Accommodating stake effects under prospect theory

This paper investigates how letting people predict others’ choices under risk affects subsequent own choices. We find an improvement of strong rationality (risk neutrality) for losses in own choices, but no such improvement for gains. There is no improvement of weak rationality (avoiding preference reversals). Overall, risk aversion in own choices increases. Conversely, for the effects of own choices on predicting for others, the risk aversion predicted in others’ choices is reduced if preceded by own choices, for both gains and losses. Remarkably, we find a new probability matching paradox at the group level. Relative to preceding studies on the effects of predicting others’ choices, we added real incentives, pure framing effects, and simplicity of stimuli. Our stimuli were maximally targeted towards our research questions.     

Dynamics in risk taking with a low-probability hazard

The experiment reported in this paper identifies the effects of experience on revealed attitudes toward risk. Subjects in the experiment encountered an uncertain risk of experiencing a negative income shock over multiple periods and were able to purchase insurance at the start of each period. Subjects engaged in greater risk taking, insuring less frequently, when faced with the same risk over multiple periods. Subjects weighted experienced outcomes proportionately, in a manner consistent with rational Bayesian inference and contrary to the theory that individuals exhibit recency bias. On the other hand, subjects assigned a greater weight to outcomes that directly impacted their earnings compared to observed outcomes that had no effect on income. Unexplained autocorrelation across subjects’ choices suggests that inertia also plays an important role in repeated risk settings. I explore the relevance of these findings to public policy aimed at influencing market outcomes in the presence of infrequent environmental hazards.     

The intrinsic value of choice: The propensity to under-delegate in the face of potential gains and losses

Human beings are often faced with a pervasive problem: whether to make their own decision or to delegate the decision task to someone else. Here, we test whether people are inclined to forgo monetary rewards in order to retain agency when faced with choices that could lead to losses and gains. In a simple choice task, we show that participants choose to pay in order to control their own payoff more than they should if they were to maximize monetary rewards and minimize monetary losses. This tendency cannot be explained by participants’ overconfidence in their own ability, as their perceived ability was elicited and accounted for. Nor can the results be explained by lack of information. Rather, the results seem to reflect an intrinsic value for choice, which emerges in the domain of both gains and of losses. Moreover, our data indicate that participants are aware that they are making suboptimal choices in the normative sense, but do so anyway, presumably for psychological gains.     

Competence effects for choices involving gains and losses

We investigate how choices for uncertain gain and loss prospects are affected by the decision maker’s perceived level of knowledge about the underlying domain of uncertainty. Specifically, we test whether Heath and Tversky’s (J Risk Uncertain 4:5–28, 1991) competence hypothesis extends from gains to losses. We predict that the commonly-observed preference for high knowledge over low knowledge prospects for gains reverses for losses. We employ an empirical setup in which participants make hypothetical choices between gain or loss prospects in which the outcome depends on whether a high or low knowledge event occurs. We infer decision weighting functions for high and low knowledge events from choices using a representative agent preference model. For gains, we replicate the results of Kilka and Weber (Manage Sci 47:1712–1726, 2001), finding that decision makers are more attracted to choices that they feel more knowledgeable about. However, for losses, we find limited support for our extension of the competence effect.     

Effects of Outcome and Probabilistic Ambiguity on Managerial Choices

Information ambiguity is prevalent in organizations and likely influences management decisions. This study examines, given imprecise probabilities and outcomes, how managers make choices when they are provided with single-figure benchmarks. Seventy-nine MBA students completed two experiments. We found that, in a decision framed as a decision under certainty involving an ambiguous outcome, the majority of the subjects were ambiguity prone in the loss condition and switched to ambiguity aversion in the gain condition. However, in the presence of probabilistic ambiguity in a decision under risk, this expected switching pattern was shown only when the difference in riskiness between the two choice options (in the loss condition) was perceived to be relatively small. In a companion study, we used a written protocol approach to identify factors that affect decision makers' investment choices when faced with ambiguous outcomes. Protocols frequently mentioned that the ambiguous outcome option was risky, even in the case which was framed as a decision under certainty in the problem statement. In a decision under risk with ambiguous outcomes, the combination of probabilistic risk and outcome ambiguity was seen as even more risky.     

Inferring beliefs as subjectively imprecise probabilities

We propose a method for estimating subjective beliefs, viewed as a subjective probability distribution. The key insight is to characterize beliefs as a parameter to be estimated from observed choices in a well-defined experimental task and to estimate that parameter as a random coefficient. The experimental task consists of a series of standard lottery choices in which the subject is assumed to use conventional risk attitudes to select one lottery or the other and then a series of betting choices in which the subject is presented with a range of bookies offering odds on the outcome of some event that the subject has a belief over. Knowledge of the risk attitudes of subjects conditions the inferences about subjective beliefs. Maximum simulated likelihood methods are used to estimate a structural model in which subjects employ subjective beliefs to make bets. We present evidence that some subjective probabilities are indeed best characterized as probability distributions with non-zero variance.     

Experimental markets for insurance

This article extends the large amount of research on double-oral auction markets to hazards that produce only losses. We report results from a series of experiments in which subjects endowed with low-probability losses can pay a premium for insurance protection. Insurers specify the price at which they are willing to assume the risk of a loss. Insurance prices approach expected value for a large range of probabilities and loss amounts. Subjects seem to realize losses are statistically independent. Prices are not affected by ambiguity about the probability of loss.     

Bias and brains: Risk aversion and cognitive ability across real and hypothetical settings

I collect data on subjects’ risk attitudes using real and hypothetical risky choices. I also measure their cognitive ability using the cognitive reflective test (CRT). On average, measured risk preferences are not significantly different across real and hypothetical settings. However, cognitive ability is inversely related to risk aversion when choices are hypothetical, but it is unrelated when the choices are real. This interaction between cognitive ability and hypothetical setting is consistent with the notion that some individuals, specifically higher-ability individuals, may treat hypothetical choices as “puzzles,” and provides one potential explanation for why some studies find that subjects indicate that they are more tolerant of risk when they make hypothetical choices than when they make real choices.     

Probability and juxtaposition effects: An experimental investigation of the common ratio effect

Several theories explain the common ratio effect as a probability effect resulting from properties of individuals' preference orderings over probability distributions of consequences. In contrast, regret theory explains it as the result of changes in the juxtaposition of consequences in the action/state matrix. This article reports an experiment that allowed probability effects and juxtaposition effects to be separately identified for some common ratio problems, some of which involved real gains and others, real losses. The main finding is that changes in the juxtaposition of consequences have systematic effects on choices in the direction predicted by regret theory.University of East Anglia     

Buying Insurance for Disaster-Type Risks: Experimental Evidence

This paper presents a series of experiments that confront subjects with low probability, high loss situations. A rich parameter set is examined and we find subjects respond to low probability, high loss risks in predictable ways. As loss events become more likely, or loss amounts get larger, or the cost of insurance falls, subjects are more likely to buy indemnifying insurance, even for the class of low probability risks that usually presents problems for standard expected utility theory. A novel application of Cameron's method to estimate willingness to pay from dichotomous choice responses allows us to estimate willingness to pay for insurance. We do not observe the bimodal distribution of bids found in other studies of similar risk situations.     

It is Whether You Win or Lose: The Importance of the Overall Probabilities of Winning or Losing in Risky Choice

Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; $0, .20; –$25, .20; –$50, .20). What happens when the opportunity to improve such a gamble is provided by a manipulation that adds value to one outcome versus another outcome, particularly when the opportunity to add value to one outcome versus another outcome changes the overall probability of a gain or the overall probability of a loss? Such a choice provides a simple test of the expected utility model (EU), original prospect theory (OPT), and cumulative prospect theory (CPT). A study of risky choices involving 375 respondents indicates that respondents were most sensitive to changes in outcome values that either increased the overall probability of a strict gain or decreased the overall probability of a strict loss. These results indicate more support for OPT rather than CPT and EU under various assumptions about the shape of the utility and value and weighting functions. Most importantly, the main difference between the various expectation models of risky choice occurs for outcomes near the reference value. A second study of risky choice involving 151 respondents again demonstrated the sensitivity of subjects to reducing the probability of a strict loss even at the cost of reduced expected value. Consequently, we argue that theories of how people choose among gambles that involve three or more consequences with both gains and losses need to include measures of the overall probabilities of a gain and of a loss.JEL Classification  D81     

Advances in prospect theory: Cumulative representation of uncertainty

We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses. Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting functions. A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability.This article has benefited from discussions with Colin Camerer, Chew Soo-Hong, David Freedman, and David H. Krantz. We are especially grateful to Peter P. Wakker for his invaluable input and contribution to the axiomatic analysis. We are indebted to Richard Gonzalez and Amy Hayes for running the experiment and analyzing the data. This work was supported by Grants 89-0064 and 88-0206 from the Air Force Office of Scientific Research, by Grant SES-9109535 from the National Science Foundation, and by the Sloan Foundation.     

Time Insensitivity for Protective Investments

Investments in protective measures involve an initial immediate cost in exchange for a stream of potential benefits accruing over time in the form of reduced expected losses. This paper describes two studies in which individuals were asked both to make choices and indicate the maximum amount they were willing to pay (WTP) for such protective measures. By varying the number of years that the measures provided protection, we observed four different decision strategies that individuals use to determine their maximum WTP. The findings from these experiments strongly suggest that most individuals do not take into account the added benefits of having a protective measure in place over a longer period of time when determining the likelihood of purchasing protection or the maximum price they are willing to pay. The behavior of a relatively small proportion of the subjects was consistent with a discounted utility model. Most subjects were either myopic in their behavior or did not change their maximum WTP as the time horizon changed. Despite the fact that most subjects were insensitive to time changes, they were generally willing to purchase the protective measures. Implications of these results for investment in risk mitigation measures are briefly explored.     

Updating beliefs with imperfect signals: Experimental evidence

We conduct an experiment on individual choice under risk in which we study belief updating when an agent receives a signal that restricts the number of possible states of the world. Subjects observe a sample drawn from an urn and form initial beliefs about the urn??s composition. We then elicit how beliefs are modified after subjects receive a signal that restricts the set of the possible urns from which the observed sample could have been drawn. We find that this type of signal increases the frequency of correct assessments and that prediction accuracy is higher for lower levels of risk. We also show that prediction accuracy is higher after invalidating signals (i.e. signals that contradict the initial belief). This pattern is explained by the lower level of risk associated with invalidating signals. Finally, we find evidence for a lack of persistence of choices under high risk.     

Choosing a gambling partner: testing a model of mutual insurance in the lab

In this study, we investigate how economic agents choose gambling partners and how paired risky choices differ from individual ones. To this aim, we develop a simple model and design a laboratory experiment that allows us to compare individual versus paired decisions across two treatments, where pairs are, respectively, exogenously and endogenously formed. In both treatments, paired subjects decide individually and independently how to allocate their wealth over a portfolio of lotteries and fully commit to share any winnings. The main result from our experiment is that whenever agents are allowed to choose a gambling partner they decide to team up with other agents who display the same degree of risk aversion as themselves. Moreover, paired choices consistently involve higher risk taking than individual choices. This finding is more evident when information on subjects’ risk attitudes is made available and when subjects team up in homogeneous pairs, thereby confirming that subjects successfully exploit the benefits of mutual insurance.     

Failing to learn from experience about catastrophes: The case of hurricane preparedness

This paper explores the question of whether there are inherent limits to our ability to learn from experience about the value of protection against low-probability, high-consequence, events. Findings are reported from two controlled experiments in which participants have a monetary incentive to learn from experience making investments to protect against hurricane risks. A central finding is that investments display a short-term forgetting effect consistent with the use of reinforcement learning rules, where a significant driver of investments in a given period is whether storm losses were incurred in the precious period. Given the relative rarity of such losses, this reinforcement process produces a mean investment level below that which would be optimal for most storm threats. Investments are also found to be insensitive to the censoring effect of protection itself, implying that the size of experienced losses??rather than losses that are avoided??is the primary driver of investment decisions.     

Subjective Probability Weighting and the Discovered Preference Hypothesis

Numerous studies have convincingly shown that prospect theory can better describe risky choice behavior than the classical expected utility model because it makes the plausible assumption that risk aversion is driven not only by the degree of sensitivity toward outcomes, but also by the degree of sensitivity toward probabilities. This article presents the results of an experiment aimed at testing whether agents become more sensitive toward probabilities over time when they repeatedly face similar decisions, receive feedback on the consequences of their decisions, and are given ample incentives to reflect on their decisions, as predicted by Plott’s Discovered Preference Hypothesis (DPH). The results of a laboratory experiment with N = 62 participants support this hypothesis. The elicited subjective probability weighting function converges significantly toward linearity when respondents are asked to make repeated choices and are given direct feedback after each choice. Such convergence to linearity is absent in an experimental treatment where respondents are asked to make repeated choices but do not experience the resolution of risk directly after each choice, as predicted by the DPH. I thank Peter P. Wakker for useful comments and suggestions.     

Why the Young Do Not Buy Long-Term Care Insurance

This paper explains why those individuals who purchase long-term care insurance usually postpone their decision until they reach the age of retirement. This behavior is shown to be rational if there are fixed costs of loading or if there is uncertainty about the costs of disability. Individuals with a low risk of becoming disabled before retirement may prefer to buy insurance late in order to avoid losses in expected income. However, if the probability of becoming disabled after retirement is uncertain, it is generally preferable to buy long-term care insurance early.     

Loss Averse Behavior

A behavioral condition of loss aversion is proposed and tested. Forty-nine students participated in experiments on binary choices among lotteries involving small scale real gains and losses. At the aggregate level, a significant proportion of the choices are in the direction predicted by loss aversion. Individuals can be classified as loss averse (28 participants), gain seeking (12), and unclassified (9). A comparison with risk behavior for binary choices on lotteries involving only gains shows that risk attitudes vary across these domains of lotteries. A gender effect is also observed: proportionally more women are loss averse. In contrast to the predictions of comonotonic independence, the size of common outcomes has systematic influence on choice behavior. JEL Classification: D81, C91