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Broad bracketing for low probability events

Shereen J. Chaudhrya*

aUniversity of Chicago Booth School of Business

Chicago, IL 60637

Phone: 210-410-9396

Email: shereen.chaudhry@chicagobooth.edu

ORCID: 0000-0002-2248-8524

Michael Handb

bU.S. General Services Administration Office of

Evaluation Sciences

1800 F St NW

Washington, DC 20405

Phone: 202-538-3113

E-Mail: michaels.hand@gsa.gov

Howard Kunreutherc

cDepartment of Operations, Information and Decisions

Wharton School, University of Pennsylvania

3730 Walnut St., #563 Huntsman Hall

Philadelphia, PA 19104-6340

Phone: 215-898-4589

E-Mail: kunreuth@wharton.upenn.edu

*Please direct correspondence to Shereen Chaudhry at the contact information listed above.

Author’s note: This research was conducted while Dr. Chaudhry was employed as a postdoctoral research fellow at

the Wharton Risk Management and Decision Processes Center, and while Dr. Hand was employed with the U.S.

Department of Agriculture, Forest Service. The views expressed in this paper are the authors’ and do not necessarily

represent the views or positions of the U.S. Department of Agriculture or the U.S. General Services Administration.

Acknowledgements: We wish to express gratitude to Allyson Barnett, Rick Larrick, and Ray Sin for feedback on

the experimental design and/or earlier drafts of this paper. We appreciate helpful discussions with our colleagues at

the Wharton Risk Center throughout the development of this paper: Jeff Czajkowski, Carol Heller, Carolyn Kousky,

Brett Lingle, Robert Meyer, Marilyn Montgomery, and Gina Tonn. We would also like to thank Nyzinga Patterson

for administrative support in running the experiments. We appreciate additional feedback from participants at the

2018 SABE sessions at the WEAI conference and the 2018 meeting of the Society for Judgment and Decision

Making.

Funding: Support for this research comes from a grant from the Sloan Foundation (G-2018-11100 / SUB18-04); a

National Science Foundation (NSF) grant (EAR-1520683 / SUB0000091) through Princeton University; the Geneva

Association; the Travelers Risk Management and Leadership project; and the Wharton Risk Management and

Decision Processes Center’s Extreme Events project.

Conflicts of interest: None of the authors has any conflicts of interest.

Availability of data and material: All data, power analyses, and pre-registrations are available on the Open

Science Framework (OSF) at: https://osf.io/mejf5/?view_only=9b8760dca9714e85bc178f290140125e.

Code availability: Data was analyzed using R and STATA. Code for data analysis available upon request.

Authors’ contributions: All authors contributed in a significant way to the manuscript.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

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Abstract: Individuals tend to underprepare for rare, catastrophic events because of biases in risk perception. A

simple form of broad bracketing—presenting the cumulative probability of loss over a longer time horizon—has the

potential to alleviate these barriers to risk perception and increase protective actions such as purchasing flood

insurance. However, it is an open question whether broad bracketing effects last over time: There is evidence that

descriptive probability information is ignored when decisions are made from “experience” (repeatedly and in the

face of feedback), which describes many protective decisions. Across six incentive-compatible experiments with

high stakes, we find that the broad bracketing effect does not disappear or change size when decisions are made

from experience. We also advance our understanding of the mechanisms underlying broad bracketing, finding that,

while cumulative probability size is a strong driver of the effect, this is dampened for larger brackets which lead

people to be less sensitive to probability size.

(156 words)

Keywords: Broad bracketing; Risk communication; Decisions from experience; Cumulative probabilities; Low-

probability high-consequence events; Insurance

JEL codes: D81; D91; D01

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1. Introduction

Companies and individuals often fail to protect themselves against rare, catastrophic events, such as

flooding form natural disasters. Protective decisions can generally be thought of as choices that present two options,

both with negative expected value: incur some cost for sure (e.g., purchasing flood insurance) or take a chance of

incurring a much larger cost than the sure cost (e.g., a flood occurs causing one to pay out of pocket to repair flood

damage to their house). In flood-prone areas in the United States, relatively few individuals have voluntarily taken

the protective action of purchasing insurance, opting to take their chances with flood to their detriment. For

example, the Federal Emergency Management Agency (FEMA) estimated that only 17 percent of residents most

affected by Hurricane Harvey had flood insurance (Long, 2017), and as a result, the U.S. federal government paid

out over $1.2 billion dollars in housing assistance (FEMA, 2018). Thus, not taking protective actions harms flood

victims and creates substantial negative externalities for governments.

Individuals’ failure to engage in protective behaviors against rare catastrophes like flood likely stems from

a combination of two judgment processes that defy the assumptions of expected utility theory. First, people often

consider small probability events too rare to pay attention to, or below their threshold level of concern (Kunreuther,

1996; Robinson & Wouter Botzen, 2018; Slovic et al., 1977). Though in many cases people overweight small

probabilities as suggested by prospect theory (Kahneman & Tversky, 1979), in situations like natural disasters,

where people receive feedback over time and update their beliefs about likelihoods of occurrence, people tend to

underweight small probabilities (Barron & Erev, 2003; Hertwig, Barron, Weber, & Erev, 2004; Weber, Shafir, &

Blais, 2004). Because the probability of events like natural disasters are often communicated in one-year increments

(e.g., 1% annual chance), this may lead people to take their chances with the rare event rather than incur the upfront

cost of protection without really considering what is optimal given their risk preferences.

Second, people exhibit myopia, or narrow bracketing, in their judgments and decisions—that is, people

tend to focus on short time periods rather than acknowledging long-term exposure (Kahneman & Lovallo, 1993;

Read, Loewenstein, & Rabin, 1999; Redelmeier & Tversky, 1992). Thus, a homeowner may consider the annual 1%

risk of a flood damage to their house to be too small to care about and not think about the fact that, because they

plan on living in their house for 10 years, the risk they face is actually a 10% chance of one or more floods over that

period. Furthermore, even when people do consider the long-term, they tend to underestimate the cumulative effect

of repeated exposure to risk (Doyle, 1997; Fuller, Dudley, & Blacktop, 2004; Keller, Siegrist, & Gutscher, 2006;

Knäuper, Kornik, Atkinson, Guberman, & Aydin, 2005; Linville, Fischer, & Fischhoff, 2015; Shaklee & Fischhoff,

1990; Slovic, Fischhoff, & Lichtenstein, 1978).

Previous research suggests that directly countering the narrow bracketing tendency by using broad

bracketing might lead people to behave differently and to opt more often for the protective action. Broad bracketing

involves conveying cumulative information about the distribution of possible outcomes of a gamble over a certain

broad period of time. For example, when converted to a 50-year bracket, a 1% annual chance of flood translates to

the following: 60% chance of no flood, 31% chance of one flood, 8% chance of two floods, 1% chance of three or

more floods. Researchers found that providing the broad bracket distribution for choices between a sure loss (-$4)

and a pure loss gamble played 50 times (-$0.10, 90%; -$0.50, 10%) led people to avoid the gamble and choose the

sure loss more often (Webb & Shu, 2017). This setup is analogous to protective decisions and so we will use the

terms protective decisions and pure loss choices interchangeably, as well as the terms protective action and sure

loss.

We suggest that applying broad bracketing to the context of protective decisions will lead more people to

opt for the protective action. In this investigation, we implement and test whether this effect holds for a simpler,

more scalable version of broad bracketing: We present only the cumulative loss probability, or the cumulative

probability of one or more losses over a given time horizon (e.g., 39% chance of one or more floods over 50 years).

Two studies have examined the effect of such cumulative loss probabilities on risk perceptions (De La Maza, Davis,

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Gonzalez, & Azevedo, 2019; Keller et al., 2006), but no prior work has examined the impact on actual behavior (for

a review see Visschers, Meertens, Passchier, & de Vries, 2009).

Central to our investigation is addressing the following concern: While the evidence from Webb and Shu

(2017) suggests that broad bracketing will result in increased protective action, there is evidence that its effect will

diminish over time due to people’s experiences. Experience is a key feature of protective decisions, which often

must be made monthly or annually and after learning whether the event has occurred or not (e.g., annually

purchasing flood insurance, regularly clearing drains or checking that one’s sump pump is operational). Prior to a

disaster, individuals ignore the possibility (optimism), then reverse and pay it heightened attention after the disaster

occurs (availability), and finally, after time passes without another loss, they forget the impact the disaster had and

revert to ignoring its possibility again (amnesia; Meyer & Kunreuther, 2017). Research has shown that, even if

people learn objective probability information about disasters (i.e., descriptive information), they begin to ignore

this descriptive information (which the broad bracket is) in favor of information based on their own experience

(Jessup, Bishara, & Busemeyer, 2008; Lejarraga & Gonzalez, 2011; Newell, Rakow, Yechiam, & Sambur, 2016;

Rakow, Demes, & Newell, 2008; Yechiam & Busemeyer, 2006).

Neither the Webb and Shu article nor other research on broad bracketing (Benartzi & Thaler, 1999; Hardin

& Looney, 2012; Looney & Hardin, 2009; Thaler, Tversky, Kahneman, & Schwartz, 1997) has demonstrated that

broad bracketing would be robust to experience. We are the first to examine the role of experience and test whether

broad bracketing can shift people strongly and consistently toward protective behavior when decisions are made

repeatedly in the face of feedback. The results have implications for whether or not broad bracketing could serve as

a useful tool for risk communication regarding protective decisions.

Past research has also left unknown the mechanism underlying broad bracketing by always confounding

two features in the broad bracket manipulation: probability size and time horizon length. As a result, it is unclear

whether people only respond to the larger probabilities or whether they also consider the time horizon over which

the probabilities are calculated. We expect that, while larger probabilities will lead more people to attend to and

respond to the broad bracket, longer time horizons will moderate that response. For example, we expect that

presenting people with a 10% chance of flood over 10 years will lead to more protective behavior than presenting

them with a 10% chance over 200 years. Utilizing our more scalable version of broad bracketing allows us to isolate

the key attributes of the framing technique (probability size and time horizon length) in a way that has not been done

before and uncover the separate roles played by these two components in the broad bracket effect. These findings

provide insight into the judgment processes underlying broad bracketing and serve as guidance for how to structure

a broad bracket intervention for risk communication.

In the next section, we review previous work on risk perception and broad bracketing and discuss how the

present work contributes. In Section 3, we provide an overview of the experiments. In Sections 4-9, we describe the

rationale, methods, and results for Studies 1-6. In Section 10, we discuss the implications of our findings and

suggest future directions.

2. Risk perception and decisions from experience

People tend to underweight small probabilities when learning about probabilities through experience rather

than through explicit presentation or description (Barron & Erev, 2003; Hertwig et al., 2004; Weber et al., 2004).

People also sometimes engage in an editing process wherein they ignore events with probabilities below their

threshold level of concern (Kunreuther, 1996; Robinson & Wouter Botzen, 2018; Slovic et al., 1977). Because

individuals often assess the probability of an event, like a natural disaster, using their own experiences and because

these events tend to be rare, it is likely that these facts helps explain why people choose not prepare for future

disasters (Kunreuther et al., 1978; McClelland, Schulze, & Coursey, 1993; Meyer & Kunreuther, 2017). The upfront

cost may not seem worth it given the infrequent observation of occurrence.

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Even if a person faces a one-time risk that is very small, the chances of a negative event occurring increases

with repeated exposure to that risk. In an extreme example, a 1% annual chance of flood translates to an

approximately 0.0027% daily chance of flood. On any given day the miniscule chance of flood likely falls below the

threshold of concern for most people. But a 1% annual chance of flood also translates to a 26% chance of at least

one flood over 30 years, the term of a typical mortgage. If the broad bracket (i.e., 30-year time horizon) more closely

aligns with how people make residential location and home purchase decisions than the narrow bracket (i.e., one

year or one day), then the broad-bracket probability information may be more relevant for aligning protective

actions to risk preferences.

Evidence suggests people tend to narrowly bracket their judgments and decisions, thinking only about the

near term (Kahneman & Lovallo, 1993; Read et al., 1999; Redelmeier & Tversky, 1992). This myopic behavior is

reinforced by the external choice context, which often frames risks over a small time period. For example, flood

risks are normally presented in terms of the annual chance of a disaster, rather than a time horizon that is likely to be

relevant for residential location decisions.

People also underappreciate how risks accumulate over time because of the complexity of the calculation

(Doyle, 1997; Fuller et al., 2004; Keller et al., 2006; Knäuper et al., 2005; Linville et al., 2015; Shaklee & Fischhoff,

1990; Slovic et al., 1978). Thus, even when asked to consider a larger bracket, unless people are given explicit

probabilities, they tend to underestimate how much aggregation over time or repeated exposure affects the

probabilities (Redelmeier & Tversky, 1992).

2.1 Broad bracketing and cumulative probabilities

Given people’s tendencies to underweight low probability events, to narrowly bracket their decisions, and to

underappreciate the aggregation of risk over time, explicitly presenting cumulative probabilities over a longer span

of time—such as broad bracketing—could help people consider rare, catastrophic events as within their threshold

level of concern. Broadly bracketed information about the risk of a catastrophic loss may reduce risk seeking

(preference for the pure loss gamble) and encourage protective actions (preference for the sure loss) like purchasing

insurance. While previous work on broad bracketing involved presenting the full distribution of outcomes, we

present only a single number representing the probability of at least one loss within that time, i.e., the cumulative

loss probability, which is most relevant for considering losses from natural disasters. The risk perception literature

has documented people’s difficulty in recognizing and understanding cumulative risk (Doyle, 1997; Fuller et al.,

2004; Keller et al., 2006; Knäuper et al., 2005; Linville et al., 2015; Shaklee & Fischhoff, 1990; Slovic et al., 1978),

and two papers have examined the effect of cumulative loss probabilities on judgments (De La Maza et al., 2019;

Keller et al., 2006), but no work has examined how explicitly presenting cumulative risk can be helpful in changing

behavior.

Hypothesis 1 (H1). Compared to narrow bracketing, broad bracketing operationalized as the “cumulative

loss probability” will increase the likelihood that people will choose a sure loss over a pure loss gamble.

2.2 Broad bracketing in the face of repetition and feedback

Critically, many real-world decisions, especially those involving protective actions, require people to make

decisions repeatedly. In other words, the decisions cannot be locked in for the period of the broad bracket and,

furthermore, the decision maker cannot avoid receiving feedback over time. For instance, homeowners normally

decide annually whether to purchase a one-year flood insurance policy, even if they plan to live in their homes for

several years. This decision is then followed by feedback about whether or not a flood has actually occurred during

that year.

Because evidence suggests that broad bracketing will not hold up in the context of experience with

repetition and feedback, this an important feature to test. Research suggests that decisions makers tend to lean more

on information they get from experience and less on the explicit descriptions of event probabilities they are given

(with broad bracketing falling into the description category; Jessup et al., 2008; Lejarraga & Gonzalez, 2011; Newell

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et al., 2016; Rakow et al., 2008; Yechiam & Busemeyer, 2006). For low probability events, people may begin to

ignore the broadly bracketed information over time and switch from sure loss (protective action) to the pure loss

gamble (no protective action).

This is the first investigation to examine whether broad bracketing lasts and is robust to experience. Most

empirical research on broad bracketing has focused on decisions in which participants make a single decision that is

locked in for the period of the broad bracket (Benartzi & Thaler, 1999; Gneezy & Potters, 1997; Redelmeier &

Tversky, 1992; Webb & Shu, 2017). Studies with repeated decisions and feedback have not explicitly examined the

effect of experience or asked whether it interacted with broad bracketing (Hardin & Looney, 2012; Looney &

Hardin, 2009). These designs also did not involve rare events, which are the types of events for which one is most

likely to observe a deviation in experience-based and description-based decisions over time.

To formulate our hypothesis about the interaction of broad bracketing with experience, we account for two

strong patterns of behavior that have been observed in decisions from experience: the recency effect and the

gambler’s fallacy (Croson & Sundali, 2005; Plonsky, Teodorescu, & Erev, 2015; Yin, Chen, Kunreuther, & Michel-

kerjan, 2017). The recency effect leads to excessive focus on recent experiences (Barron & Yechiam, 2009; Hertwig

et al., 2004; Hogarth & Einhorn, 1992) or those experiences most available in memory (Tversky & Kahneman,

1973). In a flood context, the recency effect suggests that a homeowner is likely to be more risk averse and engage

in protective behavior in the immediate aftermath of a flood; as time passes without the occurrence of another flood,

the emotional effects of the experience fade and the homeowner may decide to take their chances and drop their

insurance. Behavior depends on events in the decision maker’s most recent memories, which changes over time.

The gambler’s fallacy is due to a mistaken understanding of what randomness looks like: People assume

that events that have just occurred are less likely to occur next period but are more likely to occur as time passes

since their last occurrence (Ayton & Fischer, 2004; Jarvik, 1951; Kahneman & Tversky, 1972). The gambler’s

fallacy suggests that a homeowner would be less likely to purchase insurance immediately after experiencing a flood

but would start to worry that another flood is “due” to happen and invest in protection as time passes without the

occurrence of a flood.

Both the recency effect and gambler’s fallacy imply that experience-based choices exhibit some baseline

variation over time. To account for the possible influence of the recency effect or gambler’s fallacy on broad

bracketing, we consider two timeframes: the “immediate” behavioral response following the occurrence of a

negative event and the “delayed” behavioral response over time.

Hypothesis 2 (H2). The effect of broad bracketing on choice of the sure loss will not be eliminated with

experience (i.e., repetition and feedback). In particular, the effect will not be eliminated either (H2A) as time passes

without the occurrence of a loss as a result of the recency effect, or (H2B) in the immediate aftermath of a loss as

the result of the gambler’s fallacy.

2.3 Judgment processes behind broad bracketing

There is no guidance in the literature on how best to construct a broad bracket because very little work has

explored the underlying mechanism. It is currently unclear whether only the probability of loss drives the broad

bracket effect or whether decision makers also attend to the size of the bracket (which in this case corresponds to the

length of the time horizon). That is, we do not know whether people respond the same to a 26% chance of at least

one flood over 30 years as they would to a 26% over 200 years, despite the former representing a risk that is almost

seven times larger (i.e., a 1% per year compared to a 0.15% per year).

Recent research suggests, but does not explicitly test, that people ignore the time horizon and attend

primarily to the probability size. People weight losses more when receiving broadly bracketed information about a

pure loss gamble, and this leads to increased risk aversion and more frequent selection of the sure loss (Webb &

Shu, 2017). The authors further show that the framing technique does not operate by getting people to focus more on

the number of times they will be exposed to a risk. If people focus primarily on probability size, then a longer time

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horizon (i.e., a broader bracket) will have a greater impact on protective behavior simply because the probability of a

loss appears larger. Thus, people would be more likely to engage in protective behavior when they learn that there is

a 26% chance of at least one flood over 30 years than when they learn there is a 5% chance over 5 years, despite the

fact that the underlying likelihood of flood is the same.

Hypothesis 3 (H3). For a pure loss gamble, the larger the bracket that is used for communicating the

cumulative loss probability, the more likely people are to select the sure loss.

If people also pay attention to the time horizon, they might discount larger cumulative probabilities if those

probabilities are associated with longer time horizons. People are somewhat (but not fully) aware of how

probabilities of outcomes change with multiple exposures to a risky event, like a gamble (e.g., Redelmeier &

Tversky, 1992; Samuelson, 1963). This means that people might respond more strongly to a 26% chance of at least

one flood over 30 years than to a 26% chance over 200 years. Thus, lengthening the bracket, i.e., increasing both the

cumulative probability and time horizon, as suggested in H3 might not necessarily increase protective behavior.

Instead, the length of the time horizon may moderate the impact of cumulative probability size on behavior:

Hypothesis 4 (H4). (H4A) The greater the cumulative probability displayed, the more likely people will be

to select the sure loss, but (H4B) for a given cumulative probability, the larger the bracket associated with that

probability, the less likely people will be to select the sure loss.

3. Overview of current work

This paper investigates whether the effect of broad bracketing on choices in the loss domain is robust to

experience—i.e., repeated decision making with feedback—which is a critical feature for it to function as a robust

risk communication tool. The role that experience plays in the effect of broad bracketing has not been studied

before. We examine a simpler, more scalable form of the broad bracket, which also allows us to shed light on the

mechanism. In particular, we investigate how the size of the cumulative probability and the length of the time

horizon moderate the broad bracket effect, and doing so provides guidance to risk communicators for how to

structure a broad bracket in their messages.

We conducted six web-based experiments with real payoffs that ask study participants to make choices

between accepting a certain small loss or accepting a chance of a much larger loss. All six experiments share the

same basic experimental design, which is based on an incentive-compatible design from a previous paper on

insurance decisions (Kunreuther & Michel-Kerjan, 2015). Studies 1-5 test whether broad bracketing increases the

proportion of people who take protective action and whether that effect is robust to experience. Study 3 examines

whether the effect of broad bracketing is impacted by allowing earnings from previous choices to accumulate over

rounds and whether it is robust to increasing the cost of insurance. Study 4 tests whether the results from the

previous studies generalize to participants who actually live in highly flood-prone areas, and whether the broad

bracket effect is robust to simultaneous presentation with the narrow bracket. Study 5 examines whether the effect of

broad bracketing increases with longer time horizons and Study 6 investigates the relative contribution of probability

size and time horizon length to the broad bracket effect. All data, power analyses, and pre-registrations are available

on the Open Science Framework (OSF) at: https://osf.io/mejf5/?view_only=9b8760dca9714e85bc178f290140125e.

3.1 General study set up and procedures

Choices in the studies mimicked the structure of insurance decisions, where there is a small premium a

person can incur to avoid a much larger loss that could occur with a low probability (1% chance per round). The

survey was designed using the online survey platform Qualtrics. After reading the instructions and completing a

comprehension check, participants began a 15-round phase of the experiment (except for study 6, which was a single

round). At the beginning of each study individuals received an endowment and in each round had to select between

two options: (1) a sure loss (the insurance premium) or (2) a gamble with a 1 percent chance of losing a portion of

the endowment (pure loss gamble) in any round. Participants knew they would make this decision multiple times,

but they did not know the precise number (15 rounds in studies 1-5, 1 round in study 6), only that it would be fewer

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than 25. We did this to avoid different behavior in the last round of the experiment (i.e., round 15), a common

occurrence in multi-period experiments.

After each round, participants were given feedback about whether or not the gamble resulted in a loss,

regardless of which option they had chosen. In studies 1, 2, and 4 the $95 endowment was restored to the initial

level so that the financial impact of decisions in previous rounds would have no impact on their decision in the

current round. Studies 3 and 5 allowed the endowment from the previous round carry over to the next round (e.g., a

lower endowment if the participant experienced an uninsured loss). The exact wording of the study scenarios can be

found in the appendix.

3.2 Participant recruitment and power analysis

We recruited a planned number of participants meeting required criteria for each study from Amazon’s

Mechanical Turk (MTurk). Statistical power to detect relevant effect sizes for parameters of interest was estimated

for each study using a simulation-based procedure (conducted in Stata/IC version 14.2, code available upon request).

Simulations created 500 replicate estimates of the relevant statistical model for a given planned number of

participants and effect size. Statistical power was defined as the percent of replicates where a true effect was

detected and could rule out a null effect with 95% confidence. The power analysis for all studies assumed a baseline

rate of choice of sure loss of 0.617 (the baseline rate of choice of a sure loss from Kunreuther & Michel-Kerjan

(2015), who used a similar experimental setup). See Table A0 in the appendix for results of the power analyses,

including planned sample sizes and minimum detectible effects for Studies 1-6.

3.3 Payments

Participants were endowed with a large amount of money ($95), so the decisions would be treated as

important and consequential; study 1 (abstract context) expressed the endowment in dollars, while studies 2-6 (flood

insurance context) expressed the endowment in “talers,” a fictional currency (1,000 talers = $1) We used the

artificial currency in studies framed in an insurance context so participants would consider larger numeric amounts

as would be the case in a real flood insurance decision.

To keep the experimental costs reasonable, one in every 100 participants was selected at random to play the

experiment for real money. Earlier studies demonstrated that this strategy is just as effective in motivating behavior

as paying all participants (Charness, Gneezy, & Halladay, 2016; Clot, Grolleau, & Ibanez, 2018). We used a

procedure from Kunreuther and Michel-Kerjan (2015) to randomly select participants for real money play in a way

that was credible to them. Each participant was randomly assigned a number between 0 and 99, which was displayed

on the screen at the beginning and end of the experiment. They were given a specific future date and time of the

Florida Pick-2 Lottery (http://www.flalottery.com/pick2) and told that if their number was chosen at that time they

would be paid based on their decisions from a random round of the experiment. The number of the random round

was displayed on the screen at the end of the experiment.

4. Study 1: Abstract Context

The first experiment determines whether presenting the cumulative loss probability decreases risk-seeking

in the loss domain in the context of abstract gambles (H1) as has been shown for the standard form of broad

bracketing (Webb & Shu, 2017), and whether this change is robust to experience, i.e., repetition and feedback (H2).

In each round of this experiment, participants simply choose either a small sure loss or a pure loss gamble that has a

low probability of a large loss and a high probability of no loss. We expect that presenting the broad bracket will

increase the proportion of people selecting the sure loss and that this will not change substantially across rounds.

4.1 Methods

Procedure. At the beginning of each round, each individual was endowed with $95 and had to select

between two options: (1) a sure loss of $0.45 (sure loss) or (2) a gamble with a 1 percent chance of losing $45 (pure

loss gamble) in any round. At the beginning of each round, the $95 was restored to the initial level so that the

financial impact of decisions in previous rounds would have no impact on their decision in the current round.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

9

Design. The experimental design was a 2 (bracket size: 1-play, 30-play) x 3 (occurrence of loss: no loss,

early loss, late loss) between-subjects design so that participants were randomly assigned to one of six conditions.

Half the participants learned the 1-play likelihood of a loss occurring from selecting the gamble (1%), while the

other half learned the 30-play likelihood of at least one loss occurring (26%). One third of the participants were

assigned to witness the gamble result in a loss in round 4 (early-loss group), one third in round 11 (late-loss group),

and the last third would not observe any losses from the gamble (no-loss group).

Measures. After reading the initial instructions, but before beginning the first round of the experiment,

participants had to correctly answer three comprehension check questions in order to proceed (see the appendix for

more detail). We assessed risk preferences using two measures: one self-reported and one choice-based. The self-

reported measure was as follows: “How do you see yourself: Are you generally a person who likes to take risks or

do you try to avoid taking risks?” [sliding scale: 0 = not at all willing to take risks to 10 = very willing to take risks]

(Dohmen et al., 2011). The choice-based measure was adapted from Gneezy & Potters (1997): We endowed

participants with $0.50 and gave them an opportunity to invest in an option with a 50% chance of succeeding and a

50% chance of failing. If the option succeeded, they would receive 2.5 times the amount they invested, but if it

failed, they would lose the amount they invested. Participants had to decide how much of the 50 cents they wanted

to invest in the option (in 1-cent increments). Participants also answered questions about their age,

gender, income, education, employment status, and political identity.

1

At the end of the experiment, participants

were told whether the option succeeded or failed and how much money they earned from their decisions.

Participants. A total of 1,076 participants completed the study and passed the comprehension check (56%

female, Mage = 36 years, SD =11.2).

Payment. Participants were paid on average $1.07 (range: $0.50-$1.75), which included a bonus based on

their answer to the Gneezy-Potters investment task. Five people chosen at random were each paid $94.55 based on

their choices in the experiment.

4.2 Results and discussion

Participants made 15 rounds of choices with respect to opting for a sure loss or engaging in a pure loss

gamble. Table 1 summarizes the results of random-effects linear panel regressions with heteroscedasticity-consistent

standard errors (MacKinnon & White, 1985) for Studies 1-4, with results for Study 1 in column 1.

2

The predicted

variable is the percentage of people choosing the sure loss. The reference or baseline group in the regression is the

no-loss 1-play condition in round one, and the average proportion selecting the sure loss in this group is represented

1

To check whether behavior and perceived risk mirrored each other, after rounds 1, 4, 11, and 15, participants

indicated their perception of the likelihood that a loss would occur in the next round. We only asked this question

four times to reduce decision fatigue. We chose rounds 1 and 15 to assess participants’ initial beliefs without

experience and their final beliefs. We included rounds 4 and 11 to examine the immediate effect of witnessing a loss

on participants’ beliefs. The question was asked after participants learned the outcome for that round and was posed

as follows: “How likely do you feel it is that the lottery next period will result in a loss?” [sliding scale: 0 =

extremely unlikely to 10 = extremely likely]. Analysis of this variable available upon request.

2

In the estimates that follow, the variance-covariance matrix of the estimates is modified to account for

heteroscedasticity under general conditions with a degrees-of-freedom correction (referred to as the HC1 method,

MacKinnon and White 1985). In this case, because the number of respondents and observations is relatively large,

using HC1 standard errors (instead of the random-effects least-squares standard errors) and the choice of HC1

modification does not have a noticeable effect on inference and the qualitative results.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

10

by the constant term. Coefficients capture deviations from this level. Figure 1 displays results for Studies 1-4 (panels

a-d). Each panel displays the proportion of participants selecting the sure loss across all 15 rounds of the experiment

for the no-loss and early-loss conditions.

3

Study 1 results can be found in panel a.

Broad bracket effect (H1). As predicted by H1, we found that those who were given the 30-play cumulative

loss probability in the pure loss gamble were more likely to select the sure loss than those who learned the 1-play

probability. The Table 1, column 1 coefficient on the broad bracket condition indicator (“30-play prob.”) displays

that participants in the broad bracket condition were 11.7 percentage points more likely to select the sure loss in the

first round and prior to experiencing a loss compared to those in the narrow bracket condition. This effect holds

when controlling for sex, age, education, income, and risk tolerance (Table A1, column 1).

Interaction with experience (H2). To determine whether the broad bracket effect changed across rounds and

was influenced by witnessing or experiencing a loss (H2), we included three additional variables in the regression

(“round,” “round after loss,” and “time since loss”) and tested whether they interacted with the variable representing

the broad bracket effect. The “round” variable is a continuous variable that takes on values of the rounds (i.e., 2-15),

with round one as the reference. Its coefficient describes the effect of time passing on selection of the sure loss. In

order to capture further elements of the recency effect and gambler’s fallacy for people who witnessed a loss, we

included variables to separately capture behavior change in the round immediately after a loss and behavior change

over time after the loss. “Round af.ter loss” is an indicator variable equal to 1 for choices of early-loss participants in

round five and for choices of late-loss participants in round twelve (and equal to 0 otherwise). “Time since loss” is a

continuous variable that is always equal to 0 for participants in the no-loss conditions, but that is equal to values

between 1 to 11 for the early-loss and late-loss conditions depending on how many rounds it has been since the

gamble resulted in a loss. For instance, for the early loss group, the value of that variable would be 1 for round five,

2 for round six, and so on.

The “round” variable shows a slight recency effect: Over time more people switched from the sure loss to

the gamble (1 percentage point per round as shown in row 2). However, this round effect did not differ for people

given the cumulative loss probability (“30-play prob. x round”), so the broad bracket effect was neither eliminated

nor did it change in size for the no-loss conditions across rounds, or for the pre-loss rounds of the loss conditions.

The delayed effect of a loss (“time since loss”) was the same for the early- and late-loss conditions and was

consistent with elimination of the recency effect discussed above: After a loss, participants were no longer likely to

switch to the gamble as time passed. The immediate effect of a loss (“round after loss” and “round after loss x late

loss”) was different for the early- and late-loss groups, with people in the former group being less likely to choose

the sure loss in the round immediately after a loss and the latter group being more likely. While this may be an

interesting difference driven by the different timings of the loss, exploring it is beyond the scope of this

investigation. While witnessing a loss did cause significant changes in choice that were both immediate (“round

after loss” and “round after loss x late loss”) and delayed (“time since loss”), none of these changes interacted with

the broad bracket. Thus, the effect of a loss did not undo or even mitigate the impact the broad bracket on selection

of the sure loss. These results hold with the inclusion of controls (Table A1, column 1).

3

The conditions with a late loss shock are included in the regression analysis, but the figures exclude them for ease

of visual comparison with later studies that only include an early loss shock.

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11

TABLE 1. OLS panel regressions for Studies 1-4. Predicted variable is choice of sure loss (flood insurance) over

pure loss gamble (no flood insurance).

Study 1

Study 2

Study 3

Study 4

Includes all 6

conditions

Includes all 6

conditions

Includes 4 main

conditions

Includes 4 no-loss

conditions with low &

high premiums

Includes 4 main

conditions

(1)

(2)

(3)

(4)

(5)

30-play prob.

0.117*** (0.025)

0.152*** (0.018)

0.125*** (0.026)

0.131*** (0.037)

0.057* (0.023)

Round

-0.010*** (0.002)

0.002* (0.001)

0.001 (0.002)

0.001 (0.002)

-0.005** (0.002)

Round after loss

-0.068* (0.027)

-0.067*** (0.017)

-0.044* (0.021)

-0.003 (0.023)

Time since loss

0.010*** (0.003)

0.009*** (0.002)

0.009*** (0.003)

0.009** (0.003)

Round after loss x

late loss

0.127*** (0.037)

0.063** (0.022)

Time since loss x

late loss

0.005 (0.006)

-0.011* (0.004)

30-play prob. x

round

0.0002 (0.002)

-0.003+ (0.002)

-0.001 (0.003)

-0.001 (0.003)

-0.002 (0.003)

30-play prob. x

round after loss

-0.024 (0.039)

-0.041+ (0.024)

-0.049 (0.031)

-0.077* (0.033)

30-play prob. x

time since

-0.001 (0.004)

-0.001 (0.003)

-0.002 (0.004)

-0.0002 (0.004)

30-play prob. x

round after loss x

late loss

-0.012 (0.056)

-0.015 (0.032)

30-play prob. x

time since x late

loss

0.008 (0.009)

0.004 (0.006)

High price

-0.097* (0.039)

30-play prob. x

high price

0.008 (0.053)

High price x round

-0.003 (0.003)

30-play prob. x

high price x round

-0.006 (0.004)

Constant

0.584*** (0.018)

0.622*** (0.014)

0.555*** (0.020)

0.530*** (0.028)

0.720*** (0.017)

Independent

rounds

yes

yes

no

no

yes

Floodplain sample

no

no

no

no

yes

Participants

1,076

2,076

1,051

1,062

1,019

Observations

16,140

31,140

15,765

15,930

15,285

R2

0.012

0.013

0.011

0.007

0.005

Adjusted R2

0.011

0.012

0.010

0.006

0.005

Note: Reference group is narrow bracket condition with no loss in round one. Heteroskedasticity-consistent (HC1)

standard errors in parentheses. Dependent variable is binary choice of certain loss (insurance) over loss lottery (no

insurance). +p<0.1; *p<0.05; **p<0.01; ***p<0.001

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

12

(a) (b)

(c) (d)

FIGURE 1. Proportion of participants selecting the sure loss (purchasing flood insurance) in each round for Studies 1-4 (panels a-d). The x-axis

indicates the round number. The same four conditions are shown for each study: narrow-bracket no-loss (solid grey with circle markers —●—), broad-

bracket no-loss (solid black with circle markers —●—), narrow-bracket early-loss (dashed gray with triangle markers

- -▲- -

), and broad-bracket

early-loss (dashed black with triangle markers - -▲- -). The broad bracket conditions are represented by black lines, while the narrow bracket

conditions are represented by gray lines. Solid lines with circle markers indicate conditions in which no loss occurred, while dashed lines with triangle

markers represent conditions in which a loss occurred at the end of round 4.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

13

5. Study 2: Flood Insurance Context

Study 2 enriched the experimental setup to reflect a risk communication environment, as recent research

suggests that participants exhibit behavior that differs in relevant and important ways when choices are framed

neutrally rather than in a particular context (see Jaspersen, 2016 for a review). For example, participants show more

risk aversion when making decisions about insurance than when making context-free decisions for the same risk

(Hershey & Schoemaker, 1980; Lypny, 1993). We chose a setting in which participants decide whether or not to

purchase flood insurance by paying a premium so as to map onto the structure of Study 1.

5.1 Methods

Procedure. The procedure for this experiment was identical to that of Study 1, except that the scenario now

incorporated a flood insurance context with endowments expressed in the fictional taler currency. Participants were

told in each round that they needed to decide if they should buy flood insurance at a cost of 450 talers to protect their

house from a flood that could cause 45,000 talers in damage. They would learn at the end of every year whether or

not a flood occurred and see a summary of their assets given their insurance decision.

Design. The experimental design was the same as in Study 1: It was a 2 (bracket size: 1-year, 30-year) x 3

(loss occurrence: no loss, early loss, late loss) between-subjects design so that participants were randomly assigned

to one of six conditions.

4

In this study, the narrow bracket conditions displayed the annual flood probability, while

the broad bracket conditions displayed the likelihood of experiencing at least one flood over a 30-year time horizon.

Measures. As in Study 1, we included comprehension check questions, two methods to assess risk

preferences, and four subjective probability assessments. We also included questions eliciting experience with

flooding and purchasing flood insurance to use as control variables in our regressions. The exact wording of the

scenario and these measures are detailed in the appendix.

Participants. A total of 2,076 participants completed the study and passed the comprehension check (45%

female, Mage = 35.9 years, SD =11.1).

Payment. Participants were paid on average $1.09 (range: $0.51-$1.75), which included a bonus based on

their answer to the Gneezy-Potters investment task. Additionally, 18 participants were randomly chosen to be paid

based on their choices—15 of them received $94.55, while the other three received $95.

5.2 Results and discussion

Table 1, column 2 displays the results of Study 2, where the predicted variable is the percentage of people

purchasing insurance. The reference condition in the regression is the no-flood 1-year condition in round one, and

the average proportion purchasing insurance in this group is represented by the constant term. Figure 1b displays the

proportion of participants purchasing insurance across all 15 rounds of the experiment for the conditions with no

flood damage and those with an early flood loss.

Broad bracket effect (H1). As in Study 1, broad bracketing increased the proportion of participants

purchasing flood insurance, thus supporting H1. Participants shown the cumulative loss probability over 30 years

4

We also included a manipulation that tested the effect of a policy relevant feature of real-world flood insurance

decisions: the designation of a person’s house as being located within a FEMA-named Special Flood Hazard Area,

generally considered higher risk zones. We expected this dichotomous naming convention would have an impact on

decisions. This line of inquiry was pursued for its policy relevance but was not germane to our investigation of

broad bracketing. Importantly, there was no detectable impact of this information, so we collapsed across those

conditions in our investigation of the hypotheses at hand.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

14

were 15.2 percentage points more likely to purchase insurance in the first round and prior to experiencing a loss than

those who saw the 1-year probability of a flood. This effect holds when controlling for real-life flood experience as

well as sex, age, education, income, and risk tolerance (Table A1, column 2).

Interaction with experience (H2). We used the same variables to examine the effect of experience as we did

for Study 1 (“round,” “round after loss,” “time since loss”, “round after loss x late loss”, “time since x late loss”).

The broad bracket (“30-year prob.”) did not interact with any of these experience variables, thus supporting H2.

These results hold with the inclusion of controls (Table A1, column 2).

6. Study 3: Non-independent decisions

One limitation of Studies 1 and 2 is that each of the rounds was independent from previous ones so that

participants were shielded from having a choice in an early round permanently ruin their chances to earn a large

bonus in the experiment. This treatment may have reduced the potential for experience-based effects and is not as

reflective of real-world decisions, where a past outcome impacts future choices. An additional consideration is that,

in Studies 1 and 2, the insurance premium was set equal to the expected value of the loss, but in real life, the price of

insurance is usually much higher than the risk neutral price, often worth 1-2% of a person’s income, due to

considerations like loading costs. Thus, Study 3 extends our investigation by making rounds non-independent, that

is, by structuring the experiment so that earnings carry over across all 15 rounds, and by testing whether price

interacts with the broad bracket effect.

6.1 Methods

Procedure. The procedure for this experiment was similar to Study 2, except that earnings carried across

rounds (i.e., the endowment did not reset with each round in the event of an uninsured loss), and we altered the

monetary amounts so that total assets remained the same and earnings could be cumulated across rounds: The house

value was reduced to 80,000 talers, and the savings account was increased to 15,000 talers. The potential damage

from a flood was changed to 35,000 talers with an annual probability of 1%. At the end of each round, participants

were provided with the value of their total assets based on their insurance decision and whether or not a flood had

occurred during the round. Participants’ total earnings were summarized at the end of the experiment.

Design. The experimental design was a 2 (bracket size: 1-year, 30-year) x 2 (loss occurrence: no-loss,

early-loss) between-subjects design. The cost of insurance in these conditions was the actuarially fair premium (350

talers per year). To examine the impact of the insurance premium on behavior, we included two additional no-loss

conditions in which the premium was much higher (1,000 talers per year): 1-year high price and 30-year high price.

Participants were randomly assigned to one of these six conditions.

Measures. We included the same questions as in Study 2 and additional measures designed to assess

participants’ ability to calculate probabilities across different time periods, the perceived relevance/usefulness of the

cumulative probability information, and the preferred time period over which each participant would like to know

the cumulative probability. The details of these measures can be found in the appendix.

Participants. A total of 1,591 participants completed the study and passed the comprehension check (45%

female, Mage = 35.9 years, SD =11.9).

Payment. Participants were paid on average $1.95 (range: $1-$2). In addition, 20 participants were

randomly chosen to receive payment based on their choices in the experiment, receiving between $80 and $90.

6.2 Results and discussion

Broad bracket effect (H1). For the main analyses testing H1 and H2 (Table 1, column 3), we include only

the four main conditions in which the price of insurance was set to the risk neutral price of 350 talers (N=1,051).

Figure 1c displays the proportion of participants purchasing insurance in these four conditions. We find a

significantly positive impact of the broad bracket (“30-play prob.”) on choice in the direction predicted: Participants

were 12.5 percentage points more likely to purchase insurance in the first round and prior to experiencing a loss than

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

15

those who saw the 1-year probability. This effect holds when controlling for real-life flood experience as well as

sex, age, education, income, and risk tolerance (Table A1, column 3).

Interaction with experience (H2). The broad bracket did not interact with any of the experience variables

(“round,” “round after loss,” “time since loss”). These results hold with the inclusion of controls (Table A1, column

3).

Price effects. To examine whether price interacted with the broad bracket, we conducted an analysis that

compared the 1-year and 30-year no-loss conditions from the main analysis with the two additional conditions in

which the price of insurance was set to 1,000 talers (N=1,062; Table 1, column 4). Though we found that the

increased price led to an overall drop in insurance purchasing by about 9.7 percentage points (“High price”), we

found no interaction between price and the broad bracket (“30-play prob. X high price”). We also did not find any

interaction between price and either round (“High price X round) or round and the broad bracket (“30-play prob. X

high price X round”).

Thus, we found that allowing earnings to carry across rounds did not qualitatively or quantitatively affect

the main broad bracketing effect nor did it lead to an interaction of the broad bracket with experience, offering

confirming evidence for H1 and H2. These effects hold when controlling for real-life flood experience as well as

sex, age, education, income, and risk tolerance (Table A1, column 4).

7. Study 4: Disaster-prone sample

In Study 4, we extend our investigation in two ways. First, we collect a sample of participants living in

highly flood-prone areas. We target US counties where people are likely to have more experience with both flooding

and flood insurance. Second, we test whether giving participants both the narrow and broad brackets eliminates the

effect of broad bracketing. One concern with the broad bracket is that it operates by skewing people’s preferences,

making them behave more risk averse than is consistent with their underlying risk preferences. This suggests that

decision makers primarily care about the narrow bracket, and only use the broad bracket to make inferences about

the narrow bracket, which they do imperfectly. If this is true, then presenting both versions would completely

eliminate the broad bracket effect because decision makers would ignore the broad bracket. However, if the

foregoing explanation does not entirely account for the effect—i.e., if people still find the broad bracket

informative—then the effect will not be eliminated entirely. In cases where people anticipate making multiple

decisions over an extended time period, we expect that decisions makers will still find the broadly bracketed

information useful and respond to it.

7.1 Methods

Procedure. The procedures for this experiment were identical to that of Study 2.

Design. The design was a 2 (bracket size: 1-year, 30-year) x 2 (loss occurrence: no-loss, early-loss)

between-subjects design. We also included an additional no-loss condition: the 1-and-30 condition.

5

Participants in

this condition learned both the annual flood probability and the probability of at least one flood over 30 years.

Participants were randomly assigned to one of these five conditions.

5

For two exploratory conditions not central to our investigation, we recruited 468 additional participants. Those

conditions involved presenting participants with both the 1-year probability as well as a cumulative probability over

an extremely long time horizon. In one condition, that time horizon was 100 years, and in the other it was 200 years.

Results for those two conditions are available upon request.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

16

Measures. We included the same measures as in Study 3.

Participants. We recruited participants from an online panel through which participants could be identified

by zip code. We selected participants from twelve US counties in which at least 50% of the flood insurance policies

were owned by people in areas where flood insurance was not a requirement for getting a mortgage loan (i.e.,

outside the Special Flood Hazard Areas). We made this designation to ensure we were selecting counties in which

most of the people were at risk of flood, most of the people were aware of this risk, and many people were likely to

have experienced a flood and/or purchased flood insurance previously. The list of counties we sampled from can be

found in the appendix. A total of 1,260 participants completed the study and passed at least one of the three

comprehension check questions (70% female, Mage = 43.3 years, SD =15.9).

Payment. Participants were offered a payment equivalent to those in the previous three studies and were

told there was a chance they would earn a bonus based on their decisions.

6

In addition, 14 participants were

randomly selected to be paid based on their decisions; ten participants received $94.55, and four received $95.

7.2 Results and discussion

Our sampling procedure was successful in increasing the proportion of people in the sample with flood

experience and/or experience purchasing flood insurance. Thirty seven percent indicated experiencing a flood

disaster at least one time compared to only 9% and 18.5% in Studies 2 and 3; 55% indicated that they had ever

purchased flood insurance while only 11.4% and 24.2% ever had this coverage in Studies 2 and 3 respectively.

Broad bracket effect (H1). For testing H1 and H2 (Table 1, column 5), we include only the four main

conditions in which participants received information for only one of the two brackets (N=1,019). Figure 1d displays

the proportion of participants purchasing insurance in these four conditions. We find a significantly positive impact

of the broad bracket on choice in the direction predicted. Participants were 5.7 percentage points more likely to

purchase insurance in the first round and prior to experiencing a loss than those who saw the 1-year probability,

though this was a somewhat smaller increase than in Studies 2 and 3. One possible explanation for this difference is

that people in this sample were already much more likely to purchase flood insurance in any given round. The

baseline rate of insurance purchasing for people with the narrow bracket was 72%, which is 10 percentage points

higher than in the previous two studies. This lowered the proportion of decisions (i.e., choices to not purchase

insurance) that could be influenced by broad bracketing. This effect holds when controlling for real-life flood

experience as well as sex, age, education, income, and risk tolerance (Table A1, column 5).

Interaction with Experience (H2). The broad bracket (“30-year prob.”) did not interact with either the

“round” or the “time since loss” variables (H2A supported), but it did interact with the “round after loss” variable:

6

Personal communication from the panel administrators regarding exact pay received by participants was as

follows: “Our panelists select a number of different ways they would like to be incentivized. Some choose cash,

travel miles, gift cards, points to gift card websites etc. Also, we often have to incentivize the last 10% of the

respondents more in an effort to complete the survey. What we use as a good rule of thumb is that you can assume

that ~25-35% of the total cost per complete goes to the respondent.” Since we were charged $7.50 per participant,

this means that participants received the equivalent of $1.88 to $2.63 in whatever medium they chose to receive their

incentive.

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

17

Participants in the 30-year conditions who experienced a flood loss in round 4 were 2 percentage points less likely

(summing the coefficients on “30-year prob.” and “30-year prob. x round after loss”) to purchase flood insurance in

the round right after the loss than participants who also experienced a loss in round 4 but who were in the 1-year

condition. This effectively eliminated the broad bracket effect in round 5 for participants who witnessed the loss

(H2B not supported). These effects hold when including control variables (Table A1, column 5).

Based on additional analyses (available upon request from the authors), we offer one possibility that could

have led to the different result regarding H2B in this study than in Studies 1, 2, and 3. In Study 4, participants who

experienced an early flood but were uninsured at the time behaved in a way that was opposite to participants who

experience an early flood but were insured. In earlier studies, only the insured loss experiencers displayed an

immediate reaction to loss (i.e., to drop insurance). In this case, the uninsured loss experiencers were much more

likely to take up insurance in the subsequent round than were similar participants in Studies 1-3. In the previous

studies, people who were uninsured did not differ in their behavior in round 5, whether or not they experienced a

flood loss. This may be a function of Study 4 participants being real-life residents of high-risk floodplains and being

more affected by the in-experiment floods because of it. Because the 1-year condition had a more even mix of

insured and uninsured who experienced the loss shock, and because the main regression did not control for insured

status, the counteracting effects of the insured and uninsured likely canceled each other out and led to a coefficient

on “round after loss” that was not different from zero. However, because the broad bracket conditions had higher

proportions of insured participants overall than in the narrow bracket conditions, the average effect detected is that

of the insured participants—i.e., to drop one’s insurance immediately after the flood shock. This observation

demonstrates that the fact that the broad bracketing could backfire in the case of a flood, at least in the immediate

aftermath, if it has led to a significantly larger proportion of people being insured and if those insured people

respond to a flood by dropping insurance (and uninsured people respond to a flood by taking up insurance).

Simultaneous presentation of the narrow and broad brackets. To test the impact of providing both the

narrow and broad brackets, we conducted a regression that included only participants in the three no-loss conditions:

1-year, 30-year, and 1-and-30 year (N=745; Table 2). For this analysis, the 1-and-30 condition served as the

reference condition. When controlling for round effects, we observed no statistically significant difference between

any of the conditions. The broad bracket was associated with a 5.5 percentage point higher take up of insurance in

the first round relative to the 1-and-30 condition, but this was only marginally significant. As there was no

interaction between round and condition, we ran a regression where we provided more observations by not

distinguishing between rounds so as to obtain more statistical power for differentiating between smaller effect sizes

(Table 2, column 2). We found the 1-and-30 condition led to an average 2.2 percentage point decrease in insurance

purchasing compared to the 30-year condition— some choices are indeed skewed by the broad bracket—but was

associated with a 3 percentage point increase in purchasing insurance relative to the 1-year condition—suggesting

that not all choices are skewed but rather informed by the broad bracket. The proportion of people purchasing

insurance in each of the 15 rounds for these three conditions is visualized in Figure 2. (When controls are included

in the regression collapsed across rounds, the 1-and-30 condition does not appear different from the broad bracket

condition. See Table A2.)

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

18

TABLE 2. OLS regressions for Study 4. Predicted variable is purchase of flood insurance.

Panel regression

Simple OLS collapsed across rounds

(1)

(2)

1-play prob.

-0.006 (0.035)

-0.030** (0.011)

30-play prob.

0.055+ (0.033)

0.022* (0.011)

Round

-0.002 (0.002)

1-play prob. x round

-0.003 (0.003)

30-play prob. x round

-0.004 (0.003)

Constant

0.701*** (0.024)

0.682*** (0.008)

Participants

745

745

Observations

11,175

11,175

R2

0.005

0.002

Adjusted R2

0.005

0.002

Note: Reference group is 1-and-30 condition in round one for column one and collapsed across rounds for column

two. Heteroskedasticity-consistent (HC1) standard errors in parentheses. Dependent variable is binary choice to

purchase insurance. +p<0.1; *p<0.05; **p<0.01; ***p<0.001

FIGURE 2. Proportion of participants purchasing flood insurance in each round for Study 4. The

x-axis indicates the round number. Only the three no-loss conditions are displayed: narrow-

bracket no-loss (solid grey with circle markers —●—), broad-bracket no-loss (solid black with

circle markers —●—), 1-and-30 condition no-loss (dashed black with triangle markers - -▲- -).

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

19

8. Study 5: Changing the length of the time horizon

In Study 5, we extend our investigation in two ways. First, we test whether the broad bracket effect

generalizes to other time horizons. Second, we begin to investigate the mechanism and examine whether the broad

bracket effect depends on the magnitude of the cumulative probability such that longer time horizons, which are

associated with larger cumulative probabilities, lead more people to purchase insurance (H3).

8.1 Methods

Procedure. The procedure for this experiment was the same as for Study 3, where rounds were non-

independent, i.e., earnings carried across rounds.

Design. Participants were randomly assigned to one of five, between-subjects conditions that only differed

by length of the time horizon over which the cumulative probability information was presented: 1 year, 5 years, 10

years, 30 years, or 200 years. No conditions included loss occurrences.

Measures. We included the same measures and in the same format as in Study 3.

Participants. A total of 1,081 participants completed the survey and passed the comprehension check

(55% female, Mage = 38.6 years, SD =12.2).

Payment. Participants were paid on average $1.96 (range: $1-$2). In addition, five participants were

randomly chosen to receive payment based on their choices in the experiment; these participants received between

$89.80 to $93.30.

8.2 Results and discussion

Broad bracket effect (H1). Figure 3 displays the proportion of participants purchasing insurance in each of

the five conditions. We included all five conditions for the main analysis, with the 1-year condition in round one as

the reference group (Table 3, column 1). We find a significantly positive impact of all of the broad-bracket

conditions on flood insurance take up in the first round compared to the 1-year condition: The 5-year bracket led to a

14.1 percentage point increase; the 10-year led to a 16 percentage point increase; the 30-year led to a 17.3

percentage point increase; and the 200-year led to a 16.8 percentage point increase. The magnitude of these effects

are a few percentage points smaller but still significant when controlling for real-life flood experience as well as sex,

age, education, income, and risk tolerance (Table A3, column 1).

Interaction with experience (H2). There was no average change in purchasing across rounds (“round”), nor

was there any interaction of the round variable with any of the broad bracket conditions. These results hold with the

inclusion of controls (Table A3, column 1).

Broad bracket size (H3). To examine whether the length of the time horizon among the broad bracket

conditions was positively associated with purchasing insurance (H3), we ran two regressions that excluded the 1-

year condition and used the 5-year condition as the reference condition (N=861; Table 3, columns 2 and 3). In

column 2, we ran a regression with indicator variables comparing the 10-year, 30-year, and 200-year conditions to

the 5-year reference condition. In column 3, we defined a continuous variable that had four different values

corresponding to the number of years associated with each of the different time horizon lengths (i.e., 5, 10, 30, and

200). We found no difference among the broad bracket conditions in either regression (or when including controls as

in Table A3, columns 2 and 3), thus, rejecting H3. The size of the probability is, therefore, not the only driving force

behind the broad bracket effect. One potential explanation for this pattern is that probability size is positively related

to protective behavior, but there is also a counteracting time horizon effect, as proposed by H4. We examine this in

the next experiment.

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20

TABLE 3. OLS panel regressions for Study 5. Predicted variable is purchase of flood insurance.

All time horizons

(1-year as reference)

Excluding 1-year condition

(5-year as reference)

Time horizons coded as indicator

variables

Time horizon coded as

continuous variable

(1)

(2)

(3)

5-play prob.

0.141*** (0.041)

10-play prob.

0.160*** (0.040)

0.019 (0.036)

30-play prob.

0.173*** (0.040)

0.032 (0.037)

200-play prob.

0.168*** (0.041)

0.027 (0.038)

Length of time horizon

0.0001 (0.0002)

Round

-0.004 (0.002)

-0.001 (0.002)

-0.001 (0.001)

5-play prob. x round

0.002 (0.003)

10-play prob. x round

0.005+ (0.003)

0.003 (0.003)

30-play prob. x round

0.001 (0.003)

-0.001 (0.003)

200-play prob. x round

0.002 (0.003)

-0.001 (0.003)

Length of time horizon x

round

-0.00001 (0.00001)

Constant

0.556*** (0.031)

0.697*** (0.027)

0.712*** (0.016)

Participants

1,081

861

861

Observations

16,215

12,915

12,915

R2

0.004

0.001

0.0003

Adjusted R2

0.003

0.0001

0.00004

Note: Earnings carried over across rounds in Study 5 (non-independent rounds). For column one, the reference

group is the 1-year condition in round one. For columns two and three, the reference group is the 5-year condition in

round one. In column three, the “length of time horizon” variable is a continuous variable with values equal to the

length of the time horizon corresponding to the condition {1, 5, 10, 30, 200}. Heteroskedasticity-consistent (HC1)

standard errors in parentheses. Dependent variable is binary choice to purchase insurance. +p<0.1; ***p<0.001

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21

FIGURE 3. Proportion of participants purchasing flood insurance in each round for Study 5. The

x-axis indicates the round number. All five no-loss conditions are displayed: 1-year bracket

(solid grey with circle markers —●—), 30-year bracket (solid black with circle markers —●—),

5-year bracket (dashed black with triangle markers - -▲- -), 10-year bracket (dashed black with

square markers - -■- -), and 200-year bracket (dashed black with plus-sign markers - -+- -). All

the broad bracket conditions are represented in black.

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9. Study 6: Probability size versus time horizon length

Study 6 examines the roles played by probability size and time horizon length in creating the broad bracket

effect. H4 proposes that decision makers are sensitive to both probability size and time horizon length but in the

opposite direction: The greater the cumulative probability, the more people will take protective action, but the longer

the time horizon, the less people will take protective action.

9.1 Methods

Procedure. The procedure for this experiment was the same as for Study 5, except that participants made

only one decision (i.e., one round). However, participants were told that they would not be told the number of

rounds they would face ahead of time, and that they could potentially play up to 25 rounds. The talers conversion

was changed to 10,000 talers per $1 so that the maximum possible earnings from one’s decisions in the experiment

was $9.50.

Design. The design was a 2(time horizon length: 10 years, 200 years) x 5(cumulative probability size: 1%,

10%, 26%, 63%, 87%) between-subjects design, and participants were randomly assigned to one of these ten

conditions. We chose cumulative probabilities to span the range of values between 0 and 100% and included those

used in our previous studies. Note that these probability-time horizon combinations all reflect risks with different

underlying annual probabilities.

Measures. In addition to the measures used in the earlier studies, we asked each participant to guess the

one-year probability for their event (“What do you think (make your best guess) is the probability (in percent) of at

least one flood over the next 1 year?” [free response number between 0 to 100 with up to two decimal places]. We

also included a short, 3-item subjective numeracy scale (McNaughton, Cavanaugh, Kripalani, Rothman, &

Wallston, 2015).

Participants. A total of 2,303 participants completed the survey and passed the comprehension check

(57% female, Mage = 37.3 years, SD = 11.6).

Payment. Participants were paid on average $0.70 (range: $0.50-$0.75). In addition, 21 participants were

randomly chosen to receive payment based on their choices in the experiment; 16 participants received $9.40, while

five received $9.50.

9.2 Results and discussion

Table 4 displays the main results of Study 6, where the predicted variable is the percentage of people

purchasing insurance. Figure 4 displays the predicted values (lines) and observed values (points) for the proportion

of participants purchasing insurance in each condition. The cumulative probability associated with each condition is

indicated on the x-axis and the length of the time horizon is represented by the line type and point shape. The 95%

confidence bands for the predicted values are shown.

Probability size versus time horizon length (H4). To examine the relative contribution of probability size

and time horizon length on behavior, we ran an OLS regression with purchase decision as the predicted variable,

probability size (continuous), time horizon length (10 years as the reference), and the interaction of those two

variables as the predictor variables (Table 4). As expected, we found a positive association between cumulative

probability size (“Probability”) and purchasing: When the time horizon was 10 years, every one percentage point

increase in the cumulative probability shown was associated with a 0.4 percentage point increase in insurance take

up. This effect was the same size regardless of whether the interaction term was included in the regression or not.

Thus, the model predicts that going from a cumulative probability of 1% to one of 87% should result in an increase

in insurance purchasing of about 34 percentage points. We also found that, when an interaction term was included,

there was no main effect of increasing the time horizon to 200 years, but there was an interaction: The longer time

horizon did not have as much impact for smaller cumulative loss probabilities as it did for larger cumulative loss

probabilities. When the time horizon was 200 years, every one percentage point increase in the cumulative

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

23

probability was associated with a 0.2 percentage point increase in insurance take up, an effect which is half the size

of that for the 10-year time horizon. This means the model predicts only a 17.2 percentage point increase in

purchasing going from a cumulative probability of 1% to one of 87%. These results hold with control variables for

real-life flood experience as well as sex, age, education, income, and risk tolerance (Table A4).

TABLE 4. OLS regressions for Study 6. Predicted variable is purchase of flood insurance.

Main effects only

Interaction

(1)

(2)

Probability

0.004*** (0.0003)

0.004*** (0.0004)

200-year prob.

-0.096*** (0.020)

-0.030 (0.030)

Probability x 200-year prob.

-0.002** (0.001)

Constant

0.503*** (0.018)

0.470*** (0.021)

Observations

2,303

2,303

R2

0.067

0.070

Adjusted R2

0.066

0.069

Note: **p<0.01; ***p<0.001

FIGURE 4. Probability of purchasing flood insurance in each condition for Study 6. The x-axis

indicates the cumulative probability displayed to participants in a given condition. The points

represent observed values for participants in the 10-year conditions (circles ●) and the 200-year

conditions (triangles ▲). The lines represent the predicted values from the regression with the

interaction term (Table 6, column 2) for the 10-year conditions (solid line —) and the 200-year

conditions (dashed line - - -).

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10. General discussion

We investigated whether the effect of broad bracketing is present when using a simpler, more scalable

presentation format (presenting cumulative loss probabilities), and whether the effect persists over time when

choices have to be made repeatedly and in the face of feedback. Furthermore, we examined the mechanism behind

broad bracketing by isolating the relative importance of probability size and bracket size to the effect. Examining

these questions not only advances theory but also helps assess whether broad bracketing can be used as a risk

communication tool to motivate people to protect themselves from catastrophic losses. For this investigation, we

focused on the example of flooding.

The findings from our six studies have established that broad bracketing can be robust to repetition and

feedback, both of which are key features of the decision environment that people face when making protective

decisions. Specifically, we find that providing the cumulative probability of experiencing at least one negative event

over multiple periods compared to the probability of experiencing the event during one period leads more people to

take protective action against a risk. Importantly, we find that this effect lasts across 15 rounds of decisions with

feedback after each decision whether or not one experiences a loss. Study 1 demonstrates that this effect occurs in an

abstract setting and Studies 2-5 show that it also occurs in the richer, more ecologically relevant situation of insuring

one’s house and assets against a catastrophic flood. We found that all effects hold even after controlling for real-life

personal experience with floods.

We find that these effects are also robust to other features of the decision environment. In Study 3, we find

that allowing earnings and losses to carry forward across rounds (like they do in real life) does not change the main

broad bracketing result. In the same study, we find that, although increasing the cost of the protective action (i.e., the

price of flood insurance) lowers the proportion of people taking the protective action across all conditions, as one

would expect, the higher cost does not decrease the size of the broad bracket effect. In Study 4, we find that the

broad bracket increases protective behavior even among people who live in a floodplain and have experienced the

flood risk in real life. We also discover that presenting the narrow and broad brackets simultaneously reduces the

broad bracket effect, but does not eliminate it.

Our investigation also provides insights into the mechanism behind broad bracketing. The combined

findings from Studies 5 and 6 demonstrate that the broad bracket effect is not solely driven by the size of the

probabilities shown: While a larger cumulative probability leads to more people taking protective action, this effect

is dampened for longer time horizons, and this dampening effect is larger for larger probabilities. These results

indicate that both elements of broad brackets inform decision makers’ judgments.

10.1 Theoretical implicati\ons

Our findings have a number of important theoretical implications. First, we find that a more scalable,

simple form of broad bracketing that displays only the probability of loss (i.e., extending the time horizon) can have

the same effect in increasing risk averse behavior as previously demonstrated in the loss domain for the standard

form of broad bracketing (i.e., presenting a full distribution of outcomes; Webb & Shu, 2017). Second, we find that

the broad bracket effect is robust to repetition and feedback, which has not been previously demonstrated. In

particular, the effect is not only not eliminated in the face of experience, but it largely remains the same size over

time.

Third, with this latter finding, our work also contributes to the literature on decisions from description

versus decisions from experience. Past research found that decisions made in the presence of both description and

experience converged with the pattern of decisions made solely from experience, suggesting that people eventually

ignore description and place more weight on experience (Jessup et al., 2008; Lejarraga & Gonzalez, 2011; Newell

et al., 2016; Rakow et al., 2008; Yechiam & Busemeyer, 2006). This provides reasonable justification to believe the

broad bracket effect will disappear over time, or at least decrease in size. However, across all our studies, we found

this not to be the case: Even when participants have the exact same sequence of experiences, their choices are

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25

significantly influenced by the description they receive, that is, participants with the broadly bracketed probability

description make different choices than those with the narrowly bracketed probability description. Furthermore, we

observed almost no change in the size of this effect with the exception of one (Study 4) of our five studies

examining this question. Thus, we suggest that while experience may indeed influence choices when participants are

presented with both description and experience, so, too, does description. Future research is needed to rectify these

diverging observations.

Fourth, our findings advance our understanding of the mechanism underlying broad bracketing. We show

that broad bracketing is not solely the result of error-prone inference of the one-period probability, or the skewing of

risk preferences. Even when given both the one-period and cumulative probabilities, participants are still influenced

in their choices by the presence of the cumulative probability, suggesting they derive additional value from that

information. Furthermore, the size of the cumulative probability of the loss is an important driver of the effect, but it

is not the only driver. Participants discount a given cumulative probability more when it is presented using a longer

time horizon, reflecting an understanding that it represents an event with a lower underlying probability of

occurring.

10.2 Policy implications

Presenting the cumulative loss probability can be used by risk communicators as a powerful way to

motivate protective behavior against rare catastrophic events. However, the broad bracket will likely only change

behavior around rare events if it involves a cumulative probability that is large enough that people will not ignore it.

For instance, extending the time horizon over 10 years might be effective for an event with a 1% annual probability

(which translates to about a 10% probability of one or more floods over 10 years), but might not be effective for an

event with a 0.1% annual probability (which translates to a 1% probability of one or more floods over 10 years).

Indeed, Study 6 demonstrates that the “10% over 10 years” leads to more than a 55% take-up of flood insurance, but

the “1% over 10 years” leads to less than 40% take-up of flood insurance—more than a 15 percentage point

difference. This finding suggests that when an event is really rare, it may be worthwhile for risk communicators to

use longer time horizons. A 0.1% annual chance might be presented as an 18% chance of one or more floods over

200 years. If communicators are worried that doing so may bias decision makers too far away from their natural

preferences, results from Study 4 show that communicators can simultaneously present both the cumulative and one-

period probabilities without entirely eliminating a potential broad bracketing effect.

However, we advise the use of caution in providing overly long time horizons. Though we did not find any

evidence of backfiring from especially long time horizons—in Study 6, among people who received cumulative

probabilities extended over 200 years, there was still a positive trend associated with increasing the size of the

cumulative probability—we did observe a dampening effect due to longer time horizons in both Studies 5 and 6.

Further research is needed to fully confirm whether there might be some danger that participants reduce attention to

or ignore the broad bracket if it is considered unreasonably long.

Based on our findings, for events with an annual risk of 1% or more, there is little need to use time horizons

longer than five or ten years. Study 5 revealed that extending the time horizon to five years substantially increased

protective behavior compared to presenting the one-year probability of loss; however, extending the time horizon

even further (10 years, 30 years, 200 years) did not result in any additional increase in protective behavior. People

also seem to prefer shorter time periods. Exploratory questions in Studies 3-5 indicated that the majority of people

preferred a time horizon of 10 years or less in specifying the likelihood of a flood when making decisions on

purchasing insurance. The ideal time horizon will likely vary for events of different underlying probabilities of

occurring.

10.3 Future research directions

Future research is needed to deepen our understanding of the dynamics of broad bracketing and its

application to risk communication. It is currently unclear whether different types of broad bracketing presentations

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

26

will have different impacts on behavior. We tested an alternative type of broad bracketing (cumulative loss

probabilities) in which we presented only the cumulative probability of one or more losses and the timeframe over

which that probability was calculated. This differs from the standard broad bracketing presentation of the full

distribution of outcomes. Though we found similar impacts to previous work, it is unclear whether there could be

unique benefits or risks to each format.

Utilizing the cumulative loss probability format, we were able to demonstrate for the first time that the

length of the time horizon impacts the effectiveness of a broad bracketing framing. Though we did not focus on this

in our investigation, in Study 6 we found the dampening of the longer time horizon to be concentrated among

cumulative probabilities greater than 50%. Cumulative probabilities lower than 50% largely yielded similar behavior

regardless of whether they were calculated over 10 or 200 years. Furthermore, though we found a positive

relationship between cumulative probability size and the likelihood of taking protective action even for the longer

time horizon of 200 years, it is unclear whether there is a possibility that a time horizon can be so long that it

backfires. Study of this potentially non-linear relationship between attention to the broad bracket and the length of

the time horizon could provide further insights into how people respond to the risk of rare events.

Further work is also needed to examine whether the length of the time horizon interacts with the number of

decision periods a person expects to have. For instance, in Study 5, we noticed a substantial dip in protective action

for participants in the 5-year and 10-year conditions in the round immediately following these specific rounds. Some

participants who received the 5-year probability of at least one flood (5%) and did not experience a loss during the

first five rounds, seemed to believe they were “in the clear.” Thus, whether a risk communicator provides a broad

bracket that is larger or smaller than the number of decisions a decision maker will be making may influence

whether such dips in protective behavior are observed.

The extent to which the broad-bracketing effect appears in the real world, when people face real

opportunity costs, more demands on their attention, as well as real consequences to their decisions, is left to future

research. Though we observed a broad bracketing effect among people living in a floodplain, the effect was smaller

than among participants who did not live in a floodplain. The base rate of purchasing insurance was much higher

within this floodplain sample than among participants in the other studies, and based on this, one possibility is that

these individuals were convinced that they should purchase insurance whether the probability of a future flood was

presented for next year or over a 30 year period.

Another concern is that probabilities of a loss in the real world change over time, as is the case with respect

to climate change and its impact on flood-related damage in future years. If people believe that real-life event

probabilities are dynamic and change over time, they may place less weight on descriptive probability information.

Field research is needed to examine these questions.

Future work could benefit by examining variation across individuals with respect to whether and to what

extent broad bracketing impacts behavior. In particular, individual differences in numeracy and intuitive math

operations may be influential in the broad bracket effect. In Study 4, we observed that providing both the 1-year and

30-year probabilities did result in a decrease in the size of the broad bracketing effect, suggesting that this

experimental condition was helping some people correct error-prone inferences they would have made if they were

only given the broad bracket. Thus, the size of the broad bracket impact on a population may depend on the

proportion of people who make such inferences in the absence of full information.

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27

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Appendix

TABLE A0. Power analyses for Studies 1-6. Minimum detectible effects (MDE) are shown for main effects and interactions of interest.

Study

Planned N

Design

Rounds

Main effect MDE

(percentage points)

Additional effects MDE

(percentage points)

1

1,000

2 (brackets) x 3

(no loss, early

loss, late loss)

15

BB = 8.5

BB x round = 0.4

BB x round-after-loss = 4.0

BB x time-since-loss = 0.7

2

2,000

2 (brackets) x 3

(no loss, early

loss, late loss)

15

BB = 6.0

BB x round = 0.2

BB x round-after-loss = 2.5

BB x time-since-loss = 0.5

3

1,000

2 (brackets) x 2

(no loss, loss)

15

BB = 9.0

BB x round = 0.4

BB x round-after-loss = 4.0

BB x time-since-loss = 0.7

4

1,000

2 (brackets) x 2

(no loss, loss)

15

BB = 9.0

BB x round = 0.4

BB x round-after-loss = 4.0

BB x time-since-loss = 0.7

5

1,000

5 conditions

15

Time horizon = 13.5

Time horizon x round = 0.4

6

2,300

2 (time horizon)

x 5 (probabilities)

1

Time horizon = 8.5

Probability = 0.1

Time horizon x probability = 0.2

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

31

TABLE A1. OLS panel regressions for Studies 1-4 with demographic and control variables. Predicted variable is choice of sure loss (flood insurance) over pure

loss gamble (no flood insurance).

Study 1

Study 2

Study 3

Study 4

Includes all 6

conditions

Includes all 6

conditions

Includes 4 main

conditions

Includes 4 no-loss conditions with

low & high premiums

Includes 4 main

conditions

(1)

(2)

(3)

(4)

(5)

30-play prob.

0.121*** (0.025)

0.153*** (0.018)

0.115*** (0.026)

0.122*** (0.036)

0.056* (0.023)

Round

-0.010*** (0.002)

0.002* (0.001)

0.001 (0.002)

0.001 (0.002)

-0.005** (0.002)

Round after loss

-0.067* (0.027)

-0.067*** (0.017)

-0.044* (0.021)

-0.004 (0.023)

Time since loss

0.010*** (0.003)

0.009*** (0.002)

0.009*** (0.003)

0.009** (0.003)

Round after loss x late loss

0.126*** (0.037)

0.063** (0.022)

Time since loss x late loss

0.005 (0.006)

-0.011* (0.004)

30-play prob. x round

0.0004 (0.002)

-0.003+ (0.002)

-0.001 (0.003)

-0.001 (0.003)

-0.002 (0.003)

30-play prob. x round after

loss

-0.026 (0.039)

-0.041+ (0.024)

-0.049 (0.031)

-0.077* (0.033)

30-play prob. x time since

-0.002 (0.004)

-0.001 (0.003)

-0.002 (0.004)

-0.0001 (0.004)

30-play prob. x round after

loss x late loss

-0.011 (0.056)

-0.015 (0.032)

30-play prob. x time since x

late loss

0.008 (0.009)

0.004 (0.006)

High price

-0.091* (0.039)

30-play prob. x high price

0.004 (0.051)

High price x round

-0.003 (0.003)

30-play prob. x high price x

round

-0.006 (0.004)

Sex: female

-0.056* (0.023)

0.008 (0.016)

-0.046* (0.022)

-0.022 (0.022)

-0.009 (0.022)

Sex: other/no answer

-0.045 (0.154)

0.142+ (0.076)

-0.347+ (0.210)

-0.442*** (0.086)

0.125 (0.085)

Age

-0.0002 (0.001)

0.001 (0.001)

-0.002+ (0.001)

-0.0002 (0.001)

0.001 (0.001)

Educ: HS diploma/GED

0.101 (0.160)

-0.150 (0.105)

0.219 (0.203)

0.150 (0.323)

-0.017 (0.080)

Educ: some college

0.144 (0.157)

-0.141 (0.103)

0.271 (0.203)

0.193 (0.322)

0.065 (0.079)

Educ: 2-year coll. deg.

0.112 (0.158)

-0.140 (0.104)

0.219 (0.203)

0.171 (0.323)

-0.009 (0.082)

Educ: 4-year coll. deg.

0.136 (0.156)

-0.132 (0.103)

0.269 (0.202)

0.238 (0.322)

0.080 (0.079)

Educ: Master’s deg.

0.173 (0.159)

-0.125 (0.105)

0.304 (0.204)

0.240 (0.323)

0.058 (0.082)

Educ: doctrate deg.

0.376* (0.177)

0.011 (0.112)

0.223 (0.222)

0.102 (0.341)

-0.039 (0.113)

Educ: professional deg.

0.223 (0.174)

-0.071 (0.113)

0.316 (0.225)

0.286 (0.338)

0.069 (0.101)

Income

-0.0004 (0.0003)

-0.0004+ (0.0002)

-0.0002 (0.0003)

-0.0005 (0.0003)

0.0005+ (0.0002)

Self-assessed risk tolerance

-0.031*** (0.004)

-0.023*** (0.003)

-0.044*** (0.004)

-0.040*** (0.004)

-0.039*** (0.004)

Real-life Flood Experience

0.031 (0.026)

0.075** (0.028)

0.092** (0.029)

0.077*** (0.019)

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

32

Constant

0.650*** (0.160)

0.832*** (0.107)

0.572** (0.206)

0.527 (0.324)

0.769*** (0.090)

Independent rounds

yes

yes

no

no

yes

Floodplain sample

no

no

no

no

yes

Participants

1,076

2,076

1,051

1,062

1,019

Observations

16,140

31,140

15,765

15,930

15,285

R2

0.016

0.015

0.018

0.014

0.015

Adjusted R2

0.015

0.015

0.017

0.013

0.014

Note: Reference group is narrow bracket condition with no loss in round one. Heteroskedasticity-consistent (HC1) standard errors in parentheses. Dependent

variable is binary choice of certain loss (insurance) over loss lottery (no insurance). +p<0.1; *p<0.05; **p<0.01; ***p<0.001

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

33

TABLE A2. OLS regressions for Study 4. Predicted variable is purchase of flood insurance.

Panel regression

Simple OLS collapsed across rounds

(1)

(2)

1-play prob.

-0.006 (0.033)

-0.030** (0.010)

30-play prob.

0.047 (0.031)

0.014 (0.010)

Round

-0.002 (0.002)

1-play prob. x round

-0.003 (0.003)

30-play prob. x round

-0.004 (0.003)

Sex: female

0.005 (0.027)

0.005 (0.009)

Sex: other/no answer

0.247*** (0.048)

0.247* (0.115)

Age

0.001+ (0.001)

0.001*** (0.0003)

Educ: HS diploma/GED

0.299** (0.100)

0.299*** (0.052)

Educ: some college

0.327** (0.100)

0.327*** (0.052)

Educ: 2-year coll. deg.

0.316** (0.103)

0.316*** (0.053)

Educ: 4-year coll. deg.

0.374*** (0.099)

0.374*** (0.052)

Educ: Master’s deg.

0.379*** (0.105)

0.379*** (0.053)

Educ: doctrate deg.

0.349** (0.121)

0.349*** (0.059)

Educ: professional deg.

0.422*** (0.119)

0.422*** (0.057)

Income

0.001* (0.0003)

0.001*** (0.0001)

Self-assessed risk tolerance

-0.048*** (0.005)

-0.048*** (0.002)

Real-life Flood Experience

0.090*** (0.023)

0.090*** (0.009)

Constant

0.444*** (0.103)

0.426*** (0.054)

Participants

745

745

Observations

11,175

11,175

R2

0.019

0.103

Adjusted R2

0.017

0.102

Note: Reference group is 1-and-30 condition in round one for column one and collapsed across rounds for column two. Heteroskedasticity-consistent (HC1)

standard errors in parentheses. Dependent variable is binary choice to purchase insurance. +p<0.1; *p<0.05; **p<0.01; ***p<0.001

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

34

TABLE A3. OLS panel regressions for Study 5. Predicted variable is purchase of flood insurance.

All time horizons

(1-year as reference)

Excluding 1-year condition

(5-year as reference)

Time horizons coded as indicator variables

Time horizon coded as continuous variable

(1)

(2)

(3)

5-play prob.

0.109** (0.039)

10-play prob.

0.139*** (0.038)

0.029 (0.035)

30-play prob.

0.160*** (0.038)

0.051 (0.035)

200-play prob.

0.132*** (0.039)

0.024 (0.036)

Length of time horizon (continuous)

0.00001 (0.0002)

Round

-0.004 (0.002)

-0.001 (0.002)

-0.001 (0.001)

5-play prob. x round

0.002 (0.003)

10-play prob. x round

0.005+ (0.003)

0.003 (0.003)

30-play prob. x round

0.001 (0.003)

-0.001 (0.003)

200-play prob. x round

0.002 (0.003)

-0.001 (0.003)

Length of time horizon (continuous) x round

-0.00001 (0.00001)

Sex: female

-0.052* (0.020)

-0.065** (0.021)

-0.065** (0.021)

Sex: other/no answer

0.112** (0.041)

0.111* (0.045)

0.109* (0.047)

Age

0.001 (0.001)

0.001 (0.001)

0.001 (0.001)

Educ: HS diploma/GED

0.209+ (0.110)

0.196+ (0.112)

0.187+ (0.107)

Educ: some college

0.177 (0.108)

0.188+ (0.110)

0.183+ (0.104)

Educ: 2-year coll. deg.

0.188+ (0.110)

0.170 (0.112)

0.167 (0.107)

Educ: 4-year coll. deg.

0.199+ (0.108)

0.208+ (0.109)

0.205* (0.103)

Educ: Master’s deg.

0.202+ (0.111)

0.193+ (0.112)

0.189+ (0.107)

Educ: doctrate deg.

0.332** (0.125)

0.417*** (0.113)

0.408*** (0.108)

Educ: professional deg.

0.295* (0.128)

0.329* (0.130)

0.325** (0.125)

Income

-0.0002 (0.0003)

-0.0003 (0.0003)

-0.0003 (0.0003)

Self-assessed risk tolerance

-0.052*** (0.004)

-0.051*** (0.004)

-0.051*** (0.004)

Real-life Flood Experience

0.072** (0.028)

0.086** (0.029)

0.086** (0.029)

Constant

0.599*** (0.118)

0.686*** (0.118)

0.716*** (0.110)

Participants

1,081

861

861

Observations

16,215

12,915

12,915

R2

0.016

0.014

0.014

Adjusted R2

0.015

0.013

0.012

Note: Earnings carried over across rounds in Study 5 (non-independent rounds). For column one, the reference group is the 1-year condition in round one. For

columns two and three, the reference group is the 5-year condition in round one. In column three, the “length of time horizon” variable is a continuous variable

with values equal to the length of the time horizon corresponding to the condition {1, 5, 10, 30, 200}. Heteroskedasticity-consistent (HC1) standard errors in

parentheses. Dependent variable is binary choice to purchase insurance. +p<0.1; *p<0.05; **p<0.01; ***p<0.001

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

35

TABLE A4. OLS regressions for Study 6. Predicted variable is purchase of flood insurance.

Main effects only

Interaction

(1)

(2)

Probability

0.003*** (0.0003)

0.004*** (0.0004)

200-year prob.

-0.080*** (0.019)

-0.022 (0.029)

Probability x 200-year prob.

-0.002** (0.001)

Sex: female

0.066*** (0.020)

0.066*** (0.020)

Sex: other/no answer

0.207 (0.146)

0.205 (0.146)

Age

-0.001 (0.001)

-0.001 (0.001)

Educ: HS diploma/GED

-0.057 (0.189)

-0.057 (0.189)

Educ: some college

-0.017 (0.188)

-0.020 (0.188)

Educ: 2-year coll. deg.

-0.029 (0.189)

-0.033 (0.189)

Educ: 4-year coll. deg.

-0.073 (0.187)

-0.076 (0.187)

Educ: Master’s deg.

-0.007 (0.189)

-0.010 (0.188)

Educ: doctrate deg.

-0.224 (0.207)

-0.226 (0.207)

Educ: professional deg.

-0.087 (0.203)

-0.085 (0.203)

Income

-0.0004 (0.0003)

-0.0004 (0.0003)

Self-assessed risk tolerance

-0.045*** (0.004)

-0.045*** (0.004)

Real-life Flood Experience

0.155*** (0.027)

0.156*** (0.027)

Constant

0.770*** (0.191)

0.737*** (0.191)

Observations

2,303

2,303

R2

0.143

0.146

Adjusted R2

0.137

0.140

Note: **p<0.01; ***p<0.001

Scenario for Study 1

You have been given $95. You have two options:

1) CERTAIN LOSS: A certain loss of $0.45.

or

2) LOTTERY: A lottery with a chance of losing $45. The chance of losing money if you play this lottery one

time is 1% [at least once if you play this lottery 30 times is 26%].

Regardless of whether you select the CERTAIN LOSS or the LOTTERY, you will learn the outcome of the

lottery at the end of each round. Each round involves a new decision with $95. That is, at the end of each round,

your earnings will be reset. You will play multiple rounds (but fewer than 25 rounds).

Scenario for Study 2

Imagine you own a house that is worth 90,000 talers and have a savings account containing 5,000 talers.

There is a chance that a major flood will occur during the year in which case you will suffer severe damage and the

value of the house will decrease by 45,000 talers. If you have purchased insurance for the year and a major flood

occurs, you will be given a 45,000 talers claims payment by the insurer to compensate you for the loss suffered. The

chance of experiencing a flood over the next year is 1% [the next 30 years for your home is 26%].

Each round involves a new decision on a house that is worth 90,000 talers with a savings account

containing 5,000 talers. That is, at the end of each round, all your assets and wealth will be reset.

Scenario for Study 3

You own a house that is currently worth 80,000 talers and have a savings account currently containing

15,000 talers. There is a chance that a major flood will occur during the year in which case you will suffer severe

damage and the value of the house will decrease by 35,000 talers (to a value of 45,000 talers). If you have

purchased insurance for the year (at a cost of 350 talers) and a major flood occurs, the insurer will cover all your

repair costs, and your house value will be fully restored to its current value of 80,000 talers. The chance of

experiencing a flood over the next 1 year for your home is 1%. [The chance of experiencing at least one flood

over the next 30 years for your home is 26%.]

[Note: Underlined amounts updated each round based on remaining wealth from decisions.]

Comprehension check questions for Study 1

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

36

The first question was “Imagine that you chose the LOTTERY and it turned out that you lost money. How

much money would you have at the end of the round?” [options: $95, $50, $25, $10]. The second question was,

“Imagine you are selected to win the bonus and in the round that was chosen you had selected the CERTAIN LOSS.

How much would the bonus for that round be worth?” [options: $95, $94.55, $50, $49.55]. The third question

depended on which probability frame was presented. For the 1-play frame, participants answered “What is the

probability of losing money at least once if you play the LOTTERY one time?” [options: 0.01%, 0.1%, 1%, 10%].

Those with the 30-play frame saw the following question: “What is the probability of losing money at least once if

you play the LOTTERY 30 times?” [options: 20%, 24%, 26%, 30%]. Participants had to get all three questions

correct before being allowed to proceed to the experiment. They were given two chances to correctly answer the

questions. If they failed to answer at least one question correctly the first time, they were sent to the beginning of the

instructions and asked to re-read them again before being allowed to answer the questions a second time.

Comprehension check questions for Study 2

The first question was “Imagine that there is a flood and you did not purchase insurance. How many talers

is your house worth at the end of the round?” [options: 90,000 talers, 45,000 talers, 25,000 talers, 10,000 talers]. The

second question was, “Imagine you are selected to win the bonus and in the round that was chosen you had

purchased insurance at a premium of 450 talers and there was a flood. How much would the bonus be worth?”

[options: $95, $94.55, $45, $44.55]. The third question depended on which probability frame was presented. For the

1-year frame, participants answered “What is the probability of experiencing a flood over the next year?” [options:

0.01%, 0.1%, 1%, 10%]. Those with the 30-year frame saw the following question: “What is the probability of

experiencing a flood over the next 30 years?” [options: 20%, 24%, 26%, 30%]. Like in Study 1, participants had to

get all three questions correct before being allowed to proceed to the experiment, and they were given two chances

to correctly answer the questions.

Comprehension check questions for Study 3

The first question was “If a flood occurs, how much damage will it cause to your house? (That is, by how

much will your house value be reduced?)” [options: 80,000 talers, 35,000 talers, 20,000 talers, 15,000 talers]. The

second question was, “What is the cost of one year of flood insurance?” [options: [premium] talers, 100 talers,

[premium + 300] talers, 200 talers]. The third question depended on which probability frame was presented. For the

1-year frame, participants answered “What is the probability of experiencing a flood over the next year?” [options:

0.01%, 0.1%, 1%, 10%]. Those with the 30-year frame saw the following question: “What is the probability of

experiencing a flood over the next 30 years?” [options: 20%, 24%, 26%, 30%]. They were given two chances to

correctly answer the questions. Participants were allowed to proceed to the experiment even if they failed the

comprehension check, but only after they were informed of the answers to the questions they answered incorrectly.

Only those who passed the comprehension check were included in the analysis.

Comprehension check questions for Study 4

The comprehension check questions were the same as for Study 2 except that participants in the 1-and-30

year condition anwered both versions of the third question. They were given two chances to correctly answer the

questions. Participants were allowed to proceed to the experiment even if they failed the comprehension check, but

only after they were informed of the answers to the questions they answered incorrectly. Only those who passed the

comprehension check were included in the analysis.

Comprehension check questions for Study 5

The comprehension check questions were the same as for Study 3 except that the third question, which

depended on condition, had three additional variations: For the 5-year frame, participants answered “What is the

probability of experiencing at least one flood over the next 5 years?” [options: 0.05%, 0.5%, 5%, 15%]; for the 10-

year frame, participants answered “What is the probability of experiencing at least one flood over the next 10

years?” [options: 1%, 5%, 10%, 15%]; and for the 200-year frame, participants answered “What is the probability of

experiencing at least one flood over the next 200 years?” [options: 26%, 63%, 87%, 99%]. They were given two

chances to correctly answer the questions. Participants were allowed to proceed to the experiment even if they failed

the comprehension check, but only after they were informed of the answers to the questions they answered

incorrectly. Only those who passed the comprehension check were included in the analysis.

Comprehension check questions for Study 6

The first two comprehension check questions were the same as for Study 3 and 5. The third question was

“What is the probability of experiencing a flood over the next [length of assigned time horizon]?”, with options

varying depending on the probability associated with the assigned time horizon [options: [probability + 5 percentage

points]%, [probability]/10 %, [probability]%, [probability – probability/4]%]. They were given two chances to

correctly answer the questions. Participants were allowed to proceed to the experiment even if they failed the

BROAD BRACKETING FOR LOW PROBABILITY EVENTS

37

comprehension check, but only after they were informed of the answers to the questions they answered incorrectly.

Only those who passed the comprehension check were included in the analysis.

Additional measures for Study 2

We assessed experience with flood insurance by asking the following question: “How experienced are you

in buying flood insurance?” The options were: (1) I have never purchased flood insurance and do not know anything

about it; (2) I have never purchased flood insurance but am familiar with it; (3) I have purchased flood insurance

several times before; ( 4) I purchase flood insurance regularly. We also assessed whether people had experienced

flood damage previously by asking the following question: “Have you had prior experience in suffering flood-

related losses from hurricanes or floods?” The options were: (1) I do not know anyone who has suffered a flood-

related loss from a hurricane or flood; (2) I know someone (such as a friend, family member, etc.) who suffered a

flood-related loss from a hurricane or flood but I have not suffered a loss myself; (3) I suffered a flood-related loss

from a hurricane or flood once; (4) I suffered a flood-related loss from a hurricane or flood several times.

Additional measures for Study 3

We assessed understanding of cumulative probabilities with the following question: “Please answer the

following questions based on your own knowledge. For each one, enter a number between 0-100. Round to the

nearest whole number (do not include decimals or percent (%) signs). (1) If the answer to the question was explicitly

provided to you in the scenario, type in that number. (2) If it was not provided to you, please just use your own

intuition to make an educated guess. In the scenario, what was the percent chance of: (a) a flood occurring in the

next 1 year? (b) at least one flood occurring of the next ten years? (c) at least one flood occurring over the next 30

years? (d) at least one flood occurring over the next 100 years? (e) at least one flood occurring over the next 200

years?” [All responses were entered into text boxes that limited responses to be between 0-100.]

We assessed preference over time horizons with the following question: “In the box below, type in the

number that best completes the following sentence for you: ‘I would prefer to know the probability of flood over the

next _______ year(s).’ (Type only numbers.)” [Text box entry allowed only numbers between 1 and 1000.]

List of counties used for Study 4

State

County

Florida

Walton County

Florida

Broward County

Florida

Indian River County

Louisiana

St. Bernard Parish

Louisiana

Orleans Parish

Louisiana

St. Tammany Parish

Louisiana

St. John the Baptist Parish

Louisiana

Plaquemines Parish

Louisiana

Terrebonne Parish

Maryland

Worcester County

New Jersey

Somerset County

Texas

Galveston County

Texas

Brazoria County

Virginia

Virginia Beach city

Data on insurance take-up in each county can be found on the OSF page at:

https://osf.io/mejf5/?view_only=9b8760dca9714e85bc178f290140125e