A 6-10 dossier articles

The effect of ambiguity on risk management choices: An experimental study

Vickie Bajtelsmit1 & Jennifer C. Coats1 & Paul Thistle2

Published online: 24 July 2015 # Springer Science+Business Media New York 2015

Abstract We introduce a model of the decision between precaution and insurance under an ambiguous probability of loss and employ a novel experimental design to test its predictions. Our experimental results show that the likelihood of insurance purchase increases with ambiguous increases in the probability of loss. When insurance is unavailable, individuals invest more in precaution when the probability of loss is known than when it is ambiguous. Our results suggest that sources of ambiguity surrounding liability losses may explain the documented tendency to overinsure against liability rather than meet a standard of care through precaution. The results provide support for our theoretical predictions related to risk management decisions under alternative probabilities of loss and information conditions, and have implications for liability, environmental, and catastrophe insurance markets.

Keywords Liability. Imperfect information . Design of experiments . Laboratory experiments

JEL Classifications K130 . D81 . C9 . C920

Two apparently conflicting puzzles consistently arise out of the empirical observation of insurance markets. Both involve a tendency to make suboptimal insurance decisions and have important implications for environmental risk mitigation, consumer decision making, public finance, and firm profit maximization. First, there is substantial evi- dence that individuals and businesses underinsure catastrophe risk (Kunreuther and

J Risk Uncertain (2015) 50:249–280 DOI 10.1007/s11166-015-9218-3

* Jennifer C. Coats [email protected]

Vickie Bajtelsmit [email protected]

Paul Thistle [email protected]

1 Department of Finance and Real Estate, Colorado State University, Fort Collins, CO 80523, USA 2 Department of Finance, University of Nevada Las Vegas, Las Vegas, NV 89154, USA

Pauly 2004; 2005). The devastating cost of a failure to insure against catastrophe is highlighted repeatedly with each natural disaster. Second, individuals and firms pur- chase liability insurance even when neither law nor contract requires they do so. Given that injurers are held liable under U.S. law only if they have failed to meet a reasonable standard of care, expenditure on care could be a less expensive alternative to purchasing actuarially unfair liability insurance. In the absence of the ability to take precaution against accident, theory suggests that risk-averse individuals will fully insure when actuarially fair insurance is available. In situations where insurance is not fairly priced or where precaution is an alternative, the optimal choice depends on risk aversion, insurer profit and risk loading, and the cost of precaution.

Although negligence liability can be avoided by exercising an appropriate level of care, there are many sources of uncertainty that could explain the existence of the thriving liability insurance market in the U.S. The theoretical literature suggests that insurance demand may be explained by uncertainty regarding one’s own risk type (Bajtelsmit and Thistle 2008; 2015), the mechanics of the pooling mechanism (DeDonder and Hindriks 2009), the cost of taking precaution (Bajtelsmit and Thistle 2009), potential for errors by the courts (Sarath 1991), and the risk of momentary lapses in judgment by oneself or others (Bajtelsmit and Thistle 2013). Uncertainty may be especially profound in the face of environmental risks. Riddel (2012) notes that environmental gambles involve greater uncertainty surrounding the probability, severity, and welfare loss effects of outcomes. In a comprehensive overview of environmental risk management, Anderson (2002) highlights the extensive degree of ambiguity surrounding potential environmental losses, even from the standpoint of risk-neutral corporations. In addition to the usual risks related to property, liability, life and health, environmental risks may include ethical, cultural, business, reputational, and regulatory uncertainty. Anderson also notes that the interpretation of preventive measures under environmental liability is particularly vague compared to other liability standards. Therefore, the degree of ambiguity that surrounds the court’s judgment of whether a defendant has met the standard of care is likely to be higher in environmental liability cases than under other liability cases. We view a greater understanding, in general, of precaution and insurance decisions under ambiguity as a crucial step towards understanding these tradeoffs under particular types of ambiguity, such as that created by environmental risks.

In this paper, we show theoretically that, when the probability of loss is more ambiguous, the demand for insurance increases. However, the ambiguity may increase or decrease expenditure on precaution, depending on assumptions related to the cost and benefit of precautionary spending. We test these results empirically in a laboratory experiment in which participants make decisions about insurance and precaution under different ambiguity conditions.

We extend the literature on the market for insurance in several dimensions. First, we develop a model which includes mistakes as a source of ambiguity underlying the decision between precaution and insurance and shows that ambiguity aversion increases insurance demand. Second, we employ a novel experimental design to test the predictions of the model. To our knowledge, ours is the first study to model the effect of ambiguity on precaution and insurance in this way and to use the experimental method to investigate the choice between precaution and insurance. Third, the experimental design also

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allows us to test previous theoretical findings related to the choice between precaution and insurance by individuals with heterogeneous probabilities of loss. In particular, Bajtelsmit and Thistle (2008) show that the optimal insur- ance contract leads individuals with high probability of loss to meet the standard of care and thereby avoid liability, whereas individuals with low probability of loss prefer to purchase insurance and take less precaution. Their results imply that individuals who have a preference for taking full precaution when insurance is unavailable will switch to insurance if it becomes available at a comparable cost. Finally, our design, parameters, and framing allow us to contribute additional evidence to existing mixed results related to the decision to insure against low-probability, high-severity losses.

Our primary motivation is to test whether ambiguity surrounding the prob- ability of a loss impacts the demand for precaution and insurance, as suggested by our theoretical model. To our knowledge, ours is the first laboratory study to allow a choice between buying insurance and exercising a level of precaution to achieve a desired level of risk of a loss. 1 The experimental design requires participants to make precaution and insurance decisions under different condi- tions, some of which involve risks with known probability distributions and others in which the probability of loss is unknown or ambiguous to both the experimenter and the participant. Participants make decisions under conditions of low and high probability of loss. In some treatments, participants can pay for a desired level of precaution and, in others, they can choose to buy insurance or alternative levels of precaution. To determine whether ambiguity of the loss distribution affects participants’ precaution and insurance decisions, in some treatments the participants are subject to an additional unknown risk of loss. By using a similar experimental design, as well as similar parameters and framing, we confirm the experimental results of Laury et al. (2009) that individuals are more likely to purchase insurance in the low probability treat- ments, after controlling for other factors such insurance pricing and loss severity. Empirical analysis of participant decisions under conditions of known versus ambiguous loss probabilities shows that the likelihood of insurance purchase increases with ambiguous increases in the probability of loss and that, when insurance is unavailable, individuals invest more in precaution when probability of loss is known than when it is unknown. Our results also provide support for theoretical findings in Bajtelsmit and Thistle (2008): in the absence of ambiguity, participants are more likely to purchase insurance in the low probability treatments and those who prefer full precaution when insurance is unavailable switch to insurance when it is available.

The next section reviews the theoretical and experimental literature related to the purchase of insurance against liability and catastrophe losses and presents a theoretical model to analyze the impact of ambiguity on insurance and precaution decisions. The laboratory experiment, which closely follows the theory setup, is described in Section 2. We formalize our hypotheses in Section 3, summarize the empirical analysis and results in Section 4 and provide conclusions in Section 5.

1 However, several papers do examine risk mitigation or endogenous risk, without considering the role of insurance—such as Fiore et al. (2009) and Harrison et al. (2010).

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1 Background and theory

1.1 Background

The extensive theoretical literature on insurance demand provides several explanations for the purchase of liability insurance. Under the standard model of expected utility theory, these include risk aversion of agents, uncertainty/ambiguity related to proba- bility of loss, cost of care, and operation of the legal system. This literature has generally distinguished individual insurance decisions from corporate insurance deci- sions. Theoretically, risk neutral corporations should not be willing to buy insurance at actuarially unfair prices. However, agency theory suggests that risk-averse managers might be motivated to do so on behalf of the firm, in order to protect their own employment and/or reputations (see, for example, Greenwald and Stiglitz 1990; Han 1996; Mayers and Smith 1982).

A second strand of the insurance literature, also based on standard expected utility theory, focuses on individual decision-making under ambiguity (when the probability of loss is not objectively known). Although the risk of negligence liability can be avoided by exercising an appropriate level of care, there are many sources of ambiguity related to understanding the risk, satisfying the negligence standard, and judicial enforcement of the standard. For example, potential injurers may face uncertainty about their own risk type (Bajtelsmit and Thistle 2008), the mechanics of the pooling mechanism (DeDonder and Hindriks 2009), or the cost of taking precaution to avoid risks (Bajtelsmit and Thistle 2009). Shavell (2000) illustrates that uncertainty regarding negligence standards results in a level of care that exceeds a socially optimal level. The potential for errors by the courts (Sarath 1991) and the possibility of injuries caused by momentary lapses in judgment, either one’s own mistakes or another agent’s (Bajtelsmit and Thistle 2013), theoretically have been shown to justify a market for insurance.

A more generalized stream of research investigates decision-making under risk and uncertainty according to both standard and non-standard risk preferences. While there are many potential sources of ambiguity in a liability case, as discussed above, our experimental design and analysis adopts Camerer and Weber’s (1992) definition of ambiguity: Buncertainty about probability created by missing information that is rele- vant and could be known^ (p. 330). They note further that Bif ambiguity is caused by missing information, then the number of possible distributions . . . might vary as the amount or nature of missing information varies^ (p. 331). In several treatments in our experiment, participants make decisions that depend on outcomes whose probabilities they have estimated with varying degrees of missing information, but are unknown at the time either to themselves or the experimenters.

A vast literature related specifically to risk preferences suggests that Bnonstandard^ features, not included in expected utility theory, drive behavior. Non-expected utility theories include alternative decision-weighted probability models, prospect theory by Kahneman and Tversky (1979), and Tversky and Kahneman’s cumulative prospect theory (1992), which combine probability-weighting with different risk preferences over gains and losses. 2 Prospect theory suggests that individuals underestimate or

2 See Starmer (2000) for a review.

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ignore very low probability events and the primary explanation in the literature given for underinsurance of catastrophic loss is that individuals may ignore probabilities below a certain threshold.3

Laboratory experiments on insurance purchase decisions under different risk and ambiguity conditions have been conducted under a wide variety of designs and protocols and the results are highly inconclusive. 4 A few experimental studies (Ganderton et al. 2000; Laury et al. 2009; McClelland et al. 1993; Slovic et al. 1977) test the tendency to underinsure against low-probability high-severity losses. However, the differences in designs, procedures, and parameters employed across the studies limit the ability to generalize conclusions from their results. The Laury et al. experimental design, discussed in detail below, implements a choice task to investigate the phenom- enon of underinsurance for low-probability, high-severity losses, and produces results that are counter to the notion that individuals ignore very low probabilities.5

1.2 The theoretical effect of ambiguity on precaution and insurance decisions

The underlying theory is based on the standard model of accidents in the law and economics literature. In the absence of the ability to take precaution against accident, theory suggests that risk-averse expected utility maximizers will fully insure when actuarially fair insurance is available. In general, the assumption of risk aversion implies that individuals will be willing to pay some level of load or risk premium to avoid risk. Thus, when insurance is not fairly priced, the optimal choice depends on the level of risk aversion and the insurance loading factor.

We assume that individuals are expected utility maximizers with increasing concave von Neumann-Morgenstern (vNM) utility u. Individuals have exogenous initial wealth w and face a potential loss d<w with probability π. Expenditure on precaution or care is denoted c (c ≥ 0) and the risk of a loss is a decreasing, convex function of c. Individuals have either a high or low probability of loss, where πH(c) > πL(c) for any expenditure on precaution. We assume 0 ≤ π(c) < 1, that is, it is possible to reduce the risk of loss to zero through expenditure on precaution. We also assume precaution has a lower marginal impact on the probability of loss for low-probability risks than for high probability risks, 0 > π′L(c) > π′H(c). We assume each person knows whether they face high or low risk and understands how the level of precaution affects the probability of loss. An insurance policy consists of a premium, pi, paid whether or not loss occurs, and an indemnity, qi, paid in the event that the loss occurs. The first best levels of precaution are ci* = argmin ci + πi(ci)d, i = H, L.

3 The behavioral literature also suggests that certain behavioral biases, such as overconfidence or optimism, as well as the tendency to overreact to recent events, may explain under- and overinsurance for certain types of losses. See, for example, Kunreuther et al. (2001). 4 See Jaspersen (2014) for a comprehensive review. 5 Many studies attempt to explain insurance markets by designing the experiments as auctions rather than choice tasks. See, for example, Camerer and Kunreuther (1989) and Hogarth and Kunreuther (1989). Although this design may work well as a mechanism for eliciting willingness to pay for insurance, and under a double auction, studying both sides of the insurance markets, the results are not necessarily generalizable to the insurance marketplace in which consumers face choice tasks rather than pricing tasks, as explained in Laury et al. (2009).

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If insurance is not available, then expected utility is

Ui cið Þ ¼ 1−πi cið Þð Þu w−cið Þ þ πi cið Þu w−ci−dð Þ ð1Þ

The individual chooses the level of precaution, ci 0, that maximizes expected utility.

Because the individual is risk averse, she is willing to pay some amount PiU to avoid the risk of loss. The results in Bajtelsmit and Thistle (2008) imply that the willingness to pay to avoid the risk is given by u(w − PiU)=Ui(ci

0). Willingness to pay can be written as PiU=ci

0+πi(ci 0)d+ρiU, where ρiU is a risk premium.

Now assume that insurance is available, that insurers can determine risk type ex ante, and that the expenditure on precaution is observable. In general, the insurance premium can be written as pi=λπi(ci)qi, where λ is the loading factor; the insurance premium is actuarially fair if λ=1 and unfair if λ>1. The individual who buys the insurance policy (pi, qi) and spends ci on care has expected utility given by

Ui pi; qi; cið Þ ¼ 1−πi cið Þð Þu w−pi−cið Þ þ πi cið Þu w−pi−ci−d þ qið Þ ð2Þ for i=H, L.

The risk of negligence liability presents a special case. If liability is determined by a negligence rule, individuals who exercise a Breasonable^ level of care will have a zero probability of loss. More specifically, under a negligence rule where the negligence standard of care is z, an individual is liable for damages if their level of precaution is less than the negligence standard, ci<z and is not liable for damages if their level of precaution meets the negligence standard, ci≥z. Meeting the negligence standard yields utility u(w − z). If insurance is available at actuarially fair prices, the individual can also eliminate the risk by fully insuring; this yields utility u(w − c* - π(c*)d). The individual will choose whichever alternative is less expensive. With a premium loading, the insurance decision will depend on the relationship between the cost of the insurance relative to the cost of precaution. If the premium is not actuarially fair, the individual will not choose full insurance. This yields utility u(w − ĉi − λπ (ĉi) q̂i − ρi), where ρi is the risk premium for the residual uninsured risk. The individual will choose insurance if ĉi + λπ (ĉi) q̂i + ρi<z, that is, if the cost of insurance and precaution is sufficiently less than the cost of meeting the negligence standard. Individuals will not choose to insure if the cost of doing so is greater than the cost of meeting the negligence standard. The size of the insurance loading factor relative to expected loss and cost of precaution makes a difference in the predicted decision between insurance and precaution. For example, for low frequency, low severity risks, expected loss may be so small that even a modest profit and risk charge will tilt the scale toward taking care instead of buying insurance.

In most analyses of liability, as in the analysis described above, the probability of an accident is a function of care or precaution and is deterministic. Now suppose that it is possible to make a mistake that, despite expenditure on care, can result in an accident. We can think of this as a momentary lapse in judgment, such as a driver glancing away from the road just before a dog crosses the street or an oil rig worker failing to notice a worn valve. Despite effort and expenditure on compliance, managers cannot predict precisely how the courts will assess liability and damages from environmental losses. As discussed at length in Anderson (2002), these types of losses expose firms to a great

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deal of uncertainty. Therefore, we model the case in which individuals and firms know that there is a random chance of a mistake, but they do not know exactly how it will impact the probability of loss.

Thus, denote ~m as the probability of a mistake, independent of expenditure on care or precaution, which results in loss d, and assume that the probability of a mistake is unknown. We deliberately do not distinguish the sources of this mistake. It could be one’s own mistake, the mistake of another agent, or an error by the courts. The fact that the probability of a mistake is unknown introduces ambiguity. Letting m = E ~mf g be the expected probability of a mistake, expected utility is given by:

Ui cim � �

¼ 1−m � �

1−πi cð Þð Þu w−cið Þ þ πi cið Þu w− ci−dð Þ½ � þ m u w− ci−dð Þ ð3Þ

for i=H,L. The optimal expenditure on care decreases with increasing expected prob- ability of mistake. As m approaches 1, expected utility is optimized with zero expen- diture on care. For a very small expected probability of a mistake, the problem reduces to Eq. (1) and the individual will select the level of care that minimizes total cost of loss and precaution.

If the individual is ambiguity averse, then decisions are made according to the second order expected utility function

Vi cið Þ ¼ E Φ Ui ci; ~m � �� �n o

¼ E Φ 1−~m � ��

1−πi cð Þð Þu w−cið Þ þ πi cið Þu w− ci−dð ÞÞ½ � þ ~m u w−ci−dð Þ n o ð4Þ

where the expectation is over the distribution of mistakes (Klibanoff et al. 2005; Neilson 2010). The vNM utility function u captures the attitude toward risk while Φ captures the attitude toward ambiguity. If the individual is ambiguity neutral then Φ is linear and if the individual is ambiguity averse then Φ is concave. An ambiguity-averse individual is willing to pay to eliminate the risk; the willingness to pay to avoid the risk is given by Φ(u(w − PiV)=max E{Φ(Ui(ci, ~m). We show that ambiguity aversion increases the willingness to pay to avoid the risk,

PiV ≥PiU ; ð5Þ the proof is given in Appendix 1.6 In sum, ambiguity aversion is shown to increase the demand for insurance.

The effect of ambiguity aversion on the optimal level of precaution is theoretically indeterminant and depends on the fine detail of the theoretical model. Snow (2011) shows that if individuals have unbiased beliefs (i.e., E{π(c, ~m)} equals the objective loss probability), then the loss probability must be either multiplicatively separable (π(c, ~m)=α(c)π(~m)) or additively separable (π(c, ~m)=π(~m)+β(c)). Snow further shows that multiplicative separability implies ambiguity aversion increases the expenditure on care. Snow (2011) and Alary et al. (2010) show that additive separability decreases the expenditure on care. The effect of ambiguity aversion on the expenditure on care is therefore an empirical question. However, decreased willingness to pay for small

6 Alary et al. (2010) and Snow (2011) show that ambiguity aversion increases the willingness to pay to avoid the risk when the distribution of the risk is fixed. Their result does not apply directly here because individuals can shift the distribution of risk by exercising care.

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reductions in risk seems at odds with an increased willingness to pay to avoid the risk and implies a discontinuity in behavior between small risk reductions and risk elimi- nation. This suggests that ambiguity will lead to lower expenditures on care.

Now consider the same case when insurance is available. If an individual’s proba- bility of loss depends both on risk type and the chance of mistake, then the expected utility for a person who buys the insurance policy (pi, qi) and spends ci on care is given by:

Ui pi; qi; ci; m � �

¼ 1−m � �

1−πi cið ÞÞ½ u w−pi−cið Þ þ πi cið Þu w−pi−ci−d þ qið Þ þ m u w−pi−ci−d þ qið Þ

ð6Þ

For an individual who is ambiguity averse, the second order expected utility is V(pi, qi, ci)=E{Φ(Ui(pi, qi, ci, ~m)}. Given the risk of mistakes, the actuarially fair premium is pi=(πi + m (1 − πi))d. If the premium is actuarially fair, then the individual will fully insure (q=d), and receive utility u(w − ci* − pi).

In the following section we discuss our use of the experimental method to investi- gate the theoretical predictions developed above and formally present a set of testable hypotheses in the context of the experimental design. To summarize, under a setting of a clearly-defined negligence standard with no risk of mistakes, we test the predictions that individuals will not insure if it is more efficient to simply meet the standard of care, and that individuals are less likely to insure as the size of the insurance loading factor increases. We introduce mistakes into the design, and investigate the impact of ambig- uous increases in the probability of loss on insurance and precaution decisions.

2 Experimental design and procedures

In this section we present the experimental design and briefly discuss the procedures that were used to implement the design in the laboratory. Where applicable, the design and procedures follow those used in the Laury et al. (2009) experiments. In our within- subject design, each participant made independent decisions in twenty treatments. A random draw of one treatment at the end of the experiment determined actual payoffs.

The risk of loss was implemented as a computer-generated random number— explained with the analogy of a random draw from 100 white and orange ping pong balls, where a draw of an orange ball resulted in a loss of a specific dollar amount from their experiment earnings. Participants were told the probability of loss through a description of the number of orange and white balls respectively in each treatment as well as the numerical probability of loss. In some treatments they could reduce their probability of loss by paying for units of precaution, described as the option to pay to replace orange balls with white balls. In other treatments, participants could choose between precaution, insurance, and no risk mitigation. An actuarially fair premium in a competitive insurance market is based on the expected loss in a population of policyholders in which some face higher risks of loss than others. Therefore, the insurance load associated with a single premium will vary across individuals. In our main treatments, we hold constant the loss severity, insurance premium, and cost per unit of precaution, which implies the insurance (or equivalent precaution) load will necessarily be higher in treatments with a lower initial risk of loss than treatments with

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a higher initial risk of loss, all else equal. To introduce ambiguity and determine whether the chance of mistakes changes participants’ choices over precaution and

Table 1 Experimental treatments and corresponding initial probabilities of loss prior to risk mitigation, by ambiguitya and risk type

Panel A: Main Treatments

Level of ambiguity in treatment

Loss amo- unt ($)

Risk mitigation alternatives available

Initial probability of loss

Treatment #

Low Risk/ High Load

Treatment #

High Risk/ Low Load

No ambiguity-known probability

45.00 Precaution only #1 0.10 #2 0.32

No ambiguity-known probability

45.00 Precaution or insurance

#3 0.10 #4 0.32

Ambiguity due to unknown probability of own mistake

45.00 Precaution only #5 ≥0.10 #6 ≥0.32

Ambiguity due to unknown probability of own mistake

45.00 Precaution or insurance

#7 ≥0.10 #8 ≥0.32

Ambiguity due to unknown probability of other’s mistake

45.00 Precaution only #9 ≥0.10 #10 ≥0.32

Ambiguity due to unknown probability of other’s mistake

45.00 Precaution or insurance

#11 ≥0.10 #12 ≥0.32

Panel B: Replication treatmentsb

Level of ambiguity in treatment

Loss amo- unt ($)

Risk mitigation alternatives available

Initial probability of loss

Treatment #

High Load Treatment #

Low Load

No ambiguity- known probability

45.00 Insurance only #13 0.01 #17 0.01

No ambiguity- known probability

4.50 Insurance only #14 0.10 #18 0.10

No ambiguity- known probability

60.00 Insurance only #15 0.01 #19 0.01

No ambiguity- known probability

6.00 Insurance only #16 0.10 #20 0.10

a In the no ambiguity treatments, prior to making the risk mitigation decision, participants are given the initial probabilities and the effect that their risk mitigation decision will have on the probability of loss. In the Own Mistake treatments, participants know the initial probability of loss, but are subject to an additional unknown risk of loss that depends on their own performance on the driving quiz. In the Others’ Mistake treatments, participants know the initial probability of loss, but are subject to an additional unknown risk of loss that depends on the performance of another participant on the driving quiz. Because the secondary risk is participant-specific, the probability of loss for the ambiguity treatments is not known for certain but is greater than or equal to the initial probability of loss that is given in the treatment b The replication treatments use the loss amounts and probabilities given in Laury et al. (2009). These treatments were included in the experiment for purposes of validation of the experimental design, but are not used in any of the main empirical models in this paper

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insurance, in some treatments the draw of a white ball could still result in a loss, depending on mistakes made during the earnings task. These elements of the experi- ment are described more fully in this section.

2.1 Earnings task

Similar to Laury et al. (2009), participants received earnings in several installments. We paid a $15 participation payment in cash at the start of the experiment, and collected a signed receipt from each participant. We encouraged them to put this money away and emphasized that the $15 was payment for their participation and would not be at risk in the experiment. We also clearly framed the risky environment to require decisions over losses of their earnings, rather than gambles over gains. This design feature was intended to more closely resemble decision-making in the actual insurance market. Prior to receiving any instructions or information about the risk management and insurance task, participants earned their endowment by successfully completing an earnings task, which required taking a written quiz covering basic knowledge about state driving rules. Upon completion of the driving quiz, they were asked to estimate their own score and the average score for the group. 7 Following the earnings task, they received instructions and completed an assessment to ensure that they fully understood the decisions they would be asked to make in the experiment. After the assessment, they reviewed their earnings task answers and estimated scores and entered them into computers. Lastly, they participated in the precaution and insurance decision-making task which, together with chance, determined whether they experienced a loss from the money they earned in the earnings task.

2.2 Risk management treatments

Table 1 summarizes the 20 treatments in the experiments. The baseline treatments, based on replication of Laury et al. (2009), are given in Panel B of Table 1. In the baseline treatments, insurance is the only available form of risk mitigation and the manipulations include probability of loss, loss amount, and insurance load. The combinations of treatment manipulations result in eight baseline treatments. As in Laury et al., the expected loss is set to $0.45 and $0.60 and insurance loads are set to 1 (actuarially fair insurance) and approximately 3 (3.22 and 3.25 to facilitate stating premiums in increments of $.05).

The manipulations that comprise the main precaution and insurance treatments are summarized in Panel A of Table 1. These treatments include: type of risk mitigation (precaution only or a choice between precaution and insurance), initial probability of loss and corresponding insurance load (either high probability-low load or low probability-high load), and ambiguity (none, ambiguity resulting from

7 A driving quiz was chosen for the earnings task to ensure that all participants would be familiar with the subject matter and could successfully complete the quiz but still have some risk of making mistakes. The median quiz score was 75%. The median estimates for own score and others’ scores were 85 and 78% respectively. We required that participants had a driver’s license. The questions on the quiz were similar to those that would appear on a written state driving test. Although the risk of errors was therefore clearly related to auto accident risk, all instructions and the loss scenarios were framed in neutral language and not in the context of decisions over auto insurance per se.

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unknown errors on a participant’s own driving quiz, or ambiguity resulting from unknown errors on a different, unknown participant’s driving quiz). In the treat- ments without ambiguity, the probability of loss, both before and after any precau- tionary spending, is known by the participants. In the treatments with ambiguity, the participants do not have full information about the probability of loss. In particular, participants’ own scores, other participants’ scores, and the distribution of quiz scores are all unknown to participants and also unknown to the experi- menter. For the ambiguity treatments, it was explained to participants that if an orange ball was drawn, they would experience a loss for certain. However if a white ball was drawn, then the outcome would depend on an additional random draw from an unknown distribution (driving quiz questions). In the Own-Mistakes treatments, a quiz question was randomly selected and each participant incurred a loss if their own answer to the selected question was incorrect. In the Others’- Mistakes treatments, a quiz question was randomly selected, and another participant was randomly selected, and a loss occurred if the other participant’s quiz question was answered incorrectly. Therefore, the quality of information across the ambiguity treatments varies. The combinations of type of risk mitigation, initial probability of loss, and ambiguity manipulation result in 12 main treatments. The experiment

Table 2 Losses, probabilities and expected losses under precaution only/no ambiguity treatmentsa

No loss Loss Low risk treatments (Initial probability of loss = 10%)

High risk treatments (Initial probability of loss = 32%)

Decision Total $ cost (risk mitigation)

Total $ cost (risk mitigation + actual loss)

Probability of loss after risk mitigation

Cost of risk mitigation + E(loss)

Probability of loss after risk mitigation

Cost of risk mitigation + E(loss)

A 0.00 45.00 10% 4.50 0.32 14.40

B 1.50 46.50 9% 5.55 0.28 14.10

C 3.00 48.00 8% 6.60 0.24 13.80

D 4.50 49.50 7% 7.65 0.2 13.50

E 6.00 51.00 6% 8.70 0.16 13.20

F 7.50 52.50 5% 9.75 0.12 12.90

G 9.00 54.00 4% 10.80 0.08 12.60

H 10.50 55.50 3% 11.85 0.04 12.30

I 12.00 57.00 2% 12.90 0 12.00

J 13.50 58.50 1% 13.95 NA NA

K 15.00 NA 0% 15.00 NA NA

a The table summarizes the costs and benefits of risk mitigation in alternatives in Treatments #1 and #2 in which the 60 participants were exposed to a known probability of loss (no ambiguity) and were given a menu of 11 risk mitigation alternatives (Decisions A-K). They could do nothing (Decision A) or they could reduce the probability of the bad outcome in Decisions B-K) by paying $1.50 to replace orange balls with white balls ($1.50 for one ball in the low risk treatments and $1.50 for 4 balls in the high risk treatments). Their total costs were therefore either the cost for the level of risk mitigation they selected (Decision A-K) or, in the event that an orange ball ended up being drawn, the cost of the risk mitigation plus the cost of the loss itself

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design includes a high loss severity ($45) relative to quiz earnings ($60) in order to simulate catastrophic loss.

In all of the main treatments summarized in Table 1, participants were offered a menu of precaution options, from which they could select to incrementally reduce the probability of loss at a cost of $1.50 per unit of precaution. In the treatments with both precaution and insurance, the option to purchase insurance for $14.50 was added as an alternative to the precaution choices. Table 2 summarizes the costs, probabilities, and expected losses under the menu of alternatives available in the No Ambiguity treatments.8 In the treatments with ambiguity, the menu of precaution options was the same, but probabilities and expected losses were unknown. Consistent with our theoretical model and with intuition, the Bmarginal product of care^ was higher for the high probability treatments. In the low probability treatments, each $1.50 resulted in a one percentage point reduction in probability of loss, and in the high probability treatments, it resulted in a 4 percentage point reduction in probability of loss. The reduction in probability was presented to the participants numerically and also with the analogy of Bremoving orange balls and replacing them with white balls.^ The initial 32% and 10% probabilities of loss for the High Risk and Low Risk manipulations correspond to expected losses of $14.40 and $4.50 respec- tively in the No Ambiguity treatments. Since participants could eliminate risk through buying full precaution in these treatments, the equivalent insurance loads under full precaution are 3.33 under the low risk of loss, and 0.83 under the high risk of loss. In the precaution/insurance treatments, the insurance premium was uniformly set at $14.50. This implies an insurance load of 3.22 in the low- probability treatment and a load of approximately 1 in the high-probability treatment. These insurance loads also facilitate comparison with Laury et al. (2009) who investigated behavior under a loading factor of 1 compared to a loading factor of 3.

2.3 Procedures

Participants were recruited for pay from business classes at a large university. The experiment was programmed and implemented with a Z-tree application (Fischbacher 2007) and all sessions were conducted in a networked computer lab with partitioned stations. Six sessions, each with ten participants, were conducted between June and October 2013. As in Laury et al. (2009), we conducted the experiment in a four-phase sequence: induction, earnings task, risk management decision task, and payment, as summarized in Fig. 1.

In the induction, we paid participants as described above, summarized procedures in a Power Point presentation at the front of the room, and read the instructions aloud. In the earnings task, participants earned $60 for correctly answering 8 or more out of 20 questions on the driving quiz described above. To assess confidence in their

8 Participants were presented with decisions, probabilities of loss, and total cost of precaution for each decision, separately for each scenario (presentation of a treatment). They were not presented with the expected loss. We did not use terms such as ‘precaution’ or ‘risk mitigation’, just the phrase ‘reduce your probability of loss.’

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performance and to encourage participants to develop a subjective probability estimate for their chance of mistake, they were asked after every question to indicate whether they were certain they had answered it correctly. At the end of the quiz, they were also asked to estimate their total correct score and to estimate the average score for the other participants in the session.

Studies eliciting subjective probabilities must provide salient incentives for participants to form their best estimates and, in our design, a more accurate estimate allowed for higher expected payoffs.9 We explained to participants that they needed to try to answer as many quiz questions as possible correctly because later in the experiment, answering more questions correctly would improve their chances of earning more money. We note that participants only recorded their estimated scores after they received all the instructions for the experiment and completed an instructions assessment which covered how driving quiz scores affect earnings. Therefore, participants appeared to comprehend that they were rewarded for accuracy in their estimates, and we view their reported estimates as the beliefs generating subjective probabilities in our analysis.10

Next, we read the risk management decision task instructions aloud, and the partic- ipants took an instructions-assessment to ensure that they fully understood the different treatment types (referred to as Bscenarios^ in the instructions) in which they would be making decisions. The next stage of the experiment did not begin until all participants completed the assessment correctly and indicated they had no further questions.

Participants then entered their driving quiz answers into the computer, including their confidence assessment for each answer, estimated their own total score and the average score for the group. The computer calculated their actual score, and reported their $60 earnings to them on the screen. Although participants did not know their scores, by earning $60, they necessarily knew that they had answered at least 40% of the quiz questions correctly. The participants then began the risk

9 For example, a risk-neutral subject with an initial 10% probability of loss who estimates a score of 90% on the driving quiz is better off not purchasing insurance. But if the participant’s actual score on the driving quiz is 70% then the expected payoff is higher with insurance and such a subject is, therefore, penalized for error. Scoring rules are often applied in experiments to reward accurate reports of subjective probabilities, typically in the form of a fixed reward for the estimate plus a penalty for error. Yet in these cases subjects’ risk preferences can affect their reports. See, for example, Andersen et al. (2013) and Harrison et al. (2013). 10 Technically, because we don’t reward and penalize reported scores directly, participants could estimate one score, but report a different score. However, given the 20 different treatments and careful attention to detail required throughout the experiment we note this would be very cognitively costly. Combined with the lack of financial incentive to record a particular score different from a true estimate, we view this as highly unlikely.

Induction

paid $15 participation fee that is not at risk in the experiment

Earnings Task

$60 endowment for successfully completing a Driving Quiz

estimate quiz scores for self and others.

Risk Management Decision Task

decisions for 20 Scenarios which place their earnings at risk.

Payment

select Scenario that will determine their net earnings from the experiment

•Subjects are •Subjects earn •Subjects make •Random draw to

•Subjects

Fig. 1 Sequence of experimental procedures

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management decision task in which they were required to make insurance and precaution decisions for the twenty treatments described in Tables 1 and 2 in the previous subsection. The treatments were randomized and participants were allowed to make revisions after completion of all twenty scenarios. This minimized the risk of order effects and data entry errors.11

In the final stage of the experiment, we randomly selected the scenario that would determine experiment earnings with a public draw by a participant from a basket of twenty numbered ping pong balls. All participants entered the scenario number into the program and the computer simulated the draw from the individual distributions that would determine their earnings, given their own expenditure on precaution or insurance for that scenario. Although all participants’ outcomes were determined by the same treatment, their individual decisions related to precaution and insurance resulted in participant-specific net earnings. The participants were then given an on-screen sum- mary of the outcome of the draw and their personal earnings. Finally, they completed a demographic survey and were then privately paid their net earnings in addition to the participation fee received in the induction, by the experimenters.

The sessions, including payment, lasted approximately 135 minutes and participants earned an average of $67 each, including their $15 participation fee. Although the sessions were relatively long, per hour compensation was fairly high and many partic- ipants indicated that, independent of earning the money, they enjoyed the experience.

3 Hypotheses

Based on the previous literature and the theoretical model in the previous section, we develop several hypotheses that are tested in the experiment. Hypotheses 1 and 2 are tests of theoretical results from Bajtelsmit and Thistle (2008). Hypotheses 3, 4 and 5 relate to the effect of ambiguity on insurance and precaution decisions.

Hypothesis 1 Individuals who prefer zero risk will choose the more efficient risk management method to accomplish this goal.

Discussion Bajtelsmit and Thistle (2008) show that heterogeneity of potential injurers, either in probability of loss or cost of taking precaution to reduce the risk of loss, can create a market for liability insurance. They find that for some individuals and firms— those with high cost of care and/or low probability of loss—it may be more efficient to buy insurance rather than to take optimal care. We hypothesize that rational expected utility maximizers will select the risk management choice that most efficiently achieves their desired outcome. In our experimental design, participants can reduce their risk to zero in the unambiguous precaution treatments by paying for full precaution and, in the precaution/insurance treatments, by purchasing insurance. We therefore expect that participants who prefer full precaution when insurance is unavailable will be more likely to buy insurance when it is available. Although full precaution and insurance in the treatments without ambiguity can accomplish the goal of zero risk, they are not

11 The order of presentation of the seven treatment types in Tables 1 and 2 was varied randomly for each subject. Within treatment type, the order of the treatments was also varied randomly.

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equivalent in cost. Therefore, we expect that participants who prefer zero risk will buy insurance to achieve their goal of risk reduction only if it is the lowest cost alternative for achieving that outcome i.e., in the low-probability treatments.

Hypothesis 2 Risk mitigation decisions will be consistent in otherwise similar treat- ments with and without insurance.

Discussion Similar to the discussion regarding Hypothesis 1, we expect that partici- pants will exhibit consistent risk preferences across the different insurance treatments. Therefore, those who prefer less than full precaution in the treatments in which insurance is unavailable will be less likely to purchase insurance when it is available.

Hypothesis 3 The likelihood of insurance purchase is higher under ambiguous increases in the probability of loss than the likelihood of insurance purchase (or the equivalent of full precaution) under objectively known increases in the probability of loss.

Discussion The model in Section 2 shows that ambiguity will increase the demand for insurance. In our experimental design, the risk of mistakes introduces ambiguity, but simultaneously increases the expected probability of loss. Thus, for the same initial probability (10% or 32%), the demand for insurance is both a function of ambiguity and the increase in probability that results from the ambig- uous risk of mistakes by self or others. A more direct test of this hypothesis is possible because the ambiguous precaution/insurance treatments with 10% initial probability have approximately the same expected probability as the unambiguous 32% probability of loss treatments. 12 The lower insurance load for the 32% probability treatments could make insurance more attractive as compared to the 10% probability treatments. The net effect is unknown, a priori, but a finding that an ambiguous increase in the probability of loss from 10% has a greater impact on the demand for insurance than the known increase from 10% to 32% would be a stronger result because the insurance load is three times larger under the initial probability of 10% than 32%.

Hypothesis 4 Individuals will exercise more precaution when the probability of loss is known than when it is ambiguous.

Discussion The effect of ambiguity on the amount of precaution is sensitive to the nuances of the theoretical model design. Similar to Hypothesis 2 above, participants are expected to respond to the price of precaution in the sense that a given amount spent on precaution does not reduce the probability of loss by as much in the ambiguity treatments as it does in the known-probability treatments. Even after paying for the maximum precaution, reducing the initial probability from 10% or 32% to zero, they are still subject to a positive, but ambiguous risk of loss. Therefore, as compared with treatments without ambiguity, the amount spent on precaution is expected to be lower in the ambiguity treatments. The degree of this difference should be related to

12 The average probability of mistakes on the driving quiz was 25%, resulting in an expected probability of loss of 32.5% before risk mitigation.

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participant-specific estimates of the risk of mistakes. For example, the higher the estimated driving quiz score, the lower the expected probability of loss.

Hypothesis 5 The likelihood of insurance purchase will increase with the degree of ambiguity.

Discussion The unknown risk of mistakes in our experiment design results in an ambiguous increase in the probability of loss. Although this is true for all the mistakes treatments, participants generally will have more information about their own risk of errors on the driving test than they do about the risk of errors by others. Therefore, the treatments in which losses depend on the risk of mistakes by others introduce greater ambiguity than those in which losses depend on the participant’s own mistakes.

4 Results

We begin with an overview of the risk mitigation choices made by participants in our main treatments and report corresponding nonparametric tests of the hypotheses. Next, we address our baseline treatments and discuss risk attitudes suggested by the data. Finally, we present formal statistical tests of our main hypotheses.

4.1 Overview and nonparametric results

Figure 2 summarizes the experiment participants’ precaution and insurance choices in our main treatments (#1–12) and offers strong evidence in favor of Hypotheses 1 and 2, that participants will make efficient and consistent decisions, given the risk management techniques available to them. Panel A shows the proportion choosing various levels of precaution when insurance is not available. Panel B shows the proportions for treatments in which insurance was also an option. Comparison of the p=.10 and p=.32 categories across the two figures suggests that under an initial 10% probability of loss, all participants who choose full precaution switch to the more efficient alternative of insurance when it becomes available. 13 Those who choose full precaution to reduce an initial 32% probability of loss to zero continue to do so after insurance becomes available because precaution remains the more efficient means to reduce the probability of loss to zero, although 10 participants (17%) purchase insurance. Comparison of the same categories reveals almost no change in the portion of participants choosing zero or partial precaution when insurance becomes available under an initial loss probabil- ity of 10%, though under an initial probability of 32%, six participants change their level of precaution from partial to either full or insurance. On net, these results suggest evidence in favor of Hypothesis 1, which predicts that participants will choose the more efficient risk mitigation approach; and also in favor of Hypothesis 2, which predicts that participants’ level of risk mitigation will be consistent across treatments with and without insurance. McNemar tests confirm

13 In the other categories, full precaution is no longer a perfect substitute for insurance because of the risk of mistakes.

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that the difference in risk mitigation approach between the initial loss probabilities of 32% versus 10% (no mistakes) treatments is significant (p=.002), and that there is no significant difference in proportions choosing full risk mitigation when insurance is available versus when it is not (p=0.7789). 14

We can compare within the figures to examine the impact of mistakes on precaution and insurance. We see evidence in favor of Hypothesis 4, that partici- pants are less likely to take precaution in the ambiguous mistakes treatments; and

14 Analysis of subject-level data reveals only 12 reversals between the partial and full risk mitigation decisions across 120 decisions in the four no-mistakes treatments. There are four switches when the initial probability is 10% and eight under an initial probability of 32%. Separate McNemar tests by initial probability also show no significant difference in proportions.

0

10

20

30

40

50

60

p=.10 p=.32 p=.10+own mistakes

p=.32+own mistakes

p=.10+others' mistakes

p=.32+others' mistakes

None Some Full

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60

70

80

p=.10 p=.32 p=.10+own mistakes

p=.32+own mistakes

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p=.32+others' mistakes

None Some Full Insurance

a

b

Fig. 2 Precaution and insurance choices. Panel A Percentage choosing no, some, or full precaution, precaution-only treatments (#1, 2, 5, 6, 9, and 10). Panel B Percentage choosing insurance or no, some, or full precaution, precaution and insurance treatments (#3, 4, 7, 8, 11, and 12)

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also of Hypothesis 5, that insurance uptake is higher when ambiguity is higher, i.e., when the risk of loss depends on others’ mistakes as compared to one’s own mistakes. In Panel A, for each initial loss probability, the proportion of participants taking less than full precaution increases, and the proportion taking full precaution decreases, under both mistakes treatments. Chi-square tests for differences in pro- portions of precaution/insurance levels are significant at p=0.019 for precaution only treatments and p<0.001 for the precaution/insurance treatments. Panel B reveals that, given an initial probability of loss, insurance uptake is considerably higher under the mistakes treatments (compared to the no-mistakes treatments) and the increase is higher under the others’ mistakes treatment. When a loss depends on another participant’s quiz, 67% purchase insurance, compared to 56% when a loss depends on a participant’s own quiz results. These proportions are significantly different from each other under the McNemar test (p=0.0193).

Hypothesis 3 predicts that ambiguous increases in the probability of loss will have a larger positive effect on the likelihood of insurance purchase than objectively known increases. To consider this hypothesis, we compare an initial 10% objective probability of loss to three different increases in the loss probability: the increase in objective probability to 32%, the ambiguous increase due to own mistakes, and the ambiguous increase due to others’ mistakes. Under the objective probability of 32%, insurance and full precaution are perfect substitutes, and 65% of participants reduce the probability of loss to zero through one of these approaches. When the probability of loss increases above 10% due to the own mistakes treatment, 55% of participants fully risk mitigate, but when the chance of loss depends on others’ mistakes, 67% choose full insurance. On the surface, there does not appear to be much support for Hypothesis 3, but we discuss estimates of subjective probabilities of loss and their impact on insurance purchase in greater detail below.

4.2 Tests of baseline treatment predictions

As discussed above, a common explanation given for the underinsurance of low- probability high-severity losses is that individuals ignore or underweight extremely low probabilities. In contrast to previous studies, under a given expected loss and insurance load, Laury et al. (2009) find no support for this explanation. We use nearly identical design elements and parameters in our baseline treatments (13–20, described in Table 1 Panel B) as in their study to evaluate evidence of this type of probability weighting by participants in our experiment. In particular, we test the following two predictions.

Prediction 1: Participants are equally likely to purchase insurance for low proba- bility losses as they are for high probability losses, holding constant insurance load and expected loss. Prediction 2: For a given probability and size of loss, participants are less likely to purchase insurance under a higher load than a lower load. That is, participants respond to the price of insurance.

Table 3 presents the data from the baseline replication treatments (insurance only) and McNemar tests of Prediction 1 for differences in insurance purchases

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under high and low probabilities of loss. The results suggest participants do not appear to ignore the very low probability of 1% compared to the higher probability of 10% and furthermore, that they respond in the predictable direction of purchas- ing less insurance when the price (load) increases.15 When insurance is fairly priced, they do not appear to overweight the worse outcome (a loss of $45.00 or $60.00 compared to a loss of $4.50 or $6.00 respectively). Therefore, we fail to reject Prediction 1 under actuarially fair insurance. We do reject Prediction 1 when the insurance load increases to 3, but for the reason that insurance purchase is higher under the low probability of a loss. 16 The percentage of participants purchasing insurance declines under a high insurance load compared to the low load, all else equal, but the decrease is not statistically significant in all treatments. McNemar tests of the hypotheses that participants are equally likely to purchase insurance under a load of 3 as they are under a load of 1 are significant for a probability of loss of 10% and expected losses of $0.45 and $0.60 (p<0.0001). When the probability of loss is 1%, the difference in insurance purchase is weakly significant for an expected loss of $0.60 (p=.0625), but is not significant when the expected loss is $0.45 (p=.3750). On net, the baseline results suggest evidence against probability weighting behavior by participants in this treatment.

15 These results are comparable to those found in the Laury et al. 2009 study. 16 We interpret this simply as a substitution away from insurance—insuring a realized loss has a relatively higher price increase under the low-loss event compared to the high-loss event. Given the loss occurs, then the insurance costs an additional $0.22 per dollar covered under the low loss event, but only an additional $0.02 per dollar covered under the high loss event.

Table 3 Baseline replication treatments (insurance only)a and test statistics for differences based on probability of loss

Treatment Insurance load b

Loss probability

Loss amount

E(Loss) % Buying insurance

McNemar (p-value) c

13 3 1% $45.00 $0.45 78% 10.71*** (0.0015)14 3 10% $4.50 $0.45 53%

15 3 1% $60.00 $0.60 82% 12.25*** (0.0085)16 3 10% $6.00 $0.60 58%

17 1 1% $45.00 $0.45 83% 0.4 (0.7539)18 1 10% $4.50 $0.45 87%

19 1 1% $60.00 $0.60 90% 1.28 (0.4531)20 1 10% $6.00 $0.60 85%

a The baseline treatments (#13–20) replicate treatments used in Laury et al. (2009) in which the participants are given the probability of an orange ball being drawn (no ambiguity) and have the option to purchase insurance against the risk of loss. These treatments alternatively vary the loss probability, the insurance load, and the loss amount. (n=60 for each treatment.) b Insurance load is the insurance premium divided by the expected loss (1 = fairly priced) c The last column shows the McNemar test statistic and p-value for differences in the percent of participants purchasing insurance in the otherwise-equivalent low and high probability treatments. *** represents signif- icance at the .01 level

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4.3 Risk attitudes

We now turn our attention to what the results suggest about participants’ risk attitudes under the experiment parameters. In the precaution-only treatments without ambiguity (Treatments 1 and 2), participants have menus of choices for reducing the objective probability of loss from 10% in the low-probability treatment and from 32% in the high-probability treatment. To examine whether behavior is consistent with the Bajtelsmit and Thistle (2008) predictions for liability insurance, we compare decisions from the treatment in which the expected payoff is higher without precaution (low probability) with decisions from the treatment in which expected payoff is higher with precaution (high probability). Therefore, the low probability treatment can reveal risk averse behavior by participants who exercise precaution, while the high probability treatment can reveal risk seeking behavior by participants who exercise less than full precaution. Table 4 presents the percent of participants who make each risk mitigation decision in Treatments 1 and 2.

In the aggregate, the results appear to suggest that participants are risk averse for the decisions in the experiment. Over three-quarters of the participants make risk averse

Table 4 Participant risk management decisions for treatments with known probabilities of loss (No Mistakes)a

Low risk treatments (Initial probability of loss = 10%)

High risk treatments (Initial probability of loss = 32%)

Participants’ choices (%) Participants’ choices (%)

Decision Total up- front cost

p(loss) Treatment 1: Precaution only

Treatment 3: Precaution or insurance

p(loss) Treatment 2: Precaution only

Treatment 4: Precaution or insurance

A 0.00 10% 23.3% 18.3% 0.32 0.0% 0.0%

B 1.50 9% 0.0% 0.0% 0.28 1.7% 3.3%

C 3.00 8% 3.3% 1.7% 0.24 3.3% 0.0%

D 4.50 7% 1.7% 3.3% 0.2 8.3% 3.3%

E 6.00 6% 10.0% 6.7% 0.16 3.3% 6.7%

F 7.50 5% 8.3% 11.7% 0.12 3.3% 5.0%

G 9.00 4% 3.3% 11.7% 0.08 18.3% 10.0%

H 10.50 3% 3.3% 0.0% 0.04 6.7% 6.7%

I 12.00 2% 3.3% 1.7% 0 55.0% 48.3%

J 13.50 1% 0.0% 1.7% NA NA NA

K 15.00 0% 43.3% 1.7% NA NA NA

Insure 14.50 0% NA 43.3% NA NA 16.7%

a The table summarizes the experimental outcomes for Treatments 1, 2, 3, and 4 in which the 60 participants were exposed to a known probability of loss (no ambiguity) and were given a menu of risk mitigation alternatives. In Treatments 1 and 2, they could do nothing (Decision A) or they could reduce the probability of the bad outcome in Decisions B-K by paying $1.50 to replace orange balls with white balls ($1.50 for one ball in the low risk treatments and $1.50 for 4 balls in the high risk treatments). In Treatments 3 and 4, they also had the option of buying insurance for $14.50

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choices under the low probability treatment by paying for some risk mitigation. The modal response in the high and low probability treatments is to reduce the risk of loss to zero. However, within the high probability treatments, nearly half of the participants do appear to be risk-seeking in that they opt for less than full precaution which, in this treatment, provides the highest expected payoff.

Table 5 combines the decisions made by each participant across Treatments 1 and 2 (precaution only-known probability) and allows us to better identify consistency with different risk attitudes. We find that 48% made both risk mitigation decisions consistent with risk-averse behavior, 17% made both decisions consistent with risk-seeking behavior, and 7% appear risk neutral in these decisions. However, 28% make risk- seeking decisions when the probability of loss is high but risk-averse decisions under the lower probability of loss.

In Table 6, we take a closer look at the 17 participants who appear to change their risk attitudes over treatments (those who make a risk-averse choice in Treatment 1 and a risk-seeking choice in Treatment 2.) This table shows average expected payoffs, given the actual precaution expenditures, and the average cost of that risk management decision in terms of foregone expected payoff. On average, these participants reduce the probability of loss to a level of 5–9%, but not to zero, suggesting a preference for a lower than initial, but still positive, risk of loss.17 Risk mitigation is cheaper under the initial high probability and participants consume more of it, on average reducing the initial risk of loss by half in the initial p=10% treatment, and by about three-quarters in the initial p=32% treatment. As a result, the expected forgone earnings from their risk management choices (compared to the choice which maximizes the expected payoff) is much higher in the low probability case at $5.00 than it is in the high probability case at $0.69. These participants appear to behave consistently with the predictions of cumu- lative prospect theory for behavior under losses, i.e., they are risk averse over low- probability losses and risk-seeking over high-probability losses.18 However, it may also be the case that some other behavioral effect (such as regret or illusion of control) or framing effect is influencing their decisions, or that they may simply have found some

Table 5 Participant choices and risk attitudesa in precaution only/no ambiguity treatments

High initial probability (32%)

Risk-seeking Not risk-seeking

Low initial probability (10%) Risk averse 28% (n=17) 48% (n=29)

Not risk averse 17% (n=10) 7% (n=4)

a This table summarizes the combined decisions made by 60 participants in the two precaution-only / no ambiguity treatments. In the low probability treatment (#1), expected payoff is higher without precaution, so a participant is labeled as Brisk averse^ if they choose to pay for any precaution. In the high probability treatment (#2), the expected payoff is highest with full precaution, so a participant is labeled as Brisk-seeking^ if they choose to take less than full precaution

17 We note that the 17 participants are distributed across all six sessions, with 1–4 instances in each session. 18 See, for example, Tversky and Kahneman (1992), Camerer (1998), Starmer (2000), and Harbaugh et al. (2010).

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entertainment value in preserving a small chance of loss.19 The reduction of probability of loss to 8% was by far the modal choice in the high initial probability treatment and the frame may have somehow made 8% a focal point for these individuals. Furthermore the scale of expected losses, which varies substantially between the treatments (ranging from $4.50 to $15.00 under p=10% and $12.00 to $14.40 under p=32%), combined with a choice-task frame, has been shown to impact decisions. Beauchamp et al. (2012), Harrison et al. (2007a), (2007b), and Andersen et al. (2006), among others, all find that scaling manipulations affect estimates of risk aversion. In sum, before presenting our formal analysis of decisions involving insurance and ambiguity, we note that participants’ behavior under risk, with objectively known loss probabilities, ap- pears generally in line with the existing literature, and while the question of which expected or non-expected utility specification best represents preferences is an impor- tant one, it is beyond the scope of this paper.20

4.4 Precaution and insurance decisions

We next consider the consistency of risk management decisions and whether the ambiguity with respect to the probability of loss impacts precaution and insurance decisions as suggested by our theoretical model and formalized in Hypotheses 2, 3, and 5. In the design of our experiment, there are three levels of ambiguity. There is no ambiguity in the No Mistakes treatments because the probability of loss is explicitly stated. Ambiguity was greatest in the treatments where the loss probability depended on the risk of mistake by another participant. Thus, we consider the level of ambiguity to be increasing from No Mistakes to Own Mistakes to Others’ Mistakes treatments.

19 For example, participants may anticipate that choosing some risk-mitigation will lessen regret if a loss occurs, or may have an illusion of control resulting from taking a risk-mitigating action. See Jaspersen (2014) for discussion of entertainment value in hypothetical settings. 20 Because the ranking of outcomes remains constant across the design, we are unable to rule out rank- dependent expected utility (see Quiggin 1982), even if we find support for another representation.

Table 6 Expected payoffs and foregone earnings for participants who made risk averse decisions in the low probability treatment and risk-seeking decisions in the high probability treatmenta

Low risk treatment (Initial probability of loss = 10%)

High risk treatment (Initial probability of loss = 32%)

Average Median Average Median

Probability of loss after risk mitigation 5% 5% 9% 8%

Expected payoff ($) 50.50 50.25 47.31 47.40

Foregone earnings = expected payoff with no precaution - expected payoff with full precaution ($)

5.00 5.25 −0.69 −0.60

a This table summarizes results for the 17 participants who chose to pay for any precaution in Treatment #1 (risk averse) and chose less than full precaution in Treatment #2 (risk-seeking)

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We perform a logit regression in which the dependent variable is the decision to purchase insurance in the treatments where it is available. In the No Mistakes treatments, taking full precaution and insurance are perfect substitutes with respect to the impact on risk, so the dependent variable is equal to 1 if the participants buy insurance OR take full precaution in those treatments. We include an inde- pendent categorical variable for the level of precaution selected in the parallel precaution-only treatment (Full precaution; Part precaution; reference category = No precaution). Based on predictions in Bajtelsmit and Thistle (2008), we expect that those who prefer full precaution when insurance is not available will switch to insurance when it becomes available. We control for the initial probability of loss (High risk = 1 in treatments where the initial probability of loss p=0.32; reference category = Low risk p=0.1) and, where applicable, the level of ambiguity (No Mistakes; Others’ Mistakes; reference category = Own mistakes). Table 7 reports the results of these estimations. The coefficients are estimated log-odds ratios of the included category to the reference category.

In the first column of Table 7, we limit the analysis to the no-ambiguity treatments in which the participants know the probability of loss. Consistent with Hypothesis 2, we find that participants who choose to pay to reduce their probability of loss when insurance is not an option are significantly more likely to buy insurance when it is available, as compared to those who took no precaution. In this regression, the participants are significantly more likely to pay to reduce their risk to zero in the high risk treatments.

In the second and third columns of Table 7, we report regression results including all treatment types, controlling for the degree of ambiguity. As compared to the treatments with known probabilities of loss, the initial probability of loss is no longer a significant factor in the decision to buy insurance. Consistent with our theoretical predictions, and Hypotheses 3 and 5, insurance take-up is increasing in the degree of ambiguity. Participants in the No Mistakes treatments were significantly less likely to buy insur- ance and those in the Others’ Mistakes treatments were significantly more likely to buy insurance, as compared to decisions in the Own Mistakes treatments.

Because the insurance load is much higher in the low risk treatments, there could be an interaction between the effects of risk treatment and ambiguity treatment. To control for this, we include interaction terms for the initial probability loss at different ambiguity levels (High Risk X No Mistakes; High Risk X Others’ Mistakes; Reference Category: High Risk X Own Mistakes). The results of this regression are reported in the third column of Table 7. Although the signs and significance of the other control variables are unchanged, the interaction term for High Risk X No Mistakes is positive and significant and we see a larger negative coefficient on No Mistakes. This implies that the positive effect of high risk is primarily found in the treatments without ambiguity.

These empirical results support our theoretical predictions as formalized in Hypotheses 3, 4 and 5. First, participants who prefer full precaution when insur- ance is unavailable are more likely to buy insurance when it is an available option for them. Second, we find evidence consistent with Hypothesis 3: ambiguity increases the demand for insurance. Finally we find that higher ambiguity is associated with a larger likelihood of insurance purchase compared to lower ambiguity, as predicted by Hypothesis 5.

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4.5 Subjective probabilities

Another potentially confounding factor is that participants have different subjective probabilities of loss, given their own risk management choice and their estimate of the risk of mistakes by themselves or others. In the previous section, we used only a categorical measure of ambiguity based on treatment type. However, the effect of the categorical measure of ambiguity may differ by participant due to individual differ- ences in subjective estimates of the probability of mistakes. Because we asked the participants to estimate their own driving quiz score and the average for others in the

Table 7 Logit regression results: Determinants of the decision to buy insurancea

Coefficient Estimates (Robust standard errors clustered at the subject level)

Independent Variables Unambiguous precaution/insurance treatments

All precaution/insurance treatments

Constant −19.203*** (1.358)

−0.491 (0.347)

−0.435 (0.387)

Full precaution when insurance unavailable

21.609*** (1.493)

4.311*** (0.670)

4.592*** (0.669)

Part precaution when insurance unavailable

17.086*** (1.499)

0.088 (0.426)

0.181 (0.462)

Initial probability of loss (High Risk=1)

1.225* (0.725)

0.079 (0.331)

−0.232 (0.379)

No Mistakes treatment=1 −1.160*** (0.297)

−2.064*** (0.448)

Others’ Mistakes treatment=1 0.603** (0.250)

0.726*** (0.240)

High Risk X No Mistakes 1.674** (0.697)

High Risk X Others’ Mistakes

−0.315 (0.429)

Subject-treatments n=120 n=360 n=360

Mean dependent variable 0.55 0.617 0.617

Log-likelihood −34.468 −156.289 −152.59 Probability > Chi-square 0.000 0.000 0.000

Adjusted R-squared 0.534 0.323 0.329

a This table reports results of logit regressions in which the dependent variable is a dummy variable equal to 1 if the participant chose to buy insurance or the equivalent. The model in the first column includes the unambiguous No Mistakes treatments only. In those, the participants face a known initial probability of loss and can take precaution only (Treatments 1 and 2) or choose between precaution and insurance (Treatments 3 and 4). We test the prediction that participants who prefer full precaution in the treatments without insurance will switch to insurance in the treatments where that is an option. The results in the two right-hand columns pool the results for all the treatments, including No Mistakes, and the ambiguity treatments Own Mistakes and Others’ Mistakes (Treatments 1–12). The dependent variable in those models is the participant’s decision to buy insurance in Treatments 3, 4, 7, 8, 11, and 12

*** significant at the .01 level ** significant at the .05 level * significant at the .1 level

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experiment, we can estimate a unique subjective probability of loss for each participant by treatment type. We calculate subjective probability of loss (SPL) according to:

Subjective Probability of Loss ¼ p þ 1−pð Þm ð7Þ where

p ¼ initial probability of drawing an orange ball; given the precaution=insurance choice m ¼ probability of mistake ¼ 0 for No Mistakes treatments;

¼ 1 − Estimated Own Quiz Score 20

� � for Own Mistakes Treatments ;

¼ 1 − Estimated Average Quiz Score 20

� � for Others’ Mistakes Treatments

We expect that those who have a higher subjective probability of loss due to higher expected risk of mistake will be more likely to purchase insurance. The calculated subjective probabilities of loss for the main treatments (precaution/insurance), prior to any spending on risk mitigation, are summarized in Table 8 below. For each participant, the SPL is the same in treatments that differ only by the addition of an insurance option (e.g., #1 and #3), so there are 6 different SPLs for each participant. For the unambig- uous treatments, we assume that the SPL is equal to the actual probability of loss, as defined in the treatment presentation. This table shows that the risk of mistakes increases the participants’ SPL relative to the unambiguous probability of loss. We expect that individuals who have a higher subjective probability of loss due to higher expected risk of mistake will be more likely to purchase insurance.

To investigate this issue empirically, we next estimate logit regressions in which the dependent variable is a dummy variable equal to one if the participant purchased insurance or paid for full precaution in the treatments where insurance was available. The subjective probability of loss (SPL), prior to making the precaution or insurance decision, is included as a control variable. We include two alternative specifications here, one with a single categorical variable for ambiguity (a dummy variable equal to

Table 8 Subjective probability of loss (SPL)a, by treatment type (n=60 participants)

Treatment Type Mean Standard Deviation Minimum Maximum

Initial Probability Ambiguity Type

Low Risk p=0.1

No Mistakes 0.100 0.000 0.100 0.100

Own Mistakes 0.263 0.093 0.145 0.505

Others’ Mistakes 0.306 0.087 0.190 0.550

High Risk p=0.32

No Mistakes 0.320 0.000 0.320 0.320

Own Mistakes 0.443 0.070 0.354 0.626

Others’ Mistakes 0.476 0.066 0.388 0.660

a SPL is the subjective probability of loss prior to any spending on risk mitigation. For the No Mistakes treatments, it is a known probability and is therefore the same for each participant. For the Own Mistakes and Others’ Mistakes treatments, SPL for each participant is calculated as p + (1 − p)m, where p is the initial probability of loss and m is the participant’s subjective estimate of the probability of mistake. The probability of mistake is calculated as one minus the participant’s estimate of their own quiz score for the Own Mistakes treatments and one minus the participant’s estimate of others’ quiz scores for the Others’ Mistakes treatments

J Risk Uncertain (2015) 50:249–280 273

one if the treatment type is either of the mistakes treatment types; reference category: No Mistakes) and the other with separate dummy variables by degree of ambiguity (reference category: Own Mistakes). The first model may be preferable because it avoids the issue of the correlation between SPL and the degree of ambiguity. As reported in Table 9, the results of this analysis show that subjective probability is a significant factor influencing precaution and insurance decisions. However, even after controlling for this factor, we find that participants are more likely to insure in the ambiguous treatments. Comparing the controls for level of ambiguity in the last column, we find that Others’ Mistakes treatments significantly increase the likelihood of insuring or taking full care. In contrast, the likelihood is significantly lower in the unambiguous, No Mistakes treatments.

Table 9 Logit regression results: Determinants of the decision to buy insurance,a controlling for subjective probability of loss (SPL)b

Independent Variables Coefficient Estimate (Robust standard errors clustered at the subject level)

Constant −2.084*** (0.442)

−1.386** (0.538)

Participant paid for full precaution when insurance was unavailable

4.038*** (0.602)

4.093*** (0.607)

Participant paid for some precaution when insurance was unavailable

−0.249 (0.387)

−0.191 (0.391)

Any Mistakes Treatment=1 0.900** (0.387)

No Mistakes Treatment=1 −0.695* (0.374)

Others’ Mistakes Treatment=1 0.496** (0.243)

Subjective Probability of Loss (SPL) Before Precaution/Insurance b

3.543** (1.689)

3.296* (1.722)

Mean dependent variable 0.617 0.617

Log-likelihood −154.245 −153.073 Probability > Chi-square 0.000 0.000

Adjusted R-squared 0.335 0.336

a This table reports results of logit regressions in which the dependent variable is a dummy variable equal to 1 if the participant chose to buy insurance or the equivalent. Alternative specifications include a general control for ambiguity in the first model (Any Mistakes=1) versus separate dummy variables by degree of ambiguity in the second model (omitted category is Own Mistakes). Both models pool all 12 precaution/insurance treatments (n=720) b For the No Mistakes treatments, SPL is the initial probability of loss prior to spending money to reduce the risk or buy insurance. For the Own Mistakes and Others’ Mistakes treatments, SPL is calculated as p + (1 − p)m, where p is the initial probability of loss and m is the participant’s subjective estimate of the probability of mistake. The probability of mistake is calculated as one minus the participant’s estimate of their own quiz score for the Own Mistakes treatments and one minus the participant’s estimate of others’ quiz scores for the Others’ Mistakes treatments

*** significant at the .01 level ** significant at the .05 level * significant at the .1 level

274 J Risk Uncertain (2015) 50:249–280

4.6 The effect of ambiguity on level of precaution

The theoretical model suggests that ambiguity should decrease the incentive to take care because, to the extent that the precaution has no impact on the additional unknown risk, the marginal benefit of taking precaution is lower. In our model and experiment design, the risk of mistakes reduces the benefit of precaution because it only affects the initial probability of loss and has no impact on the additional risk from mistakes. We hypothesize that the risk of mistakes will reduce the incentive to spend on precaution (Hypothesis 4). To investigate this issue, we estimate a tobit regression in which the dependent variable is the amount spent on precaution in the treatments where insurance is not available. A tobit regression is selected for this estimation because the dependent variable is truncated. The minimum amount spent on precaution is 0 and the maximum amount of precaution is limited by the choices offered to the participants in the given treatment. Controls are included for subjective probability of loss and mistakes treat- ment type. The results are shown in Table 10. After controlling for SPL, we find that participants spent significantly less on care in the more ambiguous treatments. How- ever, the amount spent on care in the most ambiguous Others’ Mistakes treatments is not found to be significantly different from the amount spent in the Own Mistakes treatments. As in the previous section, SPL is a significant and positive factor.

Table 10 Tobit regression results: The effect of ambiguity on precaution (precaution-only treatments)

Independent Variables Coefficient Estimates (Robust Standard Errors Clustered at the Subject Level)

Constant 6.812*** (1.134)

3.554** (1.506)

Subjective Probability of Loss (SPL)b

11.011*** (3.095)

11.418*** (3.119)

Any Mistakes Treatment=1 −3.548*** (0.596)

No Mistakes Treatment=1 3.173*** (0.643)

Others’ Mistakes Treatment=1 −0.886 (0.687)

Probability > Chi Square 0.000 0.000

Log-likelihood −1012.88 −1012.19

a This table reports results of tobit regressions in which the dependent variable is the amount spent on precaution in the precaution-only treatments (Treatments 1, 2, 5, 6, 9, and 10.) Alternative specifications include a general control for ambiguity in the first model (Any Mistakes=1) versus separate dummy variables by degree of ambiguity in the second model (omitted category is Own Mistakes). Both models pool all the precaution-only treatments (n=360) b SPL is the subjective probability of loss prior to any spending on risk mitigation. For the No Mistakes treatments, it is a known probability and is therefore the same for each participant. For the Own Mistakes and Others’ Mistakes treatments, SPL for each participant is calculated as p+(1 − p)m, where p is the initial probability of loss and m is the participant’s subjective estimate of the probability of mistake. The probability of mistake is calculated as one minus the participant’s estimate of their own quiz score for the Own Mistakes treatments and one minus the participant’s estimate of others’ quiz scores for the Others’ Mistakes treatments

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5 Conclusions

We develop a theoretical model of the decision between precaution and insurance under an ambiguous probability of loss and we employ a novel experimental design to test its predictions. This is the first study which allows participants to choose between multiple levels of costly risk mitigation and insurance in a controlled environment. We find that ambiguous increases in loss probability increase insurance uptake by more than similar but known increases in loss probability, suggesting evidence in favor of ambiguity aversion.

We test whether experiment participants prefer insurance in cases when taking full precaution can also result in full risk mitigation. We also test whether participants are less responsive to lower probabilities of loss, holding constant the expected loss. Finally, we introduce ambiguity surrounding the probability of loss and examine the impact on insurance and precaution decisions. Therefore, participants make risk miti- gation decisions under conditions of both known and uncertain probabilities of loss.

Our results contribute to better understanding of risk management decision-making in the presence of ambiguity, and provide evidence that may inform two puzzling observations regarding insurance decisions: the purchase of liability insurance and underinsurance against catastrophic loss. Paying for risk management in our experi- ment is similar to investing in risk mitigation to meet the standard of care and thereby avoiding liability. We find that when the probability of loss is known, participants choose the more efficient way to achieve their desired level of risk mitigation. When the probability of loss is ambiguous, participants are more likely to buy insurance. These results suggest that the tendency to overinsure against liability rather than meet a standard of care through precaution may be partially explained, as suggested by our model, by sources of ambiguity surrounding liability losses.

Observed underinsurance against catastrophic losses has often been explained as resulting from the tendency to ignore very low probabilities. Controlling for expected loss under insurance-only treatments, we find that participants neither ignore nor underweight (known) low probability-high severity losses. Our results also reveal that participants do not overweight high-probability losses. The results lend further support to the Laury et al. (2009) findings that probability misperceptions are not an adequate explanation for observed underinsurance against catastrophe.

There are important policy implications for cases in which individuals and firms may substitute liability insurance in place of meeting a standard of care. High transparency and consistency regarding compliance with a standard of care, when possible, may increase precaution and decrease the risk of loss due to accidents, whereas unclear standards and relatively unpredictable enforcement may deter expenditure on loss prevention. This is important, especially under environmental loss liability, where investment in precaution may be more expensive and damages more extensive, and yet liability standards are relatively unclear. For example, in addition to the usual risks related to property, liability, life and health, individuals and firms facing liability from environmental risks are also exposed to ethical, cultural, business, reputational, and regulatory uncertainty. Castellano (2010) anticipates an increase in systematic risk of catastrophes, spreading through new networks between people, markets, and networks, which are particularly difficult to antic- ipate because they have never occurred in the past.

While we find evidence in favor of ambiguity aversion, our experiment is not designed to test for consistency with specific preference types. In treatments with

276 J Risk Uncertain (2015) 50:249–280

menus of risk mitigation alternatives, we find that nearly half of the participants make decisions that are consistent with risk-averse preferences, while almost a third appear risk averse under an initial lower probability of loss but slightly risk seeking under a higher initial probability of loss. Additional research is needed to carefully examine the effect of risk and ambiguity attitudes on the expenditure on care.

Acknowledgments The authors would like to thank the anonymous referee, the Editor Kip Viscusi, Glenn Harrison, James Sundali, Bill Rankin, and seminar participants at Colorado State University, Ludwig- Maximilian University, University of Münster, and at a Behavioral Insurance Workshop sponsored by the Georgia State University Center for the Economic Analysis of Risk for helpful comments on earlier drafts of this paper. They are grateful for financial support from the Colorado State University College of Business and the Nevada Insurance Education Foundation.

Appendices

Appendix 1: Ambiguity aversion increases willingness to pay

In this Appendix we show that ambiguity aversion increases the willingness to pay to avoid risk when individuals can exercise care.

The probability of a loss is π(c, ε) where ε is a random variable with distribution F. We do not restrict the dependence of π on ε, nor do we require that beliefs be unbiased. Let

Ui c; εð Þ ¼ 1−π c; εð Þð Þu w−cð Þ þ π c; εð Þu w−c−dð Þ; ðA:1Þ the argument is still valid if utility is separable in effort. The individual has the second order expected utility function

V cð Þ ¼ E F Φ U c; εð Þð Þf g ¼ E F Φ � 1−π c; εð Þ

� � u w−cð Þ þ π c; εð Þu w−c−dð ÞÞ

n o ðA:2Þ

Let c* denote the optimal value of care. The willingness to pay to avoid the risk, P, is

Φ u w−Pð Þð Þ ¼ E F Φ U c*; εð Þð Þf g ¼ Φ E F U c*; εð Þf g−Að Þ ðA:3Þ where A is an ambiguity premium. Then we have

u w−Pð Þ ¼ E F U c*; εð Þf g−A: ðA:4Þ For an ambiguity neutral individual, the ambiguity premium is zero and the optimal level of care, c0, maximizes EF{U(c, ε)}. Then

u w−P0 � �

¼ E F U c0; ε � ��

: ðA:5Þ

For an ambiguity averse individual the ambiguity premium is positive and the optimal level of care, c1, maximizes EF{Φ(U(c, ε))} Willingness to pay is given by

u w−P1 � �

¼ E F U c1; ε � ��

−A ðA:6Þ

Since EF{U(c 0, ε)}>EF{U(c

1, ε)} and A>0, we have P1>P0, ambiguity aversion increases the willingness to pay to avoid risk when an individual’s ability to take care affects the probability of a loss.

J Risk Uncertain (2015) 50:249–280 277

Now suppose that π is free of c, so that c0=c1=0. Then EF{U(c 0, ε)}=EF{U(c

1, ε)}. Then A>0 implies that P1>P0. The results in Alary et al. (2010) and Snow (2011) are special cases of the result here.

Appendix 2: Examples of precaution-only and precaution and insurance treatments

Menu of choices in a precaution-only treatment: Choose ONE of the following options below.

Decision Up-front Cost to Replace Orange Balls

New # of Orange Balls New # of White Balls Probability Orange Ball is Drawn

A $0.00 10 90 10%

B $1.50 9 91 9%

C $3.00 8 92 8%

D $4.50 7 93 7%

E $6.00 6 94 6%

F $7.50 5 95 5%

G $9.00 4 96 4%

H $10.50 3 97 3%

I $12.00 2 98 2%

J $13.50 1 99 1%

K $15.00 0 100 0

Your decision in Scenario 1 Menu of choices in a precaution/insurance treatment:

Choose ONE of the following options below.

Decision Up-front Cost to Replace Orange Balls

New # of Orange Balls New # of White Balls Probability orange ball is drawn

A $0.00 10 90 10%

B $1.50 9 91 9%

C $3.00 8 92 8%

D $4.50 7 93 7%

E $6.00 6 94 6%

F $7.50 5 95 5%

G $9.00 4 96 4%

H $10.50 3 97 3%

I $12.00 2 98 2%

J $13.50 1 99 1%

K $15.00 0 100 0

L (Insurance) $14.50 10 90 N/A

Your decision in Scenario 7

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