Observing different orders of risk aversion

A decision maker's attitude towards risk is said to be of orderi, i=1, 2, if for every given riskē with expected value zero, the risk premium the decision maker is willing to pay to avoid the risktē goes witht to zero at the same order ast ⁱ. This article presents an experiment testing the order of decision makers' attitudes toward risk. Its major result is that both attitudes exist, each in significant proportions. Moreover, two classes of first-order behavior are defined. The rank-dependent model (Quiggin, 1982) belongs to one, the disappointment aversion model (Gul, 1991) to the other. We show that only the first of these two classes appears among our subjects.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).     

Arrow-Pratt Risk Aversion, Risk Premium and Decision Weights

This paper analyzes two issues: (a) the effect of decision-weights on risk premium, and (b) whether risk-aversion characterizes most investors. We theoretically show that cumulative prospect theory decision-weights systematically increase Arrow's risk premium, and may induce a positive risk premium even in the absence of risk-aversion. However, decision-weights may either increase or reduce Pratt's risk premium. We present three experiments revealing a striking result: a large proportion of the subjects' choices contradicts risk-aversion. This may be due to non-concave preferences, or to decision-weights. This result may have a dramatic impact on equilibrium models in economics and finance.     

Prosocial risk taking and interpersonal regret in children: An individual differences study

The relations between prosocial risk taking (taking a risk to benefit another person; PSRT) and interpersonal regret (regret that one's choices have caused a poor outcome for another person) were examined in 192 children aged 7–9. PSRT was measured by children's choices within a gambling task in which one choice guaranteed participants a good prize whereas the other involved risking this prize to help a peer also win a good prize. Interpersonal regret was assessed within the same task by examining children's change in emotion when they learned they would have won a better prize for a peer had they chosen differently. Performance on this task was also examined in relation to sympathy and resource sharing. Findings indicated that the operationalizations of PSRT and interpersonal regret were meaningful. Children who took a prosocial risk were more generous in a resource sharing task. In some circumstances, children who took a prosocial risk were also more likely to experience interpersonal regret than those who did not take a prosocial risk, indicating that experiencing interpersonal regret may be related to individual differences in prosociality. However, experiencing interpersonal regret did not have a direct effect on subsequent prosocial behavior, when measured by resource sharing. We consider findings in relation to a possible distinction between outcome and process regret and the generalizability of the behavioral consequences of regret.     

On Bivariate Risk Premia

This note examines the conditions under which the bivariate risk premium for one risk may be negative even if both risks are positively correlated, using a mean variance setting. The link between the bivariate risk premium and the partial bivariate risk premia is also investigated.     

The valuation of contingent claims markets

This article studies an agent's valuation of the right to trade in a complete contingent claims market. The proposed measure generalizes the Pratt (1964) risk premium, which captures the willingness to pay to replace a given risky wealth prospect with an actuarially equivalent, nonrisky wealth. Specifically, we define ageneralized risk premium to be the willingness to pay to trade at going market prices. If state prices are actuarially fair, the Pratt premium is obtained as a special case. We derive several properties of this generalized premium and note its relationship to the option price of a public project under uncertainty.     

Regret theory with general choice sets

The regret theory of choice under uncertainty proposed by Loomes and Sugden has performed well in explaining and predicting violations of Expected Utility theory. The original version of the model was confined to pairwise choices, which limited its usefulness as an economic theory of choice. Axioms for a more general form of regret theory have been proposed by Loomes and Sugden. In this article, it is shown that a simple nonmanipulability requirement is sufficient to characterize the functional form for regret theory with general choice sets. The stochastic dominance and comparative static properties of the model are outlined. A number of special cases are derived in which regret theory is equivalent to other well-known theories of choice under uncertainty.I would like to thank Graham Loomes, Robert Sugden, Peter Wakker, and Nancy Wallace for helpful comments and criticism.     

A Parallel between Regret Theory and Outranking Methods for Multicriteria Decision Making Under Imprecise Information

Incorporation of the behavioral issues of the decision maker (DM) is among the aspects that each Multicriteria Decision Making (MCDM) method implicitly or explicitly takes into account. As postulated by regret theory, the feelings of regret and rejoice are among the behavioral issues associated with the entire decision making process. Within the context of MCDM, the DM may feel regret, when the chosen alternative is compared with another one having at least one better criterion value. PROMETHEE II is a widely known MCDM method that makes no explicit incorporation of regret attitude of the DM. In this paper, we elaborate on the applicability of regret theory to MCDM context. In particular, we investigate the findings of regret theory and explore the parallel between regret theory and PROMETHEE II method. Relying on the concepts of regret theory, we demonstrate how a decision that is made using a PROMETHEE II based outranking method conforms to the regret attitude of the DM.     

Non-expected utility risk premiums: The cases of probability ambiguity and outcome uncertainty

This paper discusses two problems. (a) What happens to the conditional risk premium that a decision maker is willing to pay out of the middle prize in a lottery to avoid uncertainty concerning the middle prize outcome, when the probabilities of other prizes change? (b) What happens to the increase that a decision maker is willing to accept in the probability of an unpleasant outcome in order to avoid ambiguity concerning this probability, when this probability increases? We discuss both problems by using anticipated utility theory, and show that the same conditions on this functional predict behavioral patterns that are consistent both with a natural extension of the concept of diminishing risk aversion and with some experimental findings.     

Recursive expected utility and the separation of attitudes towards risk and ambiguity: an experimental study

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent.      

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 over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

     

One-sided commitment in dynamic insurance contracts: Evidence from private health insurance in Germany

This paper studies long-term private health insurance (PHI) in Germany. It describes the main actuarial principles of premium calculation and relates these to existing theory. In the German PHI policyholders do not commit to renewing their insurance contracts, but insurers commit to offering renewal at a premium rate that does not reflect revealed future information about the insured risk. We show that empirical results are consistent with theoretical predictions from one-sided commitment models: front-loading in premiums generates a lock-in of consumers, and more front-loading is generally associated with lower rates of lapse. Due to a lack of consumer commitment, dynamic information revelation about risk type implies that high-risk policyholders are more likely to retain their PHI contracts than are low-risk types.     

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.     

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.     

Statistical decisions under ambiguity

This article provides unified axiomatic foundations for the most common optimality criteria in statistical decision theory. It considers a decision maker who faces a number of possible models of the world (possibly corresponding to true parameter values). Every model generates objective probabilities, and von Neumann–Morgenstern expected utility applies where these obtain, but no probabilities of models are given. This is the classic problem captured by Wald’s (Statistical decision functions, 1950) device of risk functions. In an Anscombe–Aumann environment, I characterize Bayesianism (as a backdrop), the statistical minimax principle, the Hurwicz criterion, minimax regret, and the “Pareto” preference ordering that rationalizes admissibility. Two interesting findings are that c-independence is not crucial in characterizing the minimax principle and that the axiom which picks minimax regret over maximin utility is von Neumann–Morgenstern independence.     

Endogenous risks and the risk premium

This note tries to correct a deficiency of the microeconomic literature on decision making under uncertainty. Indeed, when considering meaningful comparative statics results in situations where risks are at least partially controllable (endogenous), this literature has mostly relied upon the traditional Arrow-Pratt risk aversion functions and has paid very little attention to the definition of the risk premium. However when they defined the risk premium and the risk aversion functions, Arrow and Pratt considered only roulette gambles, i.e. risks totally exogenous to the individual. This note highlights the fact that several definitions of the risk premium may be proposed for endogenous risks. Two of them, already used in the literature, do not preserve the intuitively-appealing properties of the Arrow-Pratt risk premium. An alternative definition is then proposed. It is shown that this new definition of the risk premium applied to endogenous risks exhibits the properties generally admitted for roulette gambles.The three authors have benefitted from Ph. Caperaa's advice and from a referee's comments.     

Bystanders' Reactions to Bullying: A Cross‐cultural Analysis of Personal Correlates Among Italian and Singaporean Students

This study examined the role of attitudes against bullying and perceived peer pressure for intervention in explaining defending the victim and passive bystanding behavior in bullying. Participants were 1031 school‐age children from two culturally diverse settings, namely Italy and Singapore, which are similar on several dimensions (e.g., quality of life, child welfare) but dramatically differ on other aspects, such as individualism—collectivism orientation. Multilevel analyses showed that country and participants' gender moderated the relations between individual predictors and behavior during bullying episodes. In particular, although individual attitudes were a stronger predictor of Italian students'—especially girls'—behavior, perceived peer expectations were more strongly associated with behavior of Singaporean participants. This study contributes to the literature by being the first to provide data analyzing the association between defending and passive bystanding behavior and different correlates using a cross‐cultural approach.     

Regret aversion in reason-based choice

This research examines the moderating role of regret aversion in reason-based choice. Earlier research has shown that regret aversion and reason-based choice effects are linked through a common emphasis on decision justification, and that a simple manipulation of regret salience can eliminate the decoy effect, a well-known reason-based choice effect. We show here that the effect of regret salience varies in theory-relevant ways from one reason-based choice effect to another. For effects such as the select/reject and decoy effect, both of which were independently judged to be unreasonable bases for deciding, regret salience eliminated the effect. For the most-important attribute effect that is judged to be normatively acceptable, however, regret salience amplified the effect. Anticipated self-blame regret and perceived decision justifiability consistently predicted preferences and thus offer a parsimonious account of both attenuation and amplification of these reason-based choice effects.