Decision Theory
Plan for today (ambitious)
1. Time inconsistency problem
2. Riskiness measures and gambling wealth
 Riskiness measures – the idea and description
• Aumann, Serrano (2008) – economic index of riskiness
• Foster, Hart (2009) – operational measure of riskiness
 Buying and selling price for a lottery and the
connection to riskiness measures
• Lewandowski (2010)
 Two problems resolved by gambling wealth
a) Rabin (2000) paradox
b) Buying/selling price gap (WTA/WTP disparity)
Let’s start…
1. Time inconsistency problem
2. Riskiness measures and gambling wealth
 Riskiness measures – the idea and description
• Aumann, Serrano (2008) – economic index of riskiness
• Foster, Hart (2009) – operational measure of riskiness
 Buying and selling price for a lottery and the
connection to riskiness measures
• Lewandowski (2010)
 Two problems resolved by gambling wealth
a) Rabin (2000) paradox
b) Buying/selling price gap (WTA/WTP disparity)
A Thought Experiment
Would you like to have
A) 15 minute massage now
or
B) 20 minute massage in an hour
Would you like to have
C) 15 minute massage in a week
or
D) 20 minute massage in a week and an hour
Read and van Leeuwen (1998)
Choosing Today
If you were
deciding today,
would you choose
fruit or chocolate
for next week?
Eating Next Week
Time
Patient choices for the future:
Choosing Today
Today, subjects
typically choose
fruit for next week.
Eating Next Week
74%
choose
fruit
Time
Impatient choices for today:
Choosing and Eating
Simultaneously
Time
If you were
deciding today,
would you choose
fruit or chocolate
for today?
Time Inconsistent Preferences:
Choosing and Eating
Simultaneously
Time
70%
choose
chocolate
Read, Loewenstein & Kalyanaraman (1999)
Choose among 24 movie videos
• Some are “low brow”: Four Weddings and a Funeral
• Some are “high brow”: Schindler’s List
• Picking for tonight: 66% of subjects choose low brow.
• Picking for next Wednesday: 37% choose low brow.
• Picking for second Wednesday: 29% choose low brow.
Tonight I want to have fun…
next week I want things that are good for me.
Extremely thirsty subjects
McClure, Ericson, Laibson, Loewenstein and Cohen (2007)
• Choosing between,
juice now
or 2x juice in 5 minutes
60% of subjects choose first option.
• Choosing between
juice in 20 minutes or 2x juice in 25 minutes
30% of subjects choose first option.
• We estimate that the 5-minute discount rate is 50% and the
“long-run” discount rate is 0%.
• Ramsey (1930s), Strotz (1950s), & Herrnstein (1960s) were the
first to understand that discount rates are higher in the short
run than in the long run.
Theoretical Framework
• Classical functional form: exponential functions.
D(t) = dt
D(t) = 1, d, d2, d3, ...
Ut = ut + d ut+1 + d2 ut+2 + d3 ut+3 + ...
• But exponential function does not show instant
gratification effect.
• Discount function declines at a constant rate.
• Discount function does not decline more quickly in the
short-run than in the long-run.
Discounted value of
delayed reward
Exponential Discount Function
1
Constant rate of decline
0
1
11
21
31
41
Week (time = t)
-D'(t)/D(t) = rate of decline of a discount function
Exponential
Hyperbolic
51
An exponential discounting paradox.
Suppose people discount at least 1% between today and
tomorrow.
Suppose their discount functions were exponential.
Then 100 utils in t years are worth 100*e(-0.01)*365*t utils today.
•
•
•
•
What is 100 today worth today?
What is 100 in a year worth today?
What is 100 in two years worth today?
What is 100 in three years worth today?
100.00
2.55
0.07
0.00
Discount Functions
Slow rate of decline
in long run
1
Rapid rate
of decline
in short run
0
1
11
21
31
41
Week
Exponential
Hyperbolic
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An Alternative Functional Form
Quasi-hyperbolic discounting
(Phelps and Pollak 1968, Laibson 1997)
D(t) = 1, bd, bd2, bd3, ...
Ut = ut + bdut+1 + bd2ut+2 + bd3ut+3 + ...
Ut = ut + b [dut+1 + d2ut+2 + d3ut+3 + ...]
b uniformly discounts all future periods.
d exponentially discounts all future periods.
Building intuition
• To build intuition, assume that b = ½ and d = 1.
• Discounted utility function becomes
Ut = ut + ½ [ut+1 + ut+2 + ut+3 + ...]
• Discounted utility from the perspective of time t+1.
Ut+1 =
ut+1 + ½ [ut+2 + ut+3 + ...]
• Discount function reflects dynamic inconsistency: preferences
held at date t do not agree with preferences held at date t+1.
Application to massages
b = ½ and d = 1
NPV in
current minutes
A 15 minutes now
B 20 minutes in 1 hour
15 minutes now
10 minutes now
C 15 minutes in 1 week
D 20 minutes in 1 week plus 1 hour
7.5 minutes now
10 minutes now
Application to massages
b = ½ and d = 1
NPV in
current minutes
A 15 minutes now
B 20 minutes in 1 hour
15 minutes now
10 minutes now
C 15 minutes in 1 week
D 20 minutes in 1 week plus 1 hour
7.5 minutes now
10 minutes now
Exercise
• Assume that b = ½ and d = 1.
• Suppose exercise (current effort 6) generates delayed benefits
(health improvement 8).
• Will you exercise?
• Exercise Today:
-6 + ½ [8] = -2
• Exercise Tomorrow: 0 + ½ [-6 + 8] = +1
• Agent would like to relax today and exercise tomorrow.
• Agent won’t follow through without commitment.
Beliefs about the future?
• Sophisticates: know that their plans to be patient
tomorrow won’t pan out (Strotz, 1957).
– “I won’t quit smoking next week, though I would like to
do so.”
• Naifs: mistakenly believe that their plans to be patient will
be perfectly carried out (Strotz, 1957). Think that β=1 in
the future.
– “I will quit smoking next week, though I’ve failed to do
so every week for five years.”
• Partial naifs: mistakenly believe that β=β* in the future
where β < β* < 1 (O’Donoghue and Rabin, 2001).
Example: A model of procrastination (sophisticated)
Carroll et al (2009)
•
•
•
•
•
•
•
Agent needs to do a task (once).
– For example, switch to a lower cost cell phone.
Until task is done, agent losses θ units per period.
Doing task costs c units of effort now.
– Think of c as opportunity cost of time
Each period c is drawn from a uniform distribution on [0,1].
Agent has quasi-hyperbolic discount function with β < 1 and δ = 1.
So weighting function is: 1, β, β, β, …
Agent has sophisticated (rational) forecast of her own future
behavior. She knows that next period, she will again have the
weighting function 1, β, β, β, …
Timing of game
1. Period begins (assume task not yet done)
2. Pay cost θ (since task not yet done)
3. Observe current value of opportunity cost c (drawn from
uniform)
4. Do task this period or choose to delay again.
5. It task is done, game ends.
6. If task remains undone, next period starts.
Pay cost θ
Period t-1
Observe current
value of c
Period t
Do task or
delay again
Period t+1
Sophisticated procrastination
• There are many equilibria of this game.
• Let’s study the equilibrium in which sophisticates act whenever
c < c*. We need to solve for c*. This is sometimes called the
action threshold.
• Let V represent the expected undiscounted cost if the agent
decides not to do the task at the end of the current period t:
Likelihood of doing it in
t+1
Likelihood of not
doing it in t+1
 c*
V   +  c *   + 1  c * V
 2 
Cost you’ll pay for
certain in t+1, since
job not yet done
Expected cost
conditional on drawing
a low enough c* so that
you do it in t+1
Expected cost
starting in t+2 if
project was not
done in t+1
• In equilibrium, the sophisticate needs to be exactly
indifferent between acting now and waiting.
c*  bV  b [ + (c*)(c * /2) + (1  c*)V ]
• Solving for
c*,
we find:
• So expected delay is:
c* 

1
1

b 2
E [ delay ]  1 c * +2  1  c * c * +3  1  c * c * +
2
2

1  c *
1  c *


1
 c*
+
+
+
1  1  c * 1  1  c * 1  1  c *
1
1
1
 c *



1  1  c * 1  1  c * c *
1 1

b 2




How does introducing β<1 change the expected delay
time?
E [ delay given b  1]
E [ delay given b =1]

1 1

b 2
 
1 1

1 2

1 1

2
b 2

1
1 1
b

1 2
If β=2/3, then the delay time is scaled up by a factor of 2
Example: A model of procrastination: naifs
•
•
•
•
Same assumptions as before, but…
Agent has naive forecasts of her own future behavior.
She thinks that future selves will act as if β = 1.
So she (falsely) thinks that future selves will pick an action
threshold of
c* 

1
1

b 2
 2
• In equilibrium, the naif needs to be exactly indifferent
between acting now and waiting.
c **  bV
 b [ + (c*)(c * /2) + (1  c*)V ]
 b  +


2  2 / 2 + 1 


 b  2 + 1 

2 V 

• To solve for V, recall that:
 c*
V   +  c *   + 1  c * V
 2 


 2 + 1  2 V
 2

2 V 

• Substituting in for V:

c **  b  2 + 1 

2

2 

 b 2
• So the naif uses an action threshold (today) of
c **  b 2
• But anticipates that in the future, she will use a higher
threshold of
c*  2
• So her (naïve) forecast of delay is:
1
1
Forecast [ delay ] 

c*
2
• And her actual delay will be:
E [ delay ] 
1
1
1


c ** b 2
2
• Her actual delay time exceeds her predicted delay time by
the factor of 1/β.
Choi, Laibson, Madrian, Metrick (2002)
Self-reports about undersaving.
Survey
Mailed to 590 employees (random sample)
Matched to administrative data on actual savings behavior
Typical breakdown among 100 employees
Out of every
100
surveyed
employees
68 self-report saving
too little
24 plan to
raise savings
rate in next 2
months
3 actually follow through
Experiment in Stanford
• http://www.ted.com/index.php/talks/joachim_de_p
osada_says_don_t_eat_the_marshmallow_yet.html
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