Ross' measure of risk aversion and portfolio selection

This article shows that if Ross' definition of riskier is replaced by a more traditional definition, such as a mean-preserving spread or second-degree stochastic dominance, then the application of Ross's stronger measure of risk aversion to the portfolio problem may no longer produce the desired result. It is also shown that the stronger measure may not perform satisfactorily when applied to exponential utility functions.The authors are grateful to John Pratt for his helpful comments.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

On the intensity of downside risk aversion

The degree of downside risk aversion (or equivalently prudence) is so far usually measured by . We propose here another measure, , which has specific and interesting local and global properties. Some of these properties are to a wide extent similar to those of the classical measure of absolute risk aversion, which is not always the case for . It also appears that the two measures are not mutually exclusive. Instead, they seem to be rather complementary as shown through an economic application dealing with a simple general equilibrium model of savings.

Supermodularity and the Comparative Statics of Risk

In this article, it is shown that a wide range of comparative statics results from expected utility theory can be extended to generalized expected utility models using the tools of supermodularity theory. In particular, a range of concepts of decreasing absolute risk aversion may be formulated in terms of the supermodularity properties of certainty equivalent representations of preferences.     

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).     

Causes of ambiguity aversion: Known versus unknown preferences

Ambiguity aversion appears to have subtle psychological causes. Curley, Yates, and Abrams found that the fear of negative evaluation by others (FNE) increases ambiguity aversion. This paper introduces a design in which preferences can be private information of individuals, so that FNE can be avoided entirely. Thus, we can completely control for FNE and other social factors, and can determine exactly to what extent ambiguity aversion is driven by such social factors. In our experiment ambiguity aversion, while appearing as commonly found in the presence of FNE, disappears entirely if FNE is eliminated. Implications are discussed.      

Preferences Representable by a Lower Expectation: Some Characterizations

We propose two different characterizations for preference relations representable by lower (upper) expectations with the aim of removing either fair price or completeness requirements. Moreover, we give an explicit characterization for comparative degrees of belief on a finite algebra of events representable by lower probabilities.     

The Value of a Statistical Life and the Coefficient of Relative Risk Aversion

Individuals’ risk preferences are estimated and employed in a variety of settings, notably including choices in financial, labor, and product markets. Recent work, especially in financial economics, provides estimates of individuals’ coefficients of relative risk aversion (R’s) in excess of one, and often significantly higher. However, it can be shown that high R’s imply equally high values for the income elasticity of the value of a statistical life. Yet estimates of this elasticity, derived from labor and product markets, are in the range of 0.5 to 0.6. Furthermore, it turns out that even an R below one is difficult to reconcile with these elasticity estimates. Thus, there appears to be an important (additional) anomaly involving individuals’ risk-taking behavior in different market settings.JEL Classification: D80, G11, G12, I10, J17     

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     

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.     

The Utility of Gambling Reconsidered

The utility of gambling, which entails an intrinsic utility or disutility of risk, has been alluded to in the economics literature for over a century. This paper demonstrates that any utility of gambling almost unavoidably implies a violation of fundamental rationality properties, such as transitivity or stochastic dominance, for static choices between gambles. This result may explain why the utility of gambling, a phenomenon so widely discussed, has never been formalized in the economics literature. The model of this paper accommodates well-known deviations from expected utility, such as the Allais paradox and the coexistence of gambling and insurance, while minimally deviating from expected utility.     

The Construction of a Simple Book

A book is made for a horse race, and punters place their bets. The problem considered here is how the bookmaker should construct his book. Before this can be solved, it has to be determined how the punters will react to any proposed book. Much of the detailed discussion is confined to a race with two horses, though some results apply in the general case. The punters' problem is solved using a utility function, special attention being paid to the case of constant risk-aversion. Two solutions are provided for the bookmaker's problem, dependent on whether it is desired to maximize expected gain, or achieve the same gain whatever horse wins.     

The role of risk preferences in bargaining when acceptance of a proposal requires less than unanimous approval

The role of risk preferences in determining the outcome to bargaining is examined for the case in which acceptance of a proposal requires less than unanimous approval. Using an n-agent extension of the Ståhl-Rubinstein alternating offer model, we find that risk preferences play a fundamentally different role when bargaining is settled using a nonunanimity voting rule. Risk preferences determine not only an agent's reservation price but also the likelihood that he is made part of the winning coalition. An implication of this analysis is that when the preferences of the agents are not too diverse, it is advantageous for an agent to be relatively risk-averse.The helpful comments and suggestions of Rich McLean and an anonymous referee are gratefully acknowledged. This article was presented as a paper at the 1988 Winter Econometric Society Meetings.     

转型社会下的多重复合性风险——三城市公众风险感知状况的调查分析

在社会转型期,中国公众的主观风险感知呈现地区差异和多重复合等特点,三重复合-高度不确定性风险、两重复合-中度不确定性风险和单重-低度不确定性风险三种结构的社会风险共存。研究发现,公众普遍对低不确定性的单重风险感知最高,对高不确定性的三重风险感知最低;教育水平较高、社会地位较高的群体对高不确定性三重风险的感知程度较高,底层社会群体则对低不确定性单重风险的感知程度较高。从公众在三种感知结构的分布情况来看,中国公众的风险感知包含了对转型社会多重风险并存、交互影响和交互作用的认识,即成为中国社会转型过程的风险投射。     

The Effects of Beliefs Versus Risk Attitude on Bargaining Outcomes

In bargaining environments with uncertain disagreement or “impasse” outcomes (e.g., litigation or labor strike outcomes), there is an identification problem that confounds data interpretation. Specifically, the minimally acceptable settlement value from a risk-averse (risk-loving) but unbiased-belief bargainer is empirically indistinguishable from what one could get with risk-neutrality and pessimistically (optimistically) biased beliefs. This article reports results from a controlled bargaining experiment where data on both risk attitude and beliefs under uncertainty are generated in order to assess their relative importance in bargaining experiment outcomes. The average lab subject is risk-averse, yet optimistic with respect to uncertainty, which is consistent with existing studies that examine each in isolation. I also find that the effects of optimism dominate those of risk-aversion. Optimistic bargainers are significantly more likely to dispute and have aggressive final bargaining positions. Dispute rates are not statistically affected by risk attitude, but there is some evidence that risk aversion leads to a weakened bargaining position. Though additional research is needed to understand the limits of extending these results, a key implication follows. In uncertainty environments where optimism dominates, increased settlement rates are more likely achieved by minimizing impasse uncertainty (to limit the potential for optimism) rather than maximizing uncertainty (to weaken the reservation point of risk-averse bargainers), as has been argued in the dispute resolution literature.      

The expected likelihood of transitivity: A survey

May developed an algebraic choice model to describe pairwise comparisons from an empirical study. A probabilistic choice variation of May's model has also been developed. This study presents a survey of work that considers the expected likelihood that a subject using the probabilistic model will have transitive responses for pairwise choices on a set of three alternatives. Of particular interest is the impact that various factors that influence the probabilistic choice model have on the expected likelihood of transitivity. These factors include the degree of accuracy with which the subject perceives the attributes of the alternatives, the number of attributes of comparison, and the consistency with which alternatives are ranked across attributes.This research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

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.

Risk Assessment as a Paradox: When Actions of an IRB Become Incompatible with Ethical Principles

As anecdotal evidence mounts up of Institutional Review Boards becoming an impediment to research, I share a personal experience in this article to argue how the conduct of an Institutional Review Board was incompatible with a number of ethical principles such as Respect for Persons, Deception, Informed Consent, and Justice. I conclude by promoting Evidence-Based Ethical Problem Solving to facilitate and enhance risk assessments through empirical evidence.     

Aggregation of preferences: The fuzzy case

The basic purpose of this paper is to link both theorems of impossibility and existence by introducing fuzzy relations of preference and an exogeneous requirement, the planner's one, and then proving the fundamental part played by the extremist agents, leximin and leximax. In other words, to bring out the link between the planner's requirement and the difficulty of the transition from individual to collective, as well as the theoric relation between this requirement and the extremist agents, we define a fuzzy behavior of preference which allows us to build up two determinant fuzzy coalitions. These coalitions will be the base of the planner's requirement and the link between pessimistic results (Arrow's impossibility) and optimistic ones (May's theorem of majority choice).     

存在的悖论：摆脱束缚与逃避自由

自由意味着摆脱束缚,是具有较高价值的人学概念。随着现代社会的发展,自由的含义发生了一定的变迁,自由演变为孤独、恐惧,成为现代人不得不逃避的精神负担。弗罗姆在《逃避自由》中剖析了这种现象。逃避自由表面上是逃避孤独和无能为力,实则是逃避成为自己。此外,逃避自由也是一种由意识形态生产出来的社会现象。弗罗姆提出通过自发性的劳动和爱来实现积极自由,使个体成为自己。这一方案充满了乌托邦色彩。