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     

Comparing risk preferences over financial and environmental lotteries

This paper investigates whether preferences over environmental risks are best modeled using probability-weighted utility functions or can be reasonably approximated by expected utility (EU) or subjective EU models as is typically assumed. I elicit risk attitudes in the financial and environmental domains using multiple-price list experiment. I examine how subjects?? behavioral, attitudinal, and demographic characteristics affect their probability weighting functions first for financial risks, then for oil-spill risks. I find that most subjects tend to overweight extreme positive outcomes relative to expected utility in both the environmental and financial domains. Subjects are more likely to overemphasize low probability, extreme environmental outcomes than low probability, extreme financial outcomes, leading subjects to offer more support for mitigating environmental gambles than financial gambles with the same odds and equivalent outcomes. I conclude that EU models are likely to underestimate subjects?? willingness to pay for environmental cleanup programs or policies with uncertain outcomes.     

Probability weighting and the ‘level’ and ‘spacing’ of outcomes: An experimental study over losses

The main goal of the experimental study described in this paper is to investigate the sensitivity of probability weighting to the payoff structure of the gambling situation—namely the level of consequences at stake and the spacing between them—in the loss domain. For that purpose, three kinds of gambles are introduced: two kinds of homogeneous gambles (involving either small or large losses), and heterogeneous gambles involving both large and small losses. The findings suggest that at least for moderate/high probability of loss do both ‘level’ and ‘spacing’ effects reach significance, with the impact of ‘spacing’ being both opposite to and stronger than the impact of ‘level’. As compared to small-loss gambles, large-loss gambles appear to enhance probabilistic optimism, while heterogeneous gambles tend to increase pessimism.

Testing Prospect Theories Using Probability Tradeoff Consistency

The two versions of prospect theory, original prospect theory (OPT; Kahneman and Tversky, 1979) and cumulative prospect theory (CPT; Tversky and Kahneman, 1992), use different composition rules to combine the value function and the probability weighting function and hence value gambles with two or more non-zero outcomes differently. Previous tests of OPT and CPT have yielded mixed results, with some investigations supporting OPT and some supporting CPT. We extend the probability tradeoff consistency axiom used in Abdellaoui (2002) for CPT to OPT, and develop a critical test of the two prospect theories based on their respective probability tradeoff consistency conditions. An empirical investigation of the critical test shows that choices are consistent with OPT, but not CPT, for gambles that do not involve a certainty effect, and consistent with both CPT and OPT for gambles that do involve a certainty effect, provided that an editing operation is invoked for OPT.     

Are risk preferences stable? Comparing an experimental measure with a validated survey-based measure

We examine the stability of risk preference within subjects by comparing measures obtained from two elicitation methods, an economics experiment with real monetary rewards and a survey with questions on hypothetical gambles. The survey questions have been validated by numerous empirical studies of investment, insurance demand, smoking and alcohol use, and recent studies have shown the experimental measure is associated with several real-world risky behaviors. For the majority of subjects, we find that risk preferences are not stable across elicitation methods. In interval regression models subjects’ risk preference classifications from survey questions on job-based gambles are not associated with risk preference estimates from the experiment. However, we find that risk classifications from inheritance-based gambles are significantly associated with the experimental measure. We identify some subjects for whom risk preference estimates are more strongly correlated across elicitation methods, suggesting that unobserved subject traits like comprehension or effort influence risk preference stability.     

Assessing small non-zero perceptions of chance: The case of H1N1 (swine) flu risks

Feelings of invulnerability, seen in judgments of 0% risk, can reflect misunderstandings of risk and risk behaviors, suggesting increased need for risk communication. However, judgments of 0% risk may be given by individuals who feel invulnerable, and by individuals who are rounding from small non-zero probabilities. We examined the effect of allowing participants to give more precise responses in the 0–1% range on the validity of reported probability judgments. Participants assessed probabilities for getting H1N1 influenza and dying from it conditional on infection, using a 0–100% visual linear scale. Those responding in the 0–1% range received a follow-up question with more options in that range. This two-step procedure reduced the use of 0% and increased the resolution of responses in the 0–1% range. Moreover, revised probability responses improved predictions of attitudes and self-reported behaviors. Hence, our two-step procedure allows for more precise and more valid measurement of perceived invulnerability.     

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.     

Is Probability Weighting Sensitive to the Magnitude of Consequences? An Experimental Investigation on Losses

Experimental investigations of the probability weighting function over losses are scarce and all involve small payoffs. The paper aims to give new insight into the probability weighting function for losses, by eliciting it through a simple two-stage semi-parametric procedure over more realistic losses, and by investigating its sensitivity to the magnitude of the payoffs. Current data confirm previous evidence of convex utility functions and inverse-S-shaped weighting functions. Still, at least for small probabilities, probability weighting appears to be affected by the size of consequences: the larger the losses, the more aversive the gambles and the more pessimistic the subjects are.     

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.     

Estimating subjective probabilities

Subjective probabilities play a central role in many economic decisions and act as an immediate confound of inferences about behavior, unless controlled for. Several procedures to recover subjective probabilities have been proposed, but in order to recover the correct latent probability one must either construct elicitation mechanisms that control for risk aversion, or construct elicitation mechanisms which undertake “calibrating adjustments” to elicited reports. We illustrate how the joint estimation of risk attitudes and subjective probabilities can provide the calibration adjustments that theory calls for. We illustrate this approach using data from a controlled experiment with real monetary consequences to the subjects. This allows the observer to make inferences about the latent subjective probability, under virtually any well-specified model of choice under subjective risk, while still employing relatively simple elicitation mechanisms.     

Individual vs. couple behavior: an experimental investigation of risk preferences

In this article, we elicit both individuals’ and couples’ preferences assuming prospect theory (PT) as a general theoretical framework for decision under risk. Our experimental method, based on certainty equivalents, allows to infer measurements of utility and probability weighting at the individual level and at the couple level. Our main results are twofold. First, risk attitude for couples is compatible with PT and incorporates deviations from expected utility similar to those found in individual decision making. Second, couples’ attitudes towards risk are found to be consistent with a mix of individual attitudes, women being more influent on couples’ preferences at low probability levels.     

Risk aversion elicitation: reconciling tractability and bias minimization

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension.     

The ratio bias phenomenon: fact or artifact?

The ratio bias—according to which individuals prefer to bet on probabilities expressed as a ratio of large numbers to normatively equivalent or superior probabilities expressed as a ratio of small numbers—has recently gained momentum, with researchers especially in health economics emphasizing the policy importance of the phenomenon. Although the bias has been replicated several times, some doubts remain about its economic significance. Our two experiments show that the bias disappears once order effects are excluded, and once salient and dominant incentives are provided. This holds true for both choice and valuation tasks. Also, adding context to the decision problem does not alter this finding. No ratio bias could be found in between-subject tests either, which leads us to the conclusion that the policy relevance of the phenomenon is doubtful at best.     

Separating curvature and elevation: A parametric probability weighting function

This paper presents a preference foundation for a two-parameter family of probability weighting functions. We provide a theoretical link between the well-established notions of probabilistic risk attitudes (i.e., optimism and pessimism) used in economics and the important independent measures for individual behavior used in the psychology literature (i.e., curvature and elevation). One of the parameters in our model measures curvature and represents the diminishing effect of optimism and pessimism when moving away from extreme probabilities 0 and 1. The other parameter measures elevation and represents the relative strength of optimism vs. pessimism. Our empirical analysis indicates that the new weighting function fits elicited probability weights well, and that it can explain differences in the treatment of probabilities for gains compared to that for probabilities of losses.     

The effect of the background risk in a simple chance improving decision model

We experimentally investigate the effect of an independent and exogenous background risk to initial wealth on subjects’ risk attitudes and explore an appropriate incentive mechanism when identical or similar tasks are repeated in an experiment. Taking a simple chance improving decision model under risk where the winning probabilities are negatively related to the potential gain, we find that such a background risk tends to make risk-averse subjects behave more risk aversely. Furthermore, we find that risk-averse subjects tend to show decreasing absolute risk aversion (DARA), and that a random round payoff mechanism (RRPM) would control the possible wealth effect. This suggests that RRPM would be a better incentive mechanism for an experiment where repetition of a task is used.

An experimental test of several generalized utility theories

There is much evidence that people willingly violate expected utility theory when making choices. Several axiomatic theories have been proposed to explain some of this evidence, but there are few data that discriminate between the theories. To gather such data, an experiment was conducted using pairs of gambles with three levels of outcomes and many combinations of probabilities. Most typical findings were replicated, including the common consequence effect and different risk attitudes for gains and losses. There is evidence of both fanning out and fanning in of indifference curves, and both quasiconcavity and quasiconvexity of preferences. No theory can explain all the data, but prospect theory and the hypothesis that indifference curves fan out can explain most of them.The Wharton School,University of Pennsylvania     

Rethinking Risk Attitude: Aspiration as Pure Risk

There exists no completely satisfactory theory of risk attitude in current normative decision theories. Existing notions confound attitudes to pure risk with unrelated psychological factors such as strength of preference for certain outcomes, and probability weighting. In addition traditional measures of risk attitude frequently cannot be applied to non-numerical consequences, and are not psychologically intuitive. I develop Pure Risk theory which resolves these problems – it is consistent with existing normative theories, and both internalises and generalises the intuitive notion of risk being related to the probability of not achieving one’s aspirations. Existing models which ignore pure risk attitudes may be misspecified, and effects hitherto modelled as loss aversion or utility curvature may be due instead to Pure Risk attitudes.     

Noise and bias in eliciting preferences

In the context of eliciting preferences for decision making under risk, we analyse the features of four different elicitation methods—pairwise choice, willingness-to-pay, willingness-to-accept, and the Becker-DeGroot-Marschak mechanism—and estimate noise, bias and risk attitudes for two different preference functionals, Expected Utility and Rank-Dependent Expected Utility. It is well-known that methods differ in terms of the bias in the elicitation; it is rather less well-known that methods differ in terms of their noisiness. It has also been reported that risk attitudes are not stable across different elicitation methods. Our results suggest that elicited preferences should only be used in the context in which they were elicited, and the bias in the certainty-equivalent methods should be kept in mind when making predictions based on the elicited preferences. Moreover, conclusions should be moderated to take into account the various methods’ noise, which is generally lowest in the case of pairwise choice.     

The impact of risk sharing on efficient decision

A group of risk-averse members must choose among monetary risks and payoff-sharing rules. Departure from the status quo requires unanimous consent. Such groups drill for oil, bail out nations, and make hostile takeover bids. Assume agreement on probabilities. As is well known, if all members have identically shaped HARA utility functions, efficient group act-choices follow another such function independently of payoff sharing. We show that all other groups inevitably have complex efficient behavior, accepting gambles among individually unacceptable lotteries in almost every status quo position. We also develop proper risk aversion for groups, and treat disagreement on probabilities.Support of the Associates of the Harvard Business School (Pratt) and the Business and Government and Energy and Environmental Policy Centers (Zeckhauser) is gratefully acknowledged. We received helpful comments from Scott Johnson, Mark Machina, and a referee. Jay Patel provided assistance beyond the call of collegiality.     

Ambiguity aversion and familiarity bias: Evidence from behavioral and gene association studies

It is increasingly recognized that decision making under uncertainty depends not only on probabilities, but also on psychological factors such as ambiguity and familiarity. Using 325 Beijing subjects, we conduct a neurogenetic study of ambiguity aversion and familiarity bias in an incentivized laboratory setting. For ambiguity aversion, 49.4% of the subjects choose to bet on the 50–50 deck despite the unknown deck paying 20% more. For familiarity bias, 39.6% choose the bet on Beijing’s temperature rather than the corresponding bet with Tokyo even though the latter pays 20% more. We genotype subjects for anxiety-related candidate genes and find a serotonin transporter polymorphism being associated with familiarity bias, but not ambiguity aversion, while the dopamine D5 receptor gene and estrogen receptor beta gene are associated with ambiguity aversion only among female subjects. Our findings contribute to understanding of decision making under uncertainty beyond revealed preference.