Choices at various levels of uncertainty: An experimental test of the restated diversification theorem

Our “Restated diversification theorem” (Skogh and Wu, 2005) says that risk-averse agents may pool risks efficiently without assignment of subjective probabilities to outcomes, also at genuine uncertainty. It suffices that the agents presume that they face equal risks. Here, the theorem is tested in an experiment where the probability of loss, and the information about this probability, varies. The result supports our theorem. Moreover, it tentatively supports an evolutionary theory of the insurance industry—starting with mutual pooling at uncertainty, turning into insurance priced ex ante when actuarial information is available.     

Individual behavior under risk and under uncertainty: An experimental study

These experiments are concerned with individual behavior under risk and under uncertainty. By exploiting the isolation effect the experiments were able to offer to 134 subjects the possibility of actually gaining or losing an important sum of money.The experimental data show that under risk as well as under complete ignorance the subjects' attitudes towards prospects of gains and towards prospects of losses are totally unrelated.The data also show that when facing prospects of gains, the subjects generally take the exact probabilities of the events into account, whereas, when facing prospects of losses many of then have only recourse to coarser categories of plausibility, or even no longer use their information at all.     

Catastrophic responses to catastrophic risks

Socializing risks from catastrophic losses is difficult even in an ideal political environment, owing to different estimates of low probability risks, solvency constraints, dangers of moral hazard, and high loss correlation. However, these intrinsic contracting problems do not justify invalidating ordinary insurance contracts or forcing insurers to cover catastrophic losses. Yet, political pressures forcing insurance subsidies now induce inefficient decisions in siting and construction, with high expected social losses. Ordinary contract solutions are always imperfect, but superior to the regulatory maze. Unfortunately, patterns of legislation and court decisions are running in the wrong direction.     

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     

Arbitrage,rationality, and equilibrium

No-arbitrage is the fundamental principle of economic rationality which unifies normative decision theory, game theory, and market theory. In economic environments where money is available as a medium of measurement and exchange, no-arbitrage is synonymous with subjective expected utility maximization in personal decisions, competitive equilibria in capital markets and exchange economies, and correlated equilibria in noncooperative games. The arbitrage principle directly characterizes rationality at the market level; the appearance of deliberate optimization by individual agents is a consequence of their adaptation to the market. Concepts of equilibrium behavior in games and markets can thus be reconciled with the ideas that individual rationality is bounded, that agents use evolutionarily-shaped decision rules rather than numerical optimization algorithms, and that personal probabilities and utilities are inseparable and to some extent indeterminate. Risk-neutral probability distributions, interpretable as products of probabilities and marginal utilities, play a central role as observable quantities in economic systems.     

Rationality on the rise: Why relative risk aversion increases with stake size

How does risk tolerance vary with stake size? This important question cannot be adequately answered if framing effects, nonlinear probability weighting, and heterogeneity of preference types are neglected. We show that the observed increase in relative risk aversion over gains cannot be captured by the curvature of the value function. Rather, it is predominantly driven by a change in probability weighting of a majority group of individuals who weight probabilities of high gains more conservatively. Contrary to gains, no coherent change in relative risk aversion is observed for losses. These results not only challenge expected utility theory, but also prospect theory.     

Prospect theory and the “forgotten” fourfold pattern of risk preferences

Markowitz (Journal of Political Economy 60:151–158, 1952) identified a fourfold pattern of risk preferences in outcome magnitude: When outcomes are large, people are risk averse in gains and risk seeking in losses, but risk preferences reverse when the outcomes are small, with people exhibiting risk seeking in gains and risk aversion in losses. This fourfold pattern was not addressed by either version of prospect theory (Kahneman and Tversky Econometrica 47:363–391, 1979; Tversky and Kahneman Journal of Risk and Uncertainty 5:297–323, 1992). We show how prospect theory can accommodate the pattern by combining an overweighting of low probabilities with a decreasingly elastic value function. We then examine the performance of prospect theory with two decreasingly elastic value functions: Prospect theory performs better, both quantitatively and qualitatively, with a normalized logarithmic value function than with a normalized exponential value function. We discuss several issues, and speculate about why Tversky and Kahneman did not address Markowitz’s fourfold pattern.     

No-betting Pareto under ambiguity

It has been argued that Pareto-improving trade is not as compelling under uncertainty as it is under certainty. The former may involve agents with different beliefs, who might wish to execute trades that are no more than betting. In response, the concept of no-betting Pareto dominance was introduced, requiring that putative Pareto improvements must be rationalizable by some common probabilities, even though the participants’ beliefs may differ. In this paper, we argue that this definition might be too narrow for use when agents are not Bayesian. Agents who face ambiguity might wish to trade in ways that can be justified by common ambiguity, though not necessarily by common probabilities. We accordingly extend the notion of no-betting Pareto dominance to characterize trades than are “no-betting Pareto” ranked according to the maxmin expected utility model.

     

Risk attitudes in a social context

Many experiments have demonstrated that when evaluating payoffs, people take not only their own payoffs into account, but also the payoffs of others in their social environment. Most of this evidence is found in settings where payoffs are riskless. It is plausible that if people care about the payoffs of others, they do so not only in a riskless context, but also in a risky one. This suggests that an individual’s decision making under risk depends on the risks others in his or her environment face. This paper is the first to test whether individuals’ risk attitudes are affected by the risks others face. The results show that risk attitudes appear to be less affected by others’ risks than expected, even though the same subjects do show concerns for inequality in a riskless setting. Interestingly, we find that people prefer risks to be independent across individuals in society rather than correlated.     

When corporate social responsibility meets the challenges of rising globalisation risks

Global firms are facing great difficulties in containing new risks along their supply chains: reputation risks, contestation risks, safety risks and markets risks, as well as the risk of having to deal with new laws and more binding norms. The lack of internal control on these Global Value Chains is revealed through different social or environmental crises. Mass media are mobilised by NGOs and citizens as a means of pressure, which undermines the reputation of firms and therefore their immaterial assets. These crises show that the codes of conduct and the so-called soft law instruments fail to control the supply chain, but at the same time, these instruments begin to set new standards to face these new risks. This paper discusses this evolutionary institutional process, and stresses how CSR is not only a fiction but starts having real effects by creating new institutions to respond to risks. We will focus on reputation risks and markets risks to show that, even though CSR might not have led to a completely new system of rules, the institutional process under study has a meaningful impact on the regulatory framework.     

Information,risk perceptions,and smoking choices of youth

Conventional wisdom maintains that youths take risks because they underestimate probabilities of harm. Presumably if they knew the true probabilities, they would behave differently. We used the National Longitudinal Survey of Youth 1997 to assess whether differences between subjective and objective probabilities that an adverse outcome to self will occur are systematically related to a harmful behavior, initiating smoking. We find that youths are generally pessimistic about probabilities of their own deaths and being violent crime victims. After smoking initiation, youths increase subjective probabilities of death by more than the objective increase in mortality risk, implying recognition of potential harms. Virtually all 12–14 year-olds know that smoking causes heart disease. The minority who believe that smoking causes AIDS are less likely to become smokers; i.e., risk misperceptions deter rather than cause smoking initiation. Messages designed to deter smoking initiation should stress other disadvantages of smoking than just probabilities of harm.     

Stake effects on ambiguity attitudes for gains and losses

We test the effect of stake size on ambiguity attitudes. Compared to a baseline condition, we find subjects to be more ambiguity seeking for small-probability gains and large-probability losses under high stakes. They are also more ambiguity averse for large-probability gains and small-probability losses. We trace these effects back to stake effects on decisions under risk (known probabilities) and uncertainty (unknown probabilities). For risk, we replicate previous findings. For uncertainty, we find an increase in probabilistic insensitivity under high stakes that is driven by increased uncertainty aversion for large-probability gains and for small-probability losses.     

Estimates of Own Lethal Risks and Anchoring Effects

The paper reports on an experiment testing whether agents perceive correctly the lethal risks they face personally. The results suggest that subjects exhibit comparable biases when making predictions for their own-age-cohort, or for the entire population (i.e. agents overestimate rare risks, and underestimate common risks). The hypothesis that agents have better knowledge of their own risks, however, cannot be dismissed entirely, as responses in the own-age-cohort survey are more homogenous and better ordered. Finally, it is shown that administering surveys in succession can generate anchoring effects, which may explain why our conclusions differ markedly from a previous study. JEL Classification D8 ⋅ C9     

Framing effects in the evaluation of multiple risk reduction

We studied how evaluation of changes in low-probability risks are affected by reference points and framing effects. Subjects considered hypothetical situations with one or two low-probability risks. Different frames were used to describe changes in risk levels. In the first experiment, subjects chose between risk-reduction options that achieved the same overall risk reduction: large reduction of one risk vs. equal (smaller) reduction of two risks. When the risks were described as losses relative to the no-risk ideal, more subjects were indifferent between the options than when the same options were described as gains relative to the status quo. In the latter case subjects preferred equal reduction of both risks, unless one risk could be reduced to zero. In a related experiment, subjects were less willing to pay any price for a commodity that carried small increases in two risks than for a commodity carrying a comparable large increase in one risk. In other experiments, subjects evaluated single changes in risks rather than comparing or evaluating pairs of changes. Subjects again placed particularly high value on reducing any risks to zero, and they were even more inclined to do so when some other risk would also be reduced to zero. In a final experiment, elimination of risk was found to be less highly valued if its source was not fully eliminated, and a status-quo effect was found. The findings are interpreted in terms of reference theories of choice.     

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     

Choices Under Ambiguity With Familiar And Unfamiliar Outcomes

This paper considers a decision-making process under ambiguity in which the decision-maker is supposed to split outcomes between familiar and unfamiliar ones. She is assumed to behave differently with respect to unfamiliar gains, unfamiliar losses and customary (familiar) outcomes. In particular, she is supposed to be pessimistic on gains, optimistic on losses and ambiguity neutral on the familiar outcomes. A generalization of the usual Choquet Integral is formalized when the decision maker holds capacities and probabilities. A characterization of the decision-maker’s behavior is provided for a specific subset of capacities, in which it is shown that the decision-maker underestimates the unfamiliar outcomes while is linear in probabilities on customary ones.     

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.     

Markets with endogenous uncertainty theory and policy

Classic formulations of markets regard uncertainty as originating from acts of nature. I extend this to a formulation of markets which face risks induced by the economy itself, such as the environmental risks of atmospheric and climate change induced by CFC and CO₂ emissions.I formulate and prove the existence of a general competitive equilibrium where the state space and the probabilities of events are endogenously determined as part of the equilibrium. Traders take optimal positions with respect to the uncertainty which their own actions induce. The equilibrium allocations are efficient in a restricted sense. I show that scientific uncertainty can be fully hedged. However uncertainty induced by the unknown level of output at an equilibrium cannot be hedged fully. I discuss applications for CAT Futures, recently introduced on the Chicago Board of Trade, and to international environmental strategies.Dedicated to Tjalling C. Koopmans     

Terrorism and Probability Neglect

When strong emotions are involved, people tend to focus on the badness of the outcome, rather than on the probability that the outcome will occur. The resulting “probability neglect” helps to explain excessive reactions to low-probability risks of catastrophe. Terrorists show a working knowledge of probability neglect, producing public fear that might greatly exceed the discounted harm. As a result of probability neglect, people often are far more concerned about the risks of terrorism than about statistically larger risks that they confront in ordinary life. In the context of terrorism and analogous risks, the legal system frequently responds to probability neglect, resulting in regulation that might be unjustified or even counterproductive. But public fear is itself a cost, and it is associated with many other costs, in the form of “ripple effects” produced by fear. As a normative matter, government should reduce even unjustified fear, if the benefits of the response can be shown to outweigh the costs.     

Do financial professionals behave according to prospect theory? An experimental study

Prospect theory is increasingly used to explain deviations from the traditional paradigm of rational agents. Empirical support for prospect theory comes mainly from laboratory experiments using student samples. It is obviously important to know whether and to what extent this support generalizes to more naturally occurring circumstances. This article explores this question and measures prospect theory for a sample of private bankers and fund managers. We obtained clear support for prospect theory. Our financial professionals behaved according to prospect theory and violated expected utility maximization. They were risk averse for gains and risk seeking for losses and their utility was concave for gains and (slightly) convex for losses. They were also averse to losses, but less so than commonly observed in laboratory studies and assumed in behavioral finance. A substantial minority focused on gains and largely ignored losses, behavior reminiscent of what caused the current financial crisis.