NYU Discounting in Time and Space Discussion

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Q1: Measuring Temporal Discounting
Ben’s choices between an immediate option and several later options are marked below. This
format for eliciting indifference points (and, through them, discount rates) is called “titration”
because of its vague similarities to the titration technique you may remember using to measure
acids and bases in chemistry class.
? $100 now
? $90 in 6 months
? $100 now
? $100 in 6 months
? $100 now
? $110 in 6 months
?
$100 now
? $120 in 6 months
?
$100 now
? $130 in 6 months
?
$100 now
? $140 in 6 months
?
$100 now
? $150 in 6 months
?
$100 now
? $160 in 6 months

(a) What is an indifference point?
(b) What can we infer is Ben’s indifference point for a delay of 6 months compared to a gain
of $100 today, based on his responses to the titration above?
(c) You run a discounting study in which you present half of your subjects with a titration list
like the one above (Group A), and present the other half (Group B) with a titration list
that has all of the same numbers, but in reversed order (starting with the choice between
$100 now and $160 in 6 months, and ending with the choice between $100 now and $90
in 6 months). You find that the average indifference point for Group A is $121, and the
average indifference point for Group B is $127, and that the difference is significant.
Using concepts we’ve talked about earlier in this course (e.g., heuristics, biases,
fallacies, etc.), how could you explain why Group B appears to have a higher
indifference point?
(d) The effect illustrated in question (c) is one we actually see in studies. It shows one of the
drawbacks to using this titration method. Identify and explain one other potential
drawback of using titration as a way of eliciting people’s indifference points, and one
benefit of using titration.
(e) Identify and explain one benefit and one drawback to using matching as your way of
eliciting your participants’ indifference points? (Matching = asking Ps to fill in their
indifference point.)
(f) Another participant, Jeremy, selects the later option for every one of the choices in the
titration above (he prefers any amount in 6 months over $100 now, including $90). What
does that tell us about Jeremey’s discount rate?
(g) Another participant, Margo, fills in the same titration with the following responses:
? $100 now
? $90 in 6 months
? $100 now
? $100 in 6 months
? $100 now
? $110 in 6 months
?
$100 now
? $120 in 6 months
?
$100 now
? $130 in 6 months
? $100 now
? $140 in 6 months
? $100 now
? $150 in 6 months
? $100 now
? $160 in 6 months
What does this titration tell us about Margo’s discount rate? If you were the
experimenter, what would you do with her data?
Q2. Wait for it…
When you buy a product online, you are effectively paying money now in exchange for a
product that you won’t get until several days in the future. Let’s say there is a book for sale on
Amazon.com that costs $12, which you consider a fair price.
(a) Why might temporal discounting theory predict that it is not an ideal business model to ask
people to pay money now for a book that they will receive a few days from now? In other
words, why might the delay between paying and actually having the book in your hands
make you less willing to buy the book?
(b) Many online retailers have a policy of not charging a customer’s credit card until the order
has actually shipped (as opposed to charging the card at the moment the order is made).
This means that there is still a delay between payment and delivery of the goods—but
according to temporal discounting theory, this policy should reduce the effect described in
part (a) of this question. Why?
Q3: Query Theory and Discounting
When we talked about process theories, we saw an example of how the thought-listing
paradigm might look for a study that tested whether Query Theory could be used to make
people more or less patient for a delayed reward.
Weber et al. (2007) actually ran this study. They used a 2×2 design, meaning that they had four
different conditions, determined by two independent variables (IVs) that each had two levels:
IV1: Discounting Frame (delay vs. accelerate)
IV2: Order of Thoughts (natural vs. unnatural)
For simplicity’s sake, let’s think about each choice as being between a $50 gift certificate now,
and a larger gift certificate in three months. (We won’t fill in the amount, as that would differ
from participant to participant.)
Our DV of interest is discount rates, where higher discount rates indicate more impatience.
a) First, fill in the table below to indicate for each experimental condition whether we would
expect Ps to think first about the $50-now option, or first about the larger-later option.
natural thought order
unnatural thought order
delay frame
1. $50 now
1.
($50-now is the default)
2. larger later
2.
acceleration frame
1.
1.
2.
2.
(larger-later is the default)
b) Now, predict what the average discount rates for each condition should roughly look like,
by filling in the table below with “high discount rate,” “medium discount rate,” or “low
discount rate,” assuming that the natural/delay condition would lead to a high discount
rate. Remember, higher discount rates mean more impatience.
Hint: the two “natural” conditions have the same predictions you would make for
a plain discounting study.
natural thought order
delay frame
unnatural thought order
high discount rate
($50-now is the default)
acceleration frame
(larger-later is the default)
c) Explain why you made the predictions you did for the two “unnatural thought order”
conditions.
3/1/22
Why is it so hard for people to save for retirement,
even when they know it’s the smart thing to do?
Lecture 12 | March 1, 2022
DECISIONS ACROSS
TIME & DISTANCE
1
Today’s Topics
Temporal Discounting
1.
2.
3.
4.
5.
How do we discount?
Why do we discount?
What influences discount rates?
domain effects on discounting
discounting in complex situations
2
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Temporal Discounting
3
Now or Later?
• Which would you prefer?
A: $100 right now
B: $110 three months from now
• What about this choice?
A: $100 right now
B: $120 three months from now
• Or this one?
A: $100 right now
B: $150 three months from now
4
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1. How do we discount?
5
Normative Model:
Exponential Discounting
Present Value of 1 Util
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
Delay (in years)
40
45
50
6
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Another Discounting Problem
• Which do you prefer:
A. $100 in one year
B. $110 in 15 months
• Earlier, I asked you:
A. $100 right now
B. $110 three months from now
7
Delay, or now vs. not now?
• We need a much better later offer to make up for not getting
something now.
• BUT, when adding a constant delay to both options, our
preferences often reverse.
– If we have to wait anyway, we’ll wait a little longer in
order to get even more.
• People (and animals) are more patient if there is no
immediate option (Thaler 1981).
• This means that time is non-fungible!
– The difference between now and later is bigger than the
difference between later and even later.
8
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Descriptive Model:
Hyperbolic Discounting
Present Value of 1 Util
1.2
1
0.8
Exponential
Hyperbolic
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
50
Delay (in years)
9
Mathematical Models of Discounting
exponential
! = # $ % &'(
hyperbolic
#
!=
(1 + ,-)
V = discounted value of A
A = absolute value of money (future offer)
D = delay (in years)
k = discount rate
e = a constant
if indifferent between:
$110
$100 today vs.
$100 =
(1 + , $ 0.25)
$110 in 3 months
, = 0.40
10
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2. Why do we discount?
11
Rational Justifications for Discounting
• interest on investment
– if we take the money now, we can invest it
• inflation
– the money is worth more now than it will be in the future
• uncertainty
– If we wait, will it ever happen?
• pressing need
– My rent is due this week… I can’t afford to wait longer
• less real value from enduring goods if received later
– if we get it now, we’ll have it longer than if we wait
• resource slack
– take from the rich, give to the poor
– in the future I’ll be richer, so I need the money more now
12
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Me, Myself, and the Future I
• rational self-interest
– future self is like a different person
– I’m worried about me now! Future Me can take care of
herself!
• More “closeness to future self” predicts less discounting.
– People who expect a life-changing event feel less close to
their future self…
• …and discount more.
• Manipulating closeness to future self
decreases discounting.
– have Ps write about how they’ll
be (dis)similar to future self
– show people age-morphed
photos of themselves
13
“Non-Rational” Reasons to Discount
• impatience
– like small children, we just want it now
• self-control (or lack thereof)
– Waiting is hard!
– Walter Mischel’s “marshmallow test”
• actually a discounting problem: one treat now, or two
treats in the future?
• kids who wait longer for the
second marshmallow
discount less later in life
• self-control at Age 4 correlates
with other positive life outcomes
14
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Why do we discount?
• There are many rational reasons why future money should
be less valuable to us than money now.
– putting the “current” in “currency”
• If you add up all the rational reasons why money is less
valuable when it’s delayed, we still discount more than
you (or an economist) would expect.
– There’s also a psychological component to temporal
discounting.
15
3. Which Factors Influence
Discount Rates?
16
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4 Influences on Discount Rates
• The higher the discount rate, the less the future is worth
(we need a larger future offer in order to wait).
• Negative discount rates mean the future is worth more (we
would wait longer for less money).
• Discount rates vary systematically depending on:
1. sign of the outcomes (gain vs. loss)
2. magnitude of outcomes
3. directional frame of the problem (is the present the
status quo, or the future the status quo?)
4. whether there is an immediate option.
17
1. The Sign Effect
• (We want to delay losses: we prefer later-larger losses over
smaller-sooner ones.)
• We discount losses less than gains.
– Losses retain more of their value over time than gains do.
• lower discount rate for losses
– We want more compensation for waiting for a gain than
we are willing to pay to delay a loss.
• indifferent between $100 today vs. $120 in 1 month
• indifferent between -$100 today vs. -$108 in 1 month
18
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Willing to wait for it?
• Which payout would you prefer?
A. $10,000 today
B. $11,000 three months from now
• Earlier, I asked you:
A. $100 right now
B. $110 three months from now
19
2. The Magnitude Effect
• We discount large magnitudes less than small ones.
– more willing to delay large gains
– We want proportionally more compensation for delaying a
small amount.
• $100 now vs. $110 in 3 months?
– increase of 10% over 1 month
– most of you are unwilling to wait
• $10,000 now vs. $11,000 in 3 months?
– increase of 10% over 1 month
– most of you are willing to wait
20
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3. The Direction Effect
• Imagine I’ll give you $120 three months from now.
– Or, if you prefer, you can take $100 today.
• In an acceleration frame:
– The future is the default option.
– “$120 in the future vs. Y now”
– People are more patient!
• smaller discount rates
• lower indifference points
– Example of attribute framing!
Discount Rate
• The typical discounting study format is delay:
– “$100 now vs. X in the future”
Accelerate
Delay
21
4. Now vs. Not Now
• The typical discounting study format is between now and
the future.
– “$100 now vs. ___ in 3 months”
• If you push both options into the future, people are more
patient:
– “$100 in one year vs. ___ in 15 months”
• lower discount rates
• lower indifference points
• Something about an immediate option gives it extra
weight.
– analogue of the certainty effect in Prospect Theory
22
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Combining Multiple Effects
• In which condition would people discount the most?
(Assuming that all outcomes are gains.)
A.
B.
C.
D.
small magnitudes, choice between now and later
large magnitudes, choice between now and later
small magnitudes, choice between later and even later
large magnitudes, choice between later and even later
23
Reflection Effects
• For most of the systematic effects on discount rates,
combining them just adds their effects to each other.
• But moving from gains to losses changes more than the
discount rate.
• Magnitude effect and delay/accelerate effect are reversed
for losses.
– We see a reflection effect: in losses,
• we discount large losses more than small losses (in
gains, we discount large gains less than small gains)
• we discount more in the accelerate frame than in the
delay frame (in gains, we discount more in the delay
frame)
24
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Influences on Discount Rates
• Discount rates vary systematically depending on:
1. the sign of the outcomes (gain vs. loss)
2. the magnitude of outcomes
3. the frame of the problem (is the present the default, or
the future the default?)
4. whether there is an immediate option
• Combining the sign effect (switching from gains to losses)
with others causes the other effects to reverse.
– “reflection effect”
25
4. Domain Effects on Discounting
26
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Discounting Across Domains
•
Intertemporal choice refers
to any decisions involving
two time periods.
– now and later
– near future and far future
• In any domain:
–
–
–
–
–
$50 now or $90 in 3 months?
Study gradually all weekend, or painfully on Sunday evening?
1 marshmallow now, or 2 marshmallows in 20 minutes?
2 days of migraine starting now, or 3 days starting next week?
30 days of improved air quality starting
today, or 35 days of improved air
quality starting in two weeks?
27
Discounting Future Health
• For health outcomes:
– still see sign effect,
magnitude effect,
delay effect
– but health discounting is
independent of financial
discounting
• On average, discounting for health & money is similar.
• But the people who discount money strongly are not
necessarily the same people who discount health strongly.
– domain specificity of discounting
Chapman, 1996
28
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Discounting the Environment
• Health discounting appears distinct from money.
• But discounting of money & environmental outcomes is
similar.
Hardisty & Weber, 2009
29
5. Discounting in Complex
Situations
30
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Outcome Streams
• For your birthday, your family has gotten you two gifts: one
big, awesome present, and one smaller, side present. You
get to open one gift today on your actual birthday, and the
other gift this weekend at your party. Would you rather:
A. open the small gift today, and the big gift in a few days
B. open the big gift today, and the small gift in a few days
C. flip a coin to decide which gift to open today
31
Outcome Streams
• People typically prefer increasing payoffs to decreasing
payoffs.
• Where would you rather eat: French restaurant or Greek
restaurant?
– most students preferred French
• Would you want to eat there in 1 month or 2 months?
– most students (80%) said 1 month
• If you were allowed to eat in both restaurants, one in 1
month and the other in 2 months, which order would you
prefer?
– more students (57%) said Greek then French
– prefer their 2nd-rated option first and their 1st option later
(Loewenstein & Prelec, 1989) 32
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Income Streams
• If you signed a contract to work for 6 years at a company,
and you were allowed to allocate your salary in any of the
following four ways, which would you choose?
Note: each option has the same total salary across 6 years.
A
B
C
D
33
Income Streams
• Decreasing income streams are most rational.
– Employer: After initial training period, people are more
productive initially than later in life.
– Employee: If you quit or get laid off, better to have
already been paid more.
– Employee: If you are paid more initially, it can be
invested and produce more money overall.
• And yet, we tend to prefer increasing
income streams.
– 76% of participants in a study of
different income streams that
were increasing, flat, or decreasing
– Why?
34
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When do we want to wait?
• Why might we want to:
– postpone positive outcomes, or
– speed up negative outcomes?
• Each of those preferences would indicate negative
discounting.
Pay a $70 parking ticket now, or pay $66-69 in 3 months?
25% of you
75% of you
35
Anticipation
• Sometimes we savor the positive anticipation leading up
to an outcome.
– vacations
– nice meal out
• Sometimes we dread the negative anticipation leading
up to the outcome.
– dentist visits
– confrontation with friend or co-worker
– paying a parking ticket
36
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Anticipation
• Savoring & dread both lead to negative discounting.
– We’ll voluntarily take a good outcome later to be able to
look forward to it.
– We’ll pay extra now to get a loss over with.
37
Temporal Discounting: Conclusions
• Because we discount the utility of future outcomes, we
usually want gains now and losses later.
– We’ll take a smaller-sooner gain over a larger-later gain.
• Our temporal discount rate indicates how impatient we are:
high discount rates mean we de-value future outcomes
more heavily.
– Discount rates vary systematically based on 4 influences.
• Discounting influences choices in many domains:
– unhealthy behavior (Bickel et al., 1999)
– insufficient savings (Thaler & Benartzi, 2004)
– depletion of environmental resources (Hendrickx et al., 2001)
• Negative discounting is possible, usually in situations of
anticipation (savoring or dread).
38
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Looking Ahead
• Decision Analysis assignment available later this week.
– Due Thursday after Spring Break (March 24)
– Depends heavily on understanding decision modes
• In-class activity next Thursday will help clarify decision
modes
– important to do the reading ahead of time
– In place of videos for that class: watch an episode of a
TV show (the options are listed on the Canvas page for
Lecture 15)
– If you don’t have access to any of the shows, let us know.
39
20
3/3/22
How could I raise your value for a
mug without endowing you with it?
Lecture 13 | March 3, 2022
PROCESS MODELS OF
DECISION MAKING
1
Overview of Process Models
1. process behind the Endowment Effect
2. Query Theory
3. Query Theory’s explanation of the Endowment effect
• Preview of Decision Architecture
2
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How do we make decisions?
1.
What is the decision-making process?
2.
What are the different modes of decision making?
3
Process Accounts
• Descriptive theories explain that we choose A.
• Process accounts explain how we get to A.
A
4
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1. Process Models of the
Endowment Effect
5
Focusing on the Foregone
• Endowment Effect for hypothetical NCAA Final Four tickets
– buyers’ WTP = $166
– sellers’ WTA = $1500
• Buyers and sellers focus on different information when
estimating price.
• Each focuses on what they will have to give up:
– buyers: money
• “There are lots of other things I could do with the money.”
– sellers: tickets/experience
• “This is a once in a lifetime opportunity.”
• “I will never forgive myself if I end up missing a great game.”
Carmon & Ariely (2000)
7
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Buyers vs. Sellers
• Each group seems influenced by a different set of thoughts:
– buyers’ WTP determined by expenses related to game,
ticket face value, attitudes toward money
– sellers’ WTA determined by experiences related to
game, intensity of fandom, significance of game
• Is this a convincing explanation?
• We need empirical evidence about the decision process:
– How does endowment lead to different thoughts?
– How do different thoughts lead to different choices?
8
2. Query Theory
9
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QT Study: Introducing the Choice
10
QT Study: Thought Listings
11
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QT Study: Thought Listings
12
QT Study: Thought Listings
ht
tho ug
s
listing
13
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QT Study: Choice Elicitation
14
QT Study: Choice Elicitation
• Which of these values could be this P’s indifference point?
A.
B.
C.
D.
E.
$50
$60
$61
$63
$65
15
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QT study: Coding the Thoughts
• Here are the thoughts you entered before your choice:
– “I need $$ now!”
– “I can’t wait.”
now; positive
future; negative
• For each, please indicate whether it was about getting
money now or about getting money in the future.
• For each, please indicate whether the thought was positive
or negative.
• …so what can we learn from thought listings, and Ps’ selfcoding of those thoughts?
16
Query Theory: A Process Story
• Assuming that preferences are constructed:
– When faced with a decision, we ask ourselves a series of
questions in order to assess our preferences.
• (questions = queries)
•
•
•
•
“What do I need to buy on Amazon.com now?”
“How much was that book I looked at the other day?”
“If I waited and had more money, what else could I buy?”
“Is it worth $25 extra to wait to get my stuff?”
– We ask the questions serially.
– We naturally start with the option that is presented as the
default or status quo.
Johnson, Häubl, & Keinan (2007); Weber et al. (2007)
17
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Inhibition of Later Queries
• For a variety of reasons, we are able to come up with more
answers for our earlier questions than the later ones.
– e.g., retrieval-induced forgetting
– e.g., inhibition of later answers caused by earlier ones
• e.g.: https://youtu.be/Wu1kSXpVV8Y?t=2319
– e.g., cognitive dissonance
– e.g., confirmation bias
Johnson, Häubl, & Keinan (2007); Weber et al. (2007)
18
QT: Putting it All Together….
• We ask ourselves questions, one at a time.
• We first ask ourselves questions about the default.
– For Endowment Effect, this is “the thing we own.”
– For temporal discounting, this is either the “now” option
(in delay frames) or the “later” option (in acceleration
frames).
• We end up with more answers for earlier questions than
later questions.
– …so we have more reasons in favor of the option we
considered first,
– …so we’re more likely to choose it.
Johnson, Häubl, & Keinan (2007); Weber et al. (2007)
19
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Query Theory
Query Theory is a process theory that can help to explain
the mental process that governs:
•
•
•
•
temporal discounting
framing effects
endowment effect
default effect
– When faced with a choice between two options, we
stick with the default more often than we switch to the
alternate.
– Default: the thing we would get
if we failed to specify that we
want the other thing.
20
Thought Order Mediates
the Default Effect
thoughts about 1% milk
are more frequent,
come first
correlation
1% milk is the
default
correlation
strong correlation
weaker correlation
preference for
1% milk
21
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3. Query Theory’s
Explanation of the
Endowment Effect
22
To Buy or Not to Buy vs.
to Sell or Not to Sell
sellers
1. Query: what’s good
about the mug?
2. Query: what’s good
about money?
• end up collecting more
answers in support of
mug
• place a higher value on
the mug
– high WTA
buyers
1. Query: what’s good
about money?
2. Query: what’s good
about the mug?
• end up collecting more
answers in support of
money
• place a lower value on
the mug
– low WTP
Johnson, Häubl, & Keinan (2007)
23
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QT: predictions
• Endowment predicts order of thoughts
– thoughts about the thing you have come earlier
• Endowment predicts number of thoughts
– more thoughts about the thing you have
• Endowment predicts choice/value
– more likely to choose and/or place high value on the
thing you have
• mediation: Endowment à thoughts à choice/value
– Endowment Effect is driven by thought order
– Thought order mediates the effect of endowment on
valuation of an item.
Johnson, Häubl, & Keinan (2007)
24
Johnson, Häubl, & Keinan (2007)
25
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Johnson, Häubl, & Keinan (2007)
26
Query Theory: implications
• Influencing people’s thoughts can change their choices.
– eliminate endowment effect for sellers
– create endowment effect for buyers
• How? Make people consider the opposite first!
– natural thought order, vs.
– unnatural thought order
Johnson, Häubl, & Keinan (2007)
27
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Eliminating Endowment Effect for Sellers
• Natural thought order:
– What’s good about mug?
– What’s good about money?
• Unnatural thought order:
– What’s good about money?
– What’s good about mug?
• After thought order manipulation: earlier/more thoughts
about not owning à more likely to choose/place higher
value on money
– lower WTA price
Johnson, Häubl, & Keinan (2007)
28
Creating Endowment Effect for Buyers
• Natural thought order:
– What’s good about money?
– What’s good about mug?
• Unnatural thought order:
– What’s good about mug?
– What’s good about money?
• After thought order manipulation: earlier/more thoughts
about owning à more likely to choose/place higher value
on mug
– higher WTP price
Johnson, Häubl, & Keinan (2007)
29
14
ß $ good … mug good à
3/3/22
30
Johnson, Häubl, & Keinan (2007)
31
ß $ good … mug good à
Johnson, Häubl, & Keinan (2007)
15
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Endowment out of thin air?
• Buyers’ natural thought order:
– What’s good about money?
– What’s good about mug?
• Sellers’ natural thought order:
– What’s good about mug?
– What’s good about money?
• Assign Ps to one condition or the other.
• Condition determines thoughts and price, without any actual
endowment!
Johnson, Häubl, & Keinan (2007)
32
Johnson, Häubl, & Keinan (2007)
33
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Process Models: Conclusions
• Process is an important part of decision making.
– Just knowing what we choose isn’t enough.
– Understanding why we choose it tells us more about
thinking & decision making.
• shows us why thoughts matter so much to choices
• shows us how the choice process could be influenced
thoughts about Option A are
more frequent, come first
Ownership of
Option A
Higher value placed
on Option A
34
Preview of Decision Architecture
• Query Theory is a process theory that explains how/why
defaults (among many other things) influence our choices.
– The default effect is one example of a decision
architecture tool.
• Decision Architecture: tailoring the presentation of a choice
in such a way as to influence the decision.
– can be used for good
– can be used for bad
– can be used for profit
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Preview of Decision Architecture
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Temporal Discounting: Review I
• Which of the following is a correct statement about temporal
discounting?
A. we want all outcomes (both gains & losses) to happen
now, rather than later
B. we want all outcomes (both gains & losses) to happen
later, rather than now
C. we want to get gains now and pay losses later
D. we want to get gains later and pay losses now
E. losses loom larger than gains
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Impatience Causes Problems
• We want gains now, and losses later.
– You have to offer more in the future to make me willing to
wait for a gain (instead of having it now).
– You have to charge more in the future to make me willing
to take a loss now (instead of paying it later).
• We leave money on the table.
– e.g., take money now instead of investing for retirement
• We put off solving expensive problems.
– e.g., don’t want to pay to build stronger levees now, even
though the cost of a flood later is high
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Group Discussion
1. What are the four things that appear to systematically
influence discount rates?
2. Which of these 4 effects could you use as leverage to
make people more willing to put $ into savings?
• How could you encourage someone to forego a
smaller gain now in favor of a larger gain later?
3. Which of these 4 effects could you use as leverage to
make people willing to buy insurance?
• How could you encourage someone to accept a
sooner loss rather than incurring a larger one
later?
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3/3/22
One solution: use the “now” effect.
• Discounting is reduced if there is no “now” option.
– $30 today vs. $40 ten days from now
– $30 tomorrow vs. $40 eleven days from now
• precommitments
– “starting with next month’s paycheck…”
– insurance plan where you pay no money upfront
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Another solution: use framing!
• To encourage patience for
gains, offer two marshmallows
in the future…
– …or if you want, you can
take just one now.
• To get people to give up
money now, make the
present the default.
– I’ll charge you $100 if you pay the fee
upfront, or $120 if you pay after the
job’s done next month
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3/3/22
Looking Ahead
• Decision Analysis assignment available later this week.
– Due Thursday after Spring Break (March 24)
– Depends heavily on understanding decision modes
• In-class activity next Thursday will help clarify decision
modes
– important to do the reading ahead of time
– In place of videos for that class: watch an episode of a
TV show (the options are listed on the Canvas page for
Lecture 15)
– If you don’t have access to any of the shows, let us know.
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