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

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• http://www.ted.com/index.php/talks/joachim_de_posada_says_don_t_eat_the_marshmallow_yet.html

Experiment in Stanford