Time and risk

Intertemporal choice has obvious similarities with choice under uncertainty. However, because of technical difficulties in mapping results between the two domains, theoretical analysis of these topics has proceeded independently. In this article, we show that, using Rank Dependent Expected Utility rather than Expected Utility as the basic uncertain choice model, numerous analogies between the two fields may be identified and exploited. The key result is the derivation of a natural analogy between risk-aversion and impatience. This permits the reinterpretation of well-known results on stochastic dominance and comparative risk-aversion in the context of intertemporal choice. It is also possible to reinterpret results on intertemporal optimization in order to derive new results for portfolio choice problems under uncertainty.     

A Theory of Rational Choice under Ignorance

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.     

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.     

The influence of probabilities on the response mode bias in utility elicitation

The response mode bias, in which subjects exhibit different risk attitudes when assessing certainty equivalents versus indifference probabilities, is a well-known phenomenon in the assessment of utility functions. In this empirical study, we develop and apply a cardinal measure of risk attitudes to analyze not only the existence, but also the strength of this phenomenon. Since probability levels involved in decision problems are already known to have a strong impact on behavior, we use this approach to study the impact of probabilities on the extent of the response mode bias. We find that the direction in which probabilities influence measured risk aversion is the opposite in the certainty equivalence (CE) method versus in the probability equivalence (PE) method. Utilizing the CE elicitation approach leads to an increase of risk seeking for gambles involving high probabilities. For the PE method, subjects tend to behave risk averse with gambles of high probabilities. This behavior is reversed in the gain domain. This “tailwhip” effect is consistently replicated in several experiments, involving both loss and gain domains of lotteries.     

Boundedly rational expected utility theory

We build a satisficing model of choice under risk which embeds Expected Utility Theory (EUT) into a boundedly rational deliberation process. The decision maker accumulates evidence for and against alternative options by repeatedly sampling from her underlying set of EU preferences until the evidence favouring one option satisfies her desired level of confidence. Despite its EUT core, the model produces patterns of behaviour that violate standard EUT axioms, while at the same time capturing systematic relationships between choice probabilities, response times and confidence judgments, which are beyond the scope of theories that do not take deliberation into account.

     

Conditional expected utility

Let \({\mathcal {E}}\) be a class of events. Conditionally Expected Utility decision makers are decision makers whose conditional preferences \(\succsim _{E}\), \(E\in {\mathcal {E}}\), satisfy the axioms of Subjective Expected Utility (SEU) theory. We extend the notion of unconditional preference that is conditionally EU to unconditional preferences that are not necessarily SEU. We study a subclass of these preferences, namely those that satisfy dynamic consistency. We give a representation theorem, and show that these preferences are Invariant Bi-separable in the sense of Ghirardato et al. (Journal of Economic Theory 118:133–173, 2004). We also show that these preferences have only a trivial overlap with the class of Choquet Expected Utility preferences, but there are plenty of preferences of the \(\alpha \)-Maxmin Expected Utility type that satisfy our assumptions. We identify several concrete settings where our results could be applied. Finally, we consider the special case where the unconditional preference is itself SEU, and compare our results with those of Fishburn (Econometrica 41:1–25, 1973).     

Beneficial safety decreases

We construct a model of rational choice under risk with biased risk judgement. On its basis, we argue that sometimes, a regulator aiming at maximising social welfare should affect the environment in such a way that it becomes ‘less safe’ in common perception. More specifically, we introduce a bias into each agent’s choice of optimal risk levels: consequently, in certain environments, agents choose a behaviour that realises higher risks than intended. Individuals incur a welfare loss through this bias. We show that by deteriorating the environment, the regulator can motivate individuals to choose behaviour that is less biased, and hence realises risk levels closer to what individuals intended. We formally investigate the conditions under which such a Beneficial Safety Decrease—i.e. a deteriorating intervention that has a positive welfare effect—exists. Finally, we discuss three applications of our model.     

Taste uncertainty and status quo effects in consumer choice

We use reference-dependent expected utility theory to develop a model of status quo effects in consumer choice. We hypothesise that, when making their decisions, individuals are uncertain about the utility that will be yielded by their consumption experiences in different ‘taste states’ of the world. If individuals have asymmetric attitudes to gains and losses of utility, the model entails acyclic reference-dependent preferences over consumption bundles. The model explains why status quo effects may vary substantially from one decision context to another and why some such effects may decay as individuals gain market experience.     

Eliciting risk preferences: When is simple better?

We study the estimation of risk preferences with experimental data and focus on the trade-offs when choosing between two different elicitation methods that have different degrees of difficulty for subjects. We analyze how and when a simpler, but coarser, elicitation method may be preferred to the more complex, but finer, one. Results indicate that the more complex measure has overall superior predictive accuracy, but its downside is that subjects exhibit noisier behavior. Our main result is that subjects’ numerical skills can help better assess this tradeoff: the simpler task may be preferred for subjects who exhibit low numeracy, as it generates less noisy behavior but similar predictive accuracy. For subjects with higher numerical skills, the greater predictive accuracy of the more complex task more than outweighs the larger noise. We also explore preference heterogeneity and provide methodological suggestions for future work.     

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.     

Emotional decision-makers and anomalous attitudes towards information

We use a simple version of the Psychological Expected Utility Model (Caplin and Leahy, Q J Econ 116:55–80, 2001) to analyze the optimal choice of information accuracy by an individual who is concerned with anticipatory feeling. The individual faces the following trade-off: on the one hand information may lead to emotional costs, on the other the higher the information accuracy, the higher the efficiency of decision-making. We completely and explicitly characterize how anticipatory utility depends on information accuracy, and study the optimal amount of information acquisition. We obtain simple and explicit conditions under which the individual prefers no-information or partial information gathering. We show that anomalous attitudes towards information can be more frequent and articulated than previously thought.     

Min- and Max-induced rankings: an experimental study

The current paper is the first to report an experimental study of “Min- and Max-induced rankings” (MMIR), i.e., a family of set rankings that require preferences over sets to be induced from comparison of the best and/or worst elements within those sets. These MMIR do not perform well in predicting preferences over simple sets of monetary outcomes. In this paper, we investigate the axiomatic underpinnings of these models by means of pairwise choice experiments. From this investigation, some important conclusions can be drawn: Axioms that prevent rankings to be based on total-goodness, as well as monotonicity conditions (ensuring that replacing a set element with a better one results in a better set) cannot be refuted. Axioms that rule out any utilization of the relative difference in the values of the outcomes and axioms that prevent rankings to be based on average-goodness are all systematically violated. The Uniform Expected Utility (UEU) criterion seems to meet the apparent shortcomings of the MMIR. Some frequently occurring preference patterns, however, suggest that a significant portion of the participants uses neither a Min- or Max-induced ranking, nor UEU, but some other unspecified decision rule, possibly characterized by the tendency to prefer a diversification of uncertainty.     

On the Conditional Value-at-Risk probability-dependent utility function

The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its utility representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox.     

How to reveal people’s preferences: Comparing time consistency and predictive power of multiple price list risk elicitation methods

The question of how to measure and classify people’s risk preferences is of substantial importance in the field of economics. Inspired by the multitude of ways used to elicit risk preferences, we conduct a holistic investigation of the most prevalent method, the multiple price list (MPL) and its derivations. In our experiment, we find that revealed preferences differ under various versions of MPLs as well as yield unstable results within a 30-minute time frame. We determine the most stable elicitation method with the highest forecast accuracy by using multiple measures of within-method consistency and by using behavior in two economically relevant games as benchmarks. A derivation of the well-known method by Holt and Laury (American Economic Review 92(5):1644–1655, 2002), where the highest payoff is varied instead of probabilities, emerges as the best MPL method in both dimensions. As we pinpoint each MPL characteristic’s effect on the revealed preference and its consistency, our results have implications for preference elicitation procedures in general.     

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.     

Error Propagation in the Elicitation of Utility and Probability Weighting Functions

Elicitation methods in decision-making under risk allow us to infer the utilities of outcomes as well as the probability weights from the observed preferences of an individual. An optimally efficient elicitation method is proposed, which takes the inevitable distortion of preferences by random errors into account and minimizes the effect of such errors on the inferred utility and probability weighting functions. Under mild assumptions, the optimally efficient method for eliciting utilities and probability weights is the following three-stage procedure. First, a probability is elicited whose subjective weight is one half. Second, the utility function is elicited through the midpoint chaining certainty equivalent method using the probability elicited at the first stage. Finally, the probability weighting function is elicited through the probability equivalent method.     

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.     

Risk and time preferences of entrepreneurs: evidence from a Danish field experiment

To understand how small business entrepreneurs respond to government policy one has to know their risk and time preferences. Are they risk averse, or have high discount rates, such that they are hard to motivate? We have conducted a set of field experiments in Denmark that will allow a direct characterization of small business entrepreneurs in terms of these traits. We build on experimental tasks that are well established in the literature. The results do not suggest that small business entrepreneurs are more or less risk averse than the general population under the assumption of Expected Utility Theory. However, we generally find an S-shaped probability weighting function for both small business entrepreneurs and non-entrepreneurs, with entrepreneurs being more optimistic about the chance of occurrence for the best outcome in lotteries with real monetary outcomes. The results also point to a significant difference in individual discount rates between entrepreneurs and non-entrepreneurs: entrepreneurs are willing to wait longer for certain rewards than the general population.     

Stability of risk preference measures: results from a field experiment on French farmers

We compare two different elicitation methods for measuring risk attitudes on a sample of French farmers. We consider the lottery tasks initially proposed by Holt and Laury (Econ Rev 92:1644?C1655, 2002) and by Eckel and Grossman (Evol Hum Behav 23:281?C295, 2002; J Econ Behav Org 68:1?C7, 2008). The main empirical result from this within-subject study is that risk preference measures are affected by the type of mechanism used. We first show that this risk preference instability can be related to non-expected utility preferences of farmers. Using a risk-taking psychometric questionnaire, we then demonstrate that risk preferences of farmers are context-dependent. This may be another explanation of the observed risk preference instability.