Does Learning Diminish Violations of Independence, Coalescing and Monotonicity?

Violations of expected utility theory are sometimes attributed to imprecise preferences interacting with a lack of learning opportunity in the experimental laboratory. This paper reports an experimental test of whether a learning opportunity which engenders accurate probability assessments, by enhancing understanding of the meaning of stated probability information, causes anomalous behaviour to diminish. The data show that whilst in some cases expected utility maximising behaviour increases with the learning opportunity, so too do systematic violations. Therefore, there should be no presumption that anomalous behaviour under risk is transient and that discovered preferences will be appropriately described by expected utility theory.     

Revealed Preference and Expected Utility

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition necessary for the revelation of linear utility.     

Expected discounted utility

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann (Ann Math Stat 34(1):199–205, 1963, p. 201).

     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

A conditional expected utility model for myopic decision makers

An expected utility model is formulated and axiomatized which allows the decision maker to specify his actions in the form of partial rather than complete contingency plans, and to simultaneously choose goals and actions in end-mean pairs. Both utility and probability are conditioned on selected goals and actions, and both are defined over the same set of events. For adaptive sequential decision problems this symmetrical treatment of utility and probability permits the expected utility criterion function to be directly updated in each decision period via transitional utility assessments in a manner analogous to Bayes' rule for updating probability distributions.This study is a condensed revised version of discussion papers [14] and [15]. Research underlying this paper was supported by National Science Foundation Grants GS-31276X and GS-35682X. Valuable comments by Professors J. S. Chipman, C. Hildreth, L. Hurwicz, I. Richards, and M. K. Richter are gratefully acknowledged.     

Lottery-Dependent Utility via Stochastic Benchmarking

The possibility to interpret expected and nonexpected utility theories in purely probabilistic terms has been recently investigated. Such interpretation proposes as guideline for the Decision Maker the comparison of random variables through their probability to outperform a stochastic benchmark. We apply this type of analysis to the model of Becker and Sarin, showing that their utility functional may be seen as the probability that an opportune random variable, depending on the one to be evaluated, does not outperform a non-random benchmark. Further, the consequent choice criterion is equivalent to a sort of probability of ruin. Possible interpretations and financial examples are discussed.     

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.     

Optimal Insurance With Divergent Beliefs About Insurer Total Default Risk

This paper extends the classic expected utility theory analysis of optimal insurance contracting to the case where the insurer has a positive probability of total default and the buyer and insurer have divergent beliefs about this probability. The optimal marginal indemnity above the deductible is smaller (greater) than one if the buyer's assessment of default risk is more pessimistic (optimistic) than the insurer's. As an application of the model, we consider the market for reinsurance against catastrophic property loss and propose an expected utility theory explanation for the increasing and concave marginal indemnity schedule observed in this market.     

Common Consequence Conditions in Decision Making under Risk

We generalize the Allais common consequence effect by describing three common consequence effect conditions and characterizing their implications for the probability weighting function in rank-dependent expected utility. The three conditions—horizontal, vertical, and diagonal shifts within the probability triangle—are necessary and sufficient for different curvature properties of the probability weighting function. The first two conditions, shifts in probability mass from the lowest to middle outcomes and middle to highest outcomes respectively, are alternative conditions for concavity and convexity of the weighting function. The third condition, decreasing Pratt-Arrow absolute concavity, is consistent with recently proposed weighting functions. The three conditions collectively characterize where indifference curves fan out and where they fan in. The common consequence conditions indicate that for nonlinear weighting functions in the context of rank-dependent expected utility, there must exist a region where indifference curves fan out in one direction and fan in the other direction.     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

A parametric analysis of prospect theory’s functionals for the general population

This article presents the results of an experiment that completely measures the utility function and probability weighting function for different positive and negative monetary outcomes, using a representative sample of N = 1,935 from the general public. The results confirm earlier findings in the lab, suggesting that utility is less pronounced than what is found in classical measurements where expected utility is assumed. Utility for losses is found to be convex, consistent with diminishing sensitivity, and the obtained loss-aversion coefficient of 1.6 is moderate but in agreement with contemporary evidence. The estimated probability weighting functions have an inverse-S shape and they imply pessimism in both domains. These results show that probability weighting is also an important phenomenon in the general population. Women and lower educated individuals are found to be more risk averse, in agreement with common findings. In contrast to previous studies that ascribed gender differences in risk attitudes solely to differences in the degree utility curvature, however, our results show that this finding is primarily driven by loss aversion and, for women, also by a more pessimistic psychological response toward the probability of obtaining the best possible outcome.     

Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs.The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.     

The two faces of independence: betweenness and homotheticity

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk.

     

Viewing the future through a warped lens: Why uncertainty generates hyperbolic discounting

A large body of experimental research has demonstrated that, on average, people violate the axioms of expected utility theory as well as of discounted utility theory. In particular, aggregate behavior is best characterized by probability distortions and hyperbolic discounting. But is it the same people who are prone to these behaviors? Based on an experiment with salient monetary incentives we demonstrate that there is a strong and significant relationship between greater departures from linear probability weighting and the degree of decreasing discount rates at the level of individual behavior. We argue that this relationship can be rationalized by the uncertainty inherent in any future event, linking discounting behavior directly to risk preferences. Consequently, decreasing discount rates may be generated by people’s proneness to probability distortions.     

Perspectives on a Pair of Envelopes

The two envelopes problem has generated a significant number of publications (I have benefitted from reading many of them, only some of which I cite; see the epilogue for a historical note). Part of my purpose here is to provide a review of previous results (with somewhat simpler demonstrations). In addition, I hope to clear up what I see as some misconceptions concerning the problem. Within a countably additive probability framework, the problem illustrates a breakdown of dominance with respect to infinite partitions in circumstances of infinite expected utility. Within a probability framework that is only finitely additive, there are failures of dominance with respect to infinite partitions in circumstances of bounded utility with finitely many consequences (see the epilogue).     

Two-parameter decision models and rank-dependent expected utility

This paper extends the existing literature concerning the relationship between two parameter decision models and those based on expected utility in two main directions. The first relaxes Meyer's location and scale (or Sinn's linear class) condition and shows that a two-parameter representation of preferences over uncertain prospects and the expected utility representation yield consistent rankings of random variables when the decision maker's choice set is restricted to random variables differing by mean shifts and monotone meanpreserving spreads. The second shows that the rank-dependent expected utility model is also consistent with two-parameter ranking methods if the probability transform satisfies certain dominance conditions. The main implication of these results is that the simple two-parameter model can be used to analyze the comparative statics properties of a wide variety of economic models, including those with multiple sources of uncertainty when the random variables are comonotonic. To illustrate this point, we apply our results to the problem of optimal portfolio investment with random initial wealth. We find that it is relatively easy to obtain strong global comparative statics results even if preferences do not satisfy the independence axiom.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Emotional balance and probability weighting

We find suggestive evidence that emotional balance has an impact on probability weighting incremental to demographic controls. Specifically, low negative affectivity (implying high emotional balance) tends to be a characteristic of those whose probability weighting functions exhibit lower curvature and more neutral elevation. In other words, emotional balance seems to push people in the direction of normative expected utility theory.