthwhile
to do it
to never.
wait for
now than
2 periods
Sophisticated behavior with 3 periods
„
„
„
32
time
8
10
25
Even better to do it at the next stage.
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G.B. Asheim, ECON4260, #2
„
Backward induction
„
Let every decision node correspond to a different
“self” or “agent” of the decision-maker
Backward induction
Self a time 1 if
A sophisticate task was not
does the task done at time 0
now, since else
postponed for
2 periods.
Self a time
0
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0
1
2
24
30
Payoff of
0-self
30
25
25
Payoff
of 1-self
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G.B. Asheim, ECON4260, #2
Sophisticated behavior with odd # periods
Sophisticated behavior with 4 periods
A sophisticate
does the task
with a delay
of 1 period.
„
A sophisticate does the task now, since else
postponed for 2 periods.
Sophisticated behavior with even # periods
„
A sophisticate does the task with a delay of 1 period.
Conclusion: Multi-self model with sophisticated
behavior may not be descriptively accurate
0
1
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2
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G.B. Asheim, ECON4260, #2
Is it “right” to apply the multi-self model?
„
Introduce naivete (or partial naivete) (O’Donoghue & Rabin,
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1999, 2001)
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G.B. Asheim, ECON4260, #2
Sophisticated behavior with f periods
If we impose that behavior is the same in all
periods, conditional on the task not having
been done, there is a unique equilibrium, where
the task is done in each period with prob. ½.
„
It is another equilibrium (planning) to do the task in
periods 1, 3, 5, …, but not in periods 0, 2, 4, … .
„
It is an equilibrium (planning) to do the task in
periods 0, 2, 4, …, but not in periods 1, 3, 5, … .
„
Backw. induct. cannot be used since no last period
„
Demand for commitment
„
„
„
Utility at t of doing the task at
30
W 1
24
W 2
W 0
3
¼ prob.
½ prob.
2
25
0
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1
0
1
2
3
„
Suppose the decision maker can purchase a commitment device costing c, ensuring that the task be
done in the next period? What is the largest c?
With an odd # periods, she does the task now with
present payoff 25. Commitment to next period
yields payoff 30 c. Hence, c cannot exceed 5.
With an even # periods, she does the task in the
next period anyway. Not interested in commiting.
With f periods, she receives a payoff of 25.
Commitment to next period yields payoff 30 c.
Hence, c cannot exceed 5.
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G.B. Asheim, ECON4260, #2
Temptation (Gul & Pesendorfer 2001)
a : Doing the task
1
G.B. Asheim, ECON4260, #2
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Temptation in a dynamic setting
(Gul & Pesendorfer 2004)
a : Not doing the task
0
„
In standard consumer theory, if a1 is preferred to a0 ,
then she has the following preference over menus :
„
{a1} ~ {a0 , a1} ; {a0 }
No time-inconsistency; the future is discounted by G
Still, temptation yields a demand for commitment
Payoff when choosing a1 : 25 54 125 t
In G & P' s analysis, if tempted by a0 and gives in :
Payoff when choosing a0 :
{a1} ; {a0 , a1} ~ {a0 }
0
4
5
W0
Maximal payoff when task has not been done :
If tempted by a0 , but does not give in :
G
{a1} ; {a0 , a1} ; {a0 }
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G.B. Asheim, ECON4260, #2
W0
max{75 t , 54 W0 }
t : Cost of temptation
4
5
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G.B. Asheim, ECON4260, #2
Temptation in a dynamic setting (2)
Optimal not to give in at a low cost of temptation ( t 75 )
W0
75 t ! 0
W > 0 0
4
5
W
0
Optimal to give in at a high cost of temptation ( t t 75 )
W0
0
W0 t 75 t
Temptation and the demand for commitment
Assume that she can commit by paying c to doing
the task in the next period (without being tempted).
Wc
0c
4
5
25 54 125
60 c
Optimal to commit at a low cost of temptation ( t 75 ) if
60 c ! 75 t
Wc
W0
c t 15
Optimal to commit at a high cost of temptation ( t t 75 ) if
Wc
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G.B. Asheim, ECON4260, #2
60 c ! 0
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W0
c 60
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G.B. Asheim, ECON4260, #2
Soft paternalism
„
„
With procrastination,
the status quo matters
E.g. organ donation,
savings decisions
Organ donation:
„ Opt out. Austria:
99.98 % consent
„ Opt in. Germany:
12 % consent
„ Active choice.
US driver’s licence
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G.B. Asheim, ECON4260, #2