Decision Theory. Plan for today (ambitious) 1.Time inconsistency problem 2.Riskiness measures and...
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Transcript of Decision Theory. Plan for today (ambitious) 1.Time inconsistency problem 2.Riskiness measures and...
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 wealtha) Rabin (2000) paradoxb) 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 wealtha) Rabin (2000) paradoxb) Buying/selling price gap (WTA/WTP disparity)
A Thought Experiment
Would you like to haveA) 15 minute massage now
orB) 20 minute massage in an hour
Would you like to haveC) 15 minute massage in a week
orD) 20 minute massage in a week and an hour
Read and van Leeuwen (1998)
TimeChoosing Today Eating Next Week
If you were deciding today,would you choosefruit or chocolatefor next week?
Patient choices for the future:
TimeChoosing Today Eating Next Week
Today, subjectstypically choosefruit for next week.
74%choosefruit
Impatient choices for today:
Time
Choosing and EatingSimultaneously
If you were deciding today,would you choosefruit or chocolatefor today?
Time Inconsistent Preferences:
Time
Choosing and EatingSimultaneously
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 subjectsMcClure, 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.
Exponential Discount Function
0
1
1 11 21 31 41 51
Week (time = t)
Dis
cou
nte
d v
alu
e o
f d
elay
ed r
ewar
d
Exponential Hyperbolic
Constant rate of decline
-D'(t)/D(t) = rate of decline of a discount function
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? 100.00• What is 100 in a year worth today? 2.55• What is 100 in two years worth today? 0.07• What is 100 in three years worth today? 0.00
Discount Functions
0
1
1 11 21 31 41 51
Week
Exponential Hyperbolic
Rapid rateof decline in short run
Slow rate of decline in long run
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 massagesb = ½ and d = 1
A 15 minutes nowB 20 minutes in 1 hour
C 15 minutes in 1 weekD 20 minutes in 1 week plus 1 hour
NPV in current minutes
15 minutes now10 minutes now
7.5 minutes now10 minutes now
Application to massagesb = ½ and d = 1
A 15 minutes nowB 20 minutes in 1 hour
C 15 minutes in 1 weekD 20 minutes in 1 week plus 1 hour
NPV in current minutes
15 minutes now10 minutes now
7.5 minutes now10 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.
Period t-1 Period t Period t+1
Pay cost θ Observe current value of c
Do task or delay again
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:
*
21 **
cc cV V
Cost you’ll pay for certain in t+1, since job not yet done
Likelihood of doing it in t+1
Expected cost conditional on drawing a low enough c* so that you do it in t+1
Likelihood of not doing 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.
• Solving for c*, we find:
• So expected delay is:
* [ ( *)( * /2) (1 *) ]c V c c c V
*1 1
2
c
2
2
delay 1 * 2 1 * * 3 1 * *
1 * 1 *1*
1 1 * 1 1 * 1 1 *
1 11 1 1 2
*1 1 * 1 1 * *
E c c c c c
c cc
c c c
cc c c
How does introducing β<1 change the expected delay time?
1 11 12
delay given 1 221
1 1delay given =1 1 11 21 2
E
E
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
* 21 1
2
c
• In equilibrium, the naif needs to be exactly indifferent between acting now and waiting.
• To solve for V, recall that:
**
[ ( *)( * /2) (1 *) ]
2 2 / 2 1 2
2 1 2
c V
c c c V
V
V
2 1 2
2
1 ***
2c
c
V
VcV
• Substituting in for V:
• So the naif uses an action threshold (today) of
• But anticipates that in the future, she will use a higher threshold of
** 2 1 2 2
2
c
** 2c
* 2c
• So her (naïve) forecast of delay is:
• And her actual delay will be:
• Her actual delay time exceeds her predicted delay time by the factor of 1/β.
1 1delay
* 2Forecast
c
1 1 1delay
** 2 2E
c
Choi, Laibson, Madrian, Metrick (2002)Self-reports about undersaving.
SurveyMailed 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
32
• http://www.ted.com/index.php/talks/joachim_de_posada_says_don_t_eat_the_marshmallow_yet.html
Experiment in Stanford