http://repub.eur.nl/res/aut/1140/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/37908/ 2012-10-24T00:00:00Z This paper presents preference axiomatizations of expected utility for nonsimple lotteries while avoiding continuity constraints. We use results by Fishburn (1975), Wakker (1993), and Kopylov (2010) to generalize results by Delbaen et al. (2011). We explain the logical relations between these contributions for risk versus uncertainty, and for finite versus countable additivity, indicating what are the most general axiomatizations of expected utility existing today. http://repub.eur.nl/res/pub/34914/ 2012-09-01T00:00:00Z Experiments frequently use a random incentive system (RIS), where only tasks that are randomly selected at the end of the experiment are for real. The most common type pays every subject one out of her multiple tasks (within-subjects randomization). Recently, another type has become popular, where a subset of subjects is randomly selected, and only these subjects receive one real payment (between-subjects randomization). In earlier tests with simple, static tasks, RISs performed well. The present study investigates RISs in a more complex, dynamic choice experiment. We find that between-subjects randomization reduces risk aversion. While within-subjects randomization delivers unbiased measurements of risk aversion, it does not eliminate carry-over effects from previous tasks. Both types generate an increase in subjects' error rates. These results suggest that caution is warranted when applying RISs to more complex and dynamic tasks. http://repub.eur.nl/res/pub/37781/ 2012-07-01T00:00:00Z This paper presents a general technique for comparing the concavity of different utility functions when probabilities need not be known. It generalizes: (a) Yaari's comparisons of risk aversion by not requiring identical beliefs; (b) Kreps and Porteus' information-timing preference by not requiring known probabilities; (c) Klibanoff, Marinacci, and Mukerji's smooth ambiguity aversion by not using subjective probabilities (which are not directly observable) and by not committing to (violations of) dynamic decision principles; (d) comparative smooth ambiguity aversion by not requiring identical second-order subjective probabilities. Our technique completely isolates the empirical meaning of utility. It thus sheds new light on the descriptive appropriateness of utility to model risk and ambiguity attitudes. http://repub.eur.nl/res/pub/34700/ 2012-06-15T00:00:00Z Time discounting and quality of life are two important factors in evaluations of medical interventions. The measurement of these two factors is complicated because they interact. Existing methods either simply assume one factor given, based on heuristic assumptions, or invoke complicating extraneous factors, such as risk, that generate extra biases. The authors introduce a method for measuring discounting (and then quality of life) that involves no extraneous factors and that avoids distorting interactions. Their method is considerably simpler and more realistic for subjects than existing methods. It is entirely choice based and thus can be founded on economic rationality requirements. An experiment demonstrates the feasibility of this method and its advantages over classical methods. http://repub.eur.nl/res/pub/32048/ 2012-04-01T00:00:00Z Two experiments show that violations of expected utility due to ambiguity, found in general decision experiments, also affect belief aggregation. Hence we use modern ambiguity theories to analyze belief aggregation, thus obtaining more refined and empirically more valid results than traditional theories can provide. We can now confirm more reliably that conflicting (heterogeneous) beliefs where some agents express certainty are processed differently than informationally equivalent imprecise homogeneous beliefs. We can also investigate new phenomena related to ambiguity. For instance, agents who express certainty receive extra weight (a cognitive effect related to ambiguity-generated insensitivity) and generate extra preference value (source preference; a motivational effect related to ambiguity aversion). Hence, incentive compatible belief elicitations that prevent manipulation are especially warranted when agents express certainty. For multiple prior theories of ambiguity, our findings imply that the same prior probabilities can be treated differently in different contexts, suggesting an interest of corresponding generalizations. http://repub.eur.nl/res/pub/26324/ 2011-09-28T00:00:00Z Utility independence is a central condition in multiattribute utility theory, where attributes of outcomes are aggregated in the context of risk. The aggregation of attributes in the absence of risk is studied in conjoint measurement. In conjoint measurement, standard sequences have been widely used to empirically measure and test utility functions, and to theoretically analyze them. This paper shows that utility independence and standard sequences are closely related: utility independence is equivalent to a standard sequence invariance condition when applied to risk. This simple relation between two widely used conditions in adjacent fields of research is surprising and useful. It facilitates the testing of utility independence because standard sequences are flexible and can avoid cancelation biases that affect direct tests of utility independence. Extensions of our results to nonexpected utility models can now be provided easily. We discuss applications to the measurement of quality-adjusted life-years (QALY) in the health domain. http://repub.eur.nl/res/pub/31488/ 2011-07-01T00:00:00Z This paper finds preference reversals in measurements of ambiguity aversion, even if psychological and informational circumstances are kept constant. The reversals are of a fundamentally different nature than the reversals found before because they cannot be explained by context-dependent weightings of attributes. We offer an explanation based on Sugden's random-reference theory, with different elicitation methods generating different random reference points. Then measurements of ambiguity aversion that use willingness to pay are confounded by loss aversion and hence overestimate ambiguity aversion. http://repub.eur.nl/res/pub/23269/ 2011-04-01T00:00:00Z We often deal with uncertain events for which no probabilities are known. Several normative models have been proposed. Descriptive studies have usually been qualitative, or they estimated ambiguity aversion through one single number. This paper introduces the source method, a tractable method for quantitatively analyzing uncertainty empirically. The theoretical key is the distinction between different sources of uncertainty, within which subjective (choice-based) probabilities can still be defined. Source functions convert those subjective probabilities into willingness to bet. We apply our method in an experiment, where we do not commit to particular ambiguity attitudes but let the data speak. http://repub.eur.nl/res/pub/23267/ 2011-03-01T00:00:00Z http://repub.eur.nl/res/pub/23693/ 2011-01-01T00:00:00Z This paper discusses Jean-Yves Jaffray’s ideas on ambiguity and the views underlying his ideas. His models, developed 20 years ago, provide the most tractable separation of risk attitudes, ambiguity attitudes, and ambiguity beliefs available in the literature today. http://repub.eur.nl/res/pub/21094/ 2010-11-01T00:00:00Z This paper introduces time-tradeoff (TTO) sequences as a general tool to analyze intertemporal choice. We give several applications. For empirical purposes, we can measure discount functions without requiring any measurement of or assumption about utility. We can quantitatively measure time inconsistencies and simplify their qualitative tests. TTO sequences can be administered and analyzed very easily, using only pencil and paper. For theoretical purposes, we use TTO sequences to axiomatize (quasi-)hyperbolic discount functions. We demonstrate the feasibility of measuring TTO sequences in an experiment, in which we tested the axiomatizations. Our findings suggest rejections of several currently popular discount functions and call for the development of new ones. It is especially desirable that such discount functions can accommodate increasing impatience. http://repub.eur.nl/res/pub/21462/ 2010-08-01T00:00:00Z Abstract: When process fairness deviates from outcome fairness, dynamic inconsistencies can arise as in nonexpected utility. Resolute choice (Machina) can restore dynamic consistency under nonexpected utility without using Strotz's precommitment. It can similarly justify dynamically consistent process fairness. http://repub.eur.nl/res/pub/25744/ 2010-05-01T00:00:00Z Proper scoring rules serve to measure subjective degrees of belief. Traditional proper scoring rules are based on the assumption of expected value maximization. There are, however, many deviations from expected value, primarily due to risk aversion. Correcting techniques have been proposed in the literature for deviations due to nonlinear utility. These techniques still assumed expected utility maximization. More recently, corrections for deviations from expected utility have been proposed. The latter concerned, however, only the quadratic scoring rule, and could handle only half of the domain of subjective beliefs. Further, beliefs close to 0.5 could not be discriminated. This paper generalizes the correcting techniques to all bounded binary proper scoring rules, covers the whole domain of beliefs and, in particular, can discriminate between all degrees of belief. Thus, we fully extend the properness requirement (in the sense of identifying all degrees of subjective beliefs) to virtually all models that deviate from expected value. Copyright © 2011 John Wiley & Sons, Ltd. http://repub.eur.nl/res/pub/21096/ 2010-04-01T00:00:00Z ABSTRACT. This paper introduces time-tradeoff (TTO) sequences as a new tool to analyze time inconsistency and intertemporal choice. TTO sequences simplify the measurement of discount functions, requiring no assumption about utility. They also simplify the qualitative testing of time inconsistencies, and allow for quantitative measurements thereof. TTO sequences can easily be administered using only pencil and paper. They readily show which subjects are most prone to time inconsistencies. We further use them to axiomatically analyze and empirically test (quasi-)hyperbolic discount functions. An experiment demonstrates the feasibility of measuring TTO sequences. Our data falsify (quasi-)hyperbolic discount functions and call for the development of models that can accommodate increasing impatience. http://repub.eur.nl/res/pub/21460/ 2010-01-01T00:00:00Z ABSTRACT. In economic decisions we often have to deal with uncertain events for which no probabilities are known. Several normative models have been proposed for such decisions. Empirical studies have usually been qualitative, or they estimated ambiguity aversion through one single number. This paper introduces the source method, a tractable method for quantitatively analyzing uncertainty empirically that can capture the richness of ambiguity attitudes. The theoretical key in our method is the distinction between different sources of uncertainty, within which subjective (choice-based) probabilities can still be defined. Source functions convert those subjective probabilities into willingness to bet. We apply our method in an experiment, where we do not commit to particular ambiguity attitudes but let the data speak. http://repub.eur.nl/res/pub/21463/ 2010-01-01T00:00:00Z Abstract: This paper discusses Jean-Yves Jaffray’s ideas on ambiguity and the views underlying his ideas. His models, developed 20 years ago, provide the most tractable separation of risk attitudes, ambiguity attitudes, and ambiguity beliefs available in the literature today. http://repub.eur.nl/res/pub/17468/ 2009-11-01T00:00:00Z Similarity measures have been studied extensively in many domains, but usually with well-structured data sets. In many psychological applications, however, such data sets are not available. It often cannot even be predicted how many items will be observed, or what exactly they will entail. This paper introduces a similarity measure, called the metric-frequency (MF) measure, that can be applied to such data sets. If it is not known beforehand how many items will be observed, then the number of items actually observed in itself carries information. A typical feature of the MF is that it incorporates such information. The primary purpose of our measure is that it should be pragmatic, widely applicable, and tractable, even if data are complex. The MF generalizes Tversky's set-theoretic measure of similarity to cases where items may be present or absent and at the same time can be numerical as with Shepard's metric measure, but need not be so. As an illustration, we apply the MF to family therapy where it cannot be predicted what issues the clients will raise in therapy sessions. The MF is flexible enough to be applicable to idiographic data. http://repub.eur.nl/res/pub/17483/ 2009-09-17T00:00:00Z Proper scoring rules provide convenient and highly efficient tools for incentive-compatible elicitations of subjective beliefs. As traditionally used, however, they are valid only under expected value maximization. This paper shows how they can be generalized to modern ("non-expected utility") theories of risk and ambiguity, yielding mutual benefits: users of scoring rules can benefit from the empirical realism of non-expected utility, and analysts of ambiguity attitudes can benefit from efficient measurements using proper scoring rules. An experiment demonstrates the feasibility of our generalization. http://repub.eur.nl/res/pub/16048/ 2009-05-01T00:00:00Z The commonly used hyperbolic and quasi-hyperbolic discount functions have been developed to accommodate decreasing impatience, which is the prevailing empirical finding in intertemporal choice, in particular for aggregate behavior. However, these discount functions do not have the flexibility to accommodate increasing impatience or strongly decreasing impatience. This lack of flexibility is particularly disconcerting for fitting data at the individual level, where various patterns of increasing impatience and strongly decreasing impatience will occur for a significant fraction of subjects. This paper presents discount functions with constant absolute (CADI) or constant relative (CRDI) decreasing impatience that can accommodate any degree of decreasing or increasing impatience. In particular, they are sufficiently flexible for analyses at the individual level. The CADI and CRDI discount functions are the analogs of the well-known CARA and CRRA utility functions for decision under risk. http://repub.eur.nl/res/pub/21474/ 2009-03-01T00:00:00Z This paper introduces a parameter-free method for measuring the weighting functions of prospect theory and rank-dependent utility. These weighting functions capture risk attitudes, subjective beliefs, and ambiguity attitudes. Our method, called the midweight method, is based on a convenient way to obtain midpoints in the weighting function scale. It can be used both for risk (known probabilities) and for uncertainty (unknown probabilities). The resulting integrated treatment of risk and uncertainty is particularly useful for measuring the differences between them: ambiguity. Compared to existing methods to measure ambiguity attitudes, our method is more efficient and it can accommodate violations of expected utility under risk. An experiment demonstrates the feasibility and tractability of our method, yielding plausible results such as ambiguity aversion for moderate and high likelihoods but ambiguity seeking for low likelihoods, as predicted by Ellsberg. http://repub.eur.nl/res/pub/14145/ 2008-12-01T00:00:00Z Koopmans provided a well-known preference axiomatization for discounted utility, the most widely used model for maximizing intertemporal choice. There were, however, some technical problems in his analysis. For example, there was an unforeseen implication of bounded utility. Some partial solutions have been advanced in various fields in the literature. The technical problems in Koopmans' analysis obscure the appeal of his intuitive axioms. This paper completely resolves Koopmans' technical problems. In particular, it obtains complete flexibility concerning the utility functions that can be used. This paper, thus, provides a clean and complete preference axiomatization of discounted utility, clarifying the appeal of Koopmans' intuitive axioms. http://repub.eur.nl/res/pub/14515/ 2008-12-01T00:00:00Z This paper examines the cross-fertilizations of random utility models with the study of decision making under risk and uncertainty. We start with a description of the expected utility (EU) theory and then consider deviations from the standard EU frameworks, involving the Allais paradox and the Ellsberg paradox, inter alia. We then discuss how the resulting non-EU framework can be modeled and estimated within the framework of discrete choices in static and dynamic contexts. Our objectives in addressing risk and ambiguity in individual choice contexts are to understand the decision choice process and to use behavioral information for prediction, prescription, and policy analysis. http://repub.eur.nl/res/pub/17474/ 2008-12-01T00:00:00Z The power family, also known as the family of constant relative risk aversion (CRRA), is the most widely used parametric family for fitting utility functions to data. Its characteristics have, however, been little understood, and have led to numerous misunderstandings. This paper explains these characteristics in a manner accessible to a wide audience http://repub.eur.nl/res/pub/14619/ 2008-10-01T00:00:00Z Many traditional conjoint representations of binary preferences are additively decomposable, or additive for short. An important generalization arises under rank-dependence, when additivity is restricted to cones with a fixed ranking of components from best to worst (comonotonicity), leading to configural weighting, rank-dependent utility, and rank- and sign-dependent utility (prospect theory). This paper provides a general result showing how additive representations on an arbitrary collection of comonotonic cones can be combined into one overall representation that applies to the union of all cones considered. The result is applied to a new paradigm for decision under uncertainty developed by Duncan Luce and others, which allows for violations of basic rationality properties such as the coalescing of events and other framing conditions. Through our result, a complete preference foundation of a number of new models by Luce and others can be obtained. We also show how additive representations on different full product sets can be combined into a representation on the union of these different product sets. http://repub.eur.nl/res/pub/14901/ 2008-09-01T00:00:00Z This article is a personal account of the author's experiences as an economist working in medical decision making. He discusses the differences between economic decision theory and medical decision making and gives examples of the mutual benefits resulting from interactions. In particular, he discusses the pros and cons of different methods for measuring quality of life (or, as economists would call it, utility), including the standard gamble, the time tradeoff, and the healthy-years equivalent methods. http://repub.eur.nl/res/pub/17460/ 2008-06-01T00:00:00Z 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. http://repub.eur.nl/res/pub/29118/ 2007-05-01T00:00:00Z In an experiment, choice-based (revealed-preference) utility of money is derived from choices under risk, and choiceless (non-revealed-preference) utility from introspective strength-of-preference judgments. The well-known inconsistencies of risky utility under expected utility are resolved under prospect theory, yielding one consistent cardinal utility index for risky choice. Remarkably, however, this cardinal index also agrees well with the choiceless utilities, suggesting a relation between a choice-based and a choiceless concept. Such a relation implies that introspective judgments can provide useful data for economics, and can reinforce the revealed-preference paradigm. This finding sheds new light on the classical debate on ordinal versus cardinal utility. http://repub.eur.nl/res/pub/11073/ 2007-01-01T00:00:00Z Koopmans provided a well-known preference foundation for discounted utility, the most widely used model for intertemporal optimization. There were, however, some problems in his analysis. For example, there was an unforeseen implication of bounded utility. For some domains solutions have been advanced in the literature, primarily when particular production processes impose time-dependent restrictions on consumption. This paper completely resolves the problems mentioned, irrespective of what the restrictions on consumption are. It obtains complete flexibility concerningt he utility functions that can be used and concerning the conceivable economic growth. This paper, thus, provides a complete preference foundation of discounted utility, and clarifies the appeal of Koopmans’ intuitive axioms. http://repub.eur.nl/res/pub/11076/ 2007-01-01T00:00:00Z http://repub.eur.nl/res/pub/11077/ 2007-01-01T00:00:00Z The commonly used hyperbolic and quasi-hyperbolic discount functions imply decreasing impatience, which is the prevailing empirical phenomenon in intertemporal choice, in particular for aggregate behavior. At the individual level there is much variation, however, and there will always be some individuals who exhibit increasing impatience. Hence, to fit data at the individual level, new discount functions are needed. This paper introduces such functions, with constant absolute (CADI) or constant relative (CRDI) decreasing impatience. These functions can accommodate any degree of decreasing or increasing impatience, which makes them sufficiently flexible for analyses at the individual level. The CADI and CRDI discount functions are the analogs of the well known CARA and CRRA utility functions for decision under risk. http://repub.eur.nl/res/pub/8203/ 2006-01-01T00:00:00Z Whereas both the Allais paradox, the first empirical challenge of the classical rationality assumptions, and learning have been the focus of many experimental investigations, no experimental study exists today into learning in the pure context of the Allais paradox. This paper presents such a study. We find that choices converge to expected utility maximization if subjects are given the opportunity to learn by both thought and experience, but less so when they learn by thought only. To the extent that genuine preferences should be measured with proper learning and incentives, our study gives the first pure demonstration that irrationalities such as in the Allais-paradox are less pronounced than often thought. http://repub.eur.nl/res/pub/8206/ 2006-01-01T00:00:00Z This is a preprint of an article to be published in Management Science http://repub.eur.nl/res/pub/8207/ 2006-01-01T00:00:00Z This chapter deals with individual decision making under uncertainty (unknown probabilities). Risk (known probabilities) is not treated as a separate case, but as a subcase of uncertainty. Many results from risk naturally extend to uncertainty. The Allais paradox, commonly applied to risk, also reveals empirical deficiencies of expected utility for uncertainty. The Ellsberg paradox does not reveal deviations from expected utility in an absolute sense, but in a relative sense, giving within-person comparisons: for some events (ambiguous or otherwise) subjects deviate more from expected utility than for other events. Besides aversion, many other attitudes towards ambiguity are empirically relevant. http://repub.eur.nl/res/pub/8208/ 2006-01-01T00:00:00Z In an experiment, choice-based utilities are derived from choices under risk, and choiceless utilities from introspective strength-of-preference judgments. The well-known inconsistencies of risky utility, when analyzed through expected utility, are resolved by means of prospect theory. A consistent cardinal utility index for risky choice results. Remarkably, however, this cardinal index agrees well with the choiceless utilities. This finding suggests a relation between a choice-based and a choiceless concept. Such a relation would imply that direct judgments can provide useful data for economics, and can reinforce the revealed-preference approach. Implications for the classical debate on ordinal versus cardinal utility are discussed. http://repub.eur.nl/res/pub/8204/ 2005-02-01T00:00:00Z This paper introduces the likelihood method for decision under uncertainty. The method allows the quantitative determination of subjective beliefs or decision weights without invoking additional separability conditions, and generalizes the Savage-de Finetti betting method. It is applied to a number of popular models for decision under uncertainty. In each case, preference foundations result from the requirement that no inconsistencies are to be revealed by the version of the likelihood method appropriate for the model considered. A unified treatment of subjective decision weights results for most of the decision models popular today. Savage's derivation of subjective expected utility can now be generalized and simplified. In addition to the intuitive and empirical contributions of the likelihood method, we provide a number of technical contributions: We generalize Savage's nonatomiticy condition ("P6") and his assumption of (sigma) algebras of events, while fully maintaining his flexibility regarding the outcome set. Derivations of Choquet expected utility and probabilistic sophistication are generalized and simplified similarly. The likelihood method also reveals a common intuition underlying many other conditions for uncertainty, such as definitions of ambiguity aversion and pessimism. http://repub.eur.nl/res/pub/8205/ 2005-01-01T00:00:00Z To a considerable extent, risk aversion as it is commonly observed is caused by loss aversion. Several indexes of loss aversion have been proposed in the literature. The one proposed in this paper leads to a clear decomposition of risk attitude into three distinct components: basic utility, probability weighting, and loss aversion. The index is independent of the unit of payment. The main theorem shows how the indexes of different decision makers can be compared through observed choices. http://repub.eur.nl/res/pub/22984/ 2004-12-01T00:00:00Z 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. http://repub.eur.nl/res/pub/23023/ 2004-12-01T00:00:00Z This paper introduces anchor levels as a new tool for multiattribute utility theory. Anchor levels are attribute levels whose values are not affected by other attributes. They allow for new interpretations and generalizations of known representations and utility measurement techniques. Generalizations of earlier techniques can be obtained because cases with complex interactions between attributes can now be handled. Anchor levels serve not only to enhance the generality, but also the tractability, of utility measurements, because stimuli can better be targeted toward the perception and real situation of clients. In an application, anchor levels were applied to the measurement of quality of life during radiotherapy treatment, where there are complex interactions with what happens before and after. Using anchor levels, the measurements could be related exactly to the situation of the clients, thus simplifying the clients’ cognitive burden. http://repub.eur.nl/res/pub/23117/ 2004-01-01T00:00:00Z Prospect theory assumes nonadditive decision weights for preferences over risky gambles. Such decision weights generalize additive probabilities. This article proposes a decomposition of decision weights into a component reflecting risk attitude and a new component depending on belief. The decomposition is based on an observable preference condition and does not use other empirical primitives such as statements of judged probabilities. The preference condition is confirmed by most of the experimental findings in the literature. The implied properties of the belief component suggest that, besides the often-studied ambiguity aversion (a motivational factor reflecting a general aversion to unknown probabilities), perceptual and cognitive limitations play a role: It is harder to distinguish among various levels of likelihood, and to process them differently, when probabilities are unknown than when they are known. http://repub.eur.nl/res/pub/23024/ 2002-12-01T00:00:00Z Most empirical studies of rank-dependent utility and cumulative prospect theory have assumed power utility functions, both for gains and for losses. As it turns out, a remarkably simple preference foundation is possible for such models: Tail independence (a weakening of comonotonic independence which underlies all rank-dependent models) together with constant proportional risk aversion suffice, in the presence of common assumptions (weak ordering, continuity, and first stochastic dominance), to imply these models. Thus, sign dependence, the different treatment of gains and losses, and the separation of decision weights and utility are obtained free of charge. http://repub.eur.nl/res/pub/23025/ 2002-01-01T00:00:00Z This paper uses decision-theoretic principles to obtain new insights into the assessment and updating of probabilities. First, a new foundation of Bayesianism is given. It does not require infinite atomless uncertainties as did Savage's classical result, and can therefore be applied to any finite Bayesian network. It neither requires linear utility as did de Finetti's classical result, and therefore allows for the empirically and normatively desirable risk aversion. Finally, by identifying and fixing utility in an elementary manner, our result can readily be applied to identify methods of probability updating. Thus, a decision-theoretic foundation is given to the computationally efficient method of inductive reasoning developed by Rudolf Carnap. Finally, recent empirical findings on probability assessments are discussed. It leads to suggestions for correcting biases in probability assessments, and for an alternative to the Dempster-Shafer belief functions that avoids the reduction to degeneracy after multiple updatings. http://repub.eur.nl/res/pub/11034/ 2001-11-01T00:00:00Z This paper proposes a quantitative modification of standard utility elicitation procedures, such as the probability and certainty equivalence methods, to correct for commonly observed violations of expected utility. Traditionally, decision analysis assumes expected utility not only for the prescriptive purpose of calculating optimal decisions but also for the descriptive purpose of eliciting utilities. However, descriptive violations of expected utility bias utility elicitations. That such biases are effective became clear when systematic discrepancies were found between different utility elicitation methods that, under expected utility, should have yielded identical utilities. As it is not clear how to correct for these biases without further knowledge of their size or nature, most utility elicitations still calculate utilities by means of the expected utility formula. This paper speculates on the biases and their sizes by using the quantitative assessments of probability transformation and loss aversion suggested by prospect theory. It presents quantitative corrections for the probability and certainty equivalence methods. If interactive sessions to correct for biases are not possible, then the authors propose to use the corrected utilities rather than the uncorrected ones in prescriptions of optimal decisions. In an experiment, the discrepancies between the probability and certainty equivalence methods are removed by the authors' proposal. http://repub.eur.nl/res/pub/23027/ 2001-11-01T00:00:00Z Among the most popular models for decision under risk and uncertainty are the rank-dependent models, introduced by Quiggin and Schmeidler. Central concepts in these models are rank-dependence and comonotonicity. It has been suggested that these concepts are technical tools that have no intuitive or empirical content. This paper describes such contents. As a result, rank-dependence and comonotonicity become natural concepts upon which preference conditions, empirical tests, and improvements in utility measurement can be based. Further, a new derivation of the rank-dependent models is obtained. It is not based on observable preference axioms or on empirical data, but naturally follows from the intuitive perspective assumed. We think that the popularity of the rank-dependent theories is mainly due to the natural concepts used in these theories. http://repub.eur.nl/res/pub/23028/ 2001-09-01T00:00:00Z This paper shows how the signed Choquet integral, a generalization of the regular Choquet integral, can model violations of separability and monotonicity. Applications to intertemporal preference, asset pricing, and welfare evaluations are discussed. http://repub.eur.nl/res/pub/23029/ 2001-06-01T00:00:00Z Background—: To perform decision analyses that include stroke as one of the possible health states, the utilities of stroke states must be determined. We reviewed the literature to obtain estimates of the utility of stroke and explored the impact of the study population and the elicitation method. Summary of Review—: We searched various databases for articles reporting empirical assessment of utilities. Mean utilities of major stroke (Rankin Scale 4 to 5) and minor stroke (Rankin Scale 2 to 3) were calculated, stratified by study population and elicitation method. Additionally, the modified Rankin Scale was mapped onto the EuroQol classification system. Utilities were obtained from 23 articles. Patients at risk for stroke assigned utilities of 0.26 and 0.55 to major and minor stroke, respectively. Stroke survivors assigned higher utilities to both major (0.41) and minor stroke (0.72). The EuroQol completed by stroke survivors revealed a utility of 0.32 and 0.71 for major and minor stroke, respectively. Utilities elicited by the Standard Gamble were generally higher, while those obtained by the Visual Analogue Scale were lower than the Time Trade Off values. Remaining variation between utilities may be caused by differences in definitions of the health states. The mapped EuroQol indicated a utility of 0.64 for minor stroke and a value just below zero for major stroke. Conclusions—: For minor stroke, a utility between 0.50 and 0.70 seems to be reasonable for both decision analyses and cost-effectiveness studies. The utility of major stroke may range between 0 and 0.30 and may possibly be negative. http://repub.eur.nl/res/pub/23026/ 2001-01-01T00:00:00Z In expected utility theory, risk attitudes are modeled entirely in terms of utility. In the rank-dependent theories, a new dimension is added: chance attitude, modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility. Most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation. Only a few qualitative conditions have been proposed or tested as yet, usually quasi-concavity or quasi-convexity in the case of given probabilities. This paper presents a general method of studying qualitative properties of chance attitude such as optimism, pessimism, and the "inverse-S shape" pattern, both for risk and for uncertainty. These qualitative properties can be characterized by permitting appropriate, relatively simple, violations of the sure-thing principle. In particular, this paper solves a hitherto open problem: the preference axiomatization of convex ("pessimistic" or "uncertainty averse") nonadditive measures under uncertainty. The axioms of this paper preserve the central feature of rank-dependent theories, i.e. the separation of chance attitude and utility. http://repub.eur.nl/res/pub/23032/ 2000-09-01T00:00:00Z Machina and Schmeidler [Econometrica, 60 (1992) 745–780] gave preference conditions for probabilistic sophistication, i.e. decision making where uncertainty can be expressed in terms of (subjective) probabilities without commitment to expected utility maximization. This note shows that simpler and more general results can be obtained by combining results from qualitative probability theory with a ‘cumulative dominance’ axiom. http://repub.eur.nl/res/pub/23031/ 2000-06-01T00:00:00Z This paper shows that a “principle of complete ignorance” plays a central role in decisions based on Dempster belief functions. Such belief functions occur when, in a first stage, a random message is received and then, in a second stage, a true state of nature obtains. The uncertainty about the random message in the first stage is assumed to be probabilized, in agreement with the Bayesian principles. For the uncertainty in the second stage no probabilities are given. The Bayesian and belief function approaches part ways in the processing of the uncertainty in the second stage. The Bayesian approach requires that this uncertainty also be probabilized, which may require a resort to subjective information. Belief functions follow the principle of complete ignorance in the second stage, which permits strict adherence to objective inputs. http://repub.eur.nl/res/pub/23033/ 2000-04-01T00:00:00Z http://repub.eur.nl/res/pub/23081/ 1999-08-01T00:00:00Z Classical foundations of expected utility were provided by Ramsey, de Finetti, von Neumann and Morgenstern, Anscombe and Aumann, and others. These foundations describe preference conditions to capture the empirical content of expected utility. The assumed preference conditions, however, vary among the models and a unifying idea is not readily transparent. Providing such a unifying idea is the purpose of this paper. The mentioned derivations have in common that a cardinal utility index for outcomes, independent of the states and probabilities, can be derived. Characterizing that feature provides the unifying idea of the mentioned models. http://repub.eur.nl/res/pub/23082/ 1999-08-01T00:00:00Z Cumulative prospect theory was introduced by Tversky and Kahneman so as to combine the empirical realism of their original prospect theory with the theoretical advantages of Quiggin''s rank-dependent utility. Preference axiomatizations were provided in several papers. All those axiomatizations, however, only consider decision under uncertainty. No axiomatization has been provided as yet for decision under risk, i.e., the case in which given probabilities are transformed. Providing the latter is the purpose of this note. The resulting axiomatization is considerably simpler than that for uncertainty. http://repub.eur.nl/res/pub/23034/ 1999-02-01T00:00:00Z This paper generalizes the Debreu/Gorman characterization of additively decomposable functionals and separable preferences to infinite dimensions. The first novelty concerns the very definition of additively decomposable functionals for infinite dimensions. For decision under uncertainty, our result provides a state-dependent extension of Savage's expected utility. A characterization in terms of preference conditions identifies the empirical content of the model; it amounts to Savage's axiom system with P4 (likelihood ordering) dropped. Our approach does not require that a (probability) measure on the state space be given a priori, or can be derived from extraneous conditions outside the realm of decision theory. Bayesian updating of new information is still possible, even though no prior probabilities are given. The finding suggests that the sure-thing principle, rather than prior probability, is at the heart of Bayesian updating. http://repub.eur.nl/res/pub/23083/ 1998-10-01T00:00:00Z Temporary health states cannot be measured in the traditional way by means of techniques such as the time tradeoff (TTO) and the standard gamble (SG), where health states are chronic and are followed by death. Chained methods have been developed to solve this problem. This study assesses the feasibility of a chained TTO and a chained SG, and the consistency and concordance between the two meth ods. Patients and methods. Seventy female early-stage breast cancer patients were interviewed. In using both chained methods, the temporary health state to be evaluated was weighed indirectly with the aid of a temporary anchor health state. The patients were asked to evaluate their actual health states, a hypothetical radiotherapy scenario, and a hypothetical chemotherapy scenario. Results. Sixty-eight patients completed the interview. The use of the anchor health state yielded some problems. A significant difference between the means of the TTO and the SG was found for the anchor health state only. For the other health states, the results were remarkably close, because the design avoided some of the bias effects in traditional measurements. Conclusion. The feasibility and the consistency of the chained procedure were satisfactory for both methods. The problems regarding the anchor health state can be solved by adapting the methods and by the use of a carefully chosen anchor health state. The chained method avoids biases present in the conventional method, and thereby the TTO and the SG may be reconciled. Moreover, there are several psychological advantages to the method, which makes it useful for diseases with uncertain prognoses. Key words: utility assessment; time tradeoff; standard gamble; breast cancer; chemotherapy; ra diotherapy http://repub.eur.nl/res/pub/23085/ 1998-05-01T00:00:00Z Nonadditive expected utility models were developed for explaining preferences in settings where probabilities cannot be assigned to events. In the absence of probabilities, difficulties arise in the interpretation of likelihoods of events. In this paper we introduce a notion of revealed likelihood that is defined entirely in terms of preferences and that does not require the existence of (subjective) probabilities. Our proposal is that decision weights rather than capacities are more suitable measures of revealed likelihood in rank-dependent expected utility models and prospect theory. Applications of our proposal to the updating of beliefs and to the description of attitudes towards ambiguity are presented. http://repub.eur.nl/res/pub/11055/ 1998-01-01T00:00:00Z This paper studies the implications of the "zero-condition" for multiattribute utility theory. The zero-condition simplifies the measurement and derivation of the Quality Adjusted Life Year (QALY) measure commonly used in medical decision analysis. For general multiattribute utility theory, no simple condition has heretofore been found to characterize multiplicatively decomposable forms. When the zero-condition is satisfied, however, such a simple condition, "standard gamble invariance," becomes available. http://repub.eur.nl/res/pub/23093/ 1997-10-01T00:00:00Z Probabilistic insurance is an insurance policy involving a small probability that the consumer will not be reimbursed. Survey data suggest that people dislike probabilistic insurance and demand more than a 20% reduction in the premium to compensate for a 1% default risk. While these preferences are intuitively appealing they are difficult to reconcile with expected utility theory. Under highly plausible assumptions about the utility function, willingness to pay for probabilistic insurance should be very close to willingness to pay for standard insurance less the default risk. However, the reluctance to buy probabilistic insurance is predicted by the weighting function of prospect theory. This finding highlights the potential role of the weighting function to explain insurance. http://repub.eur.nl/res/pub/23092/ 1997-08-01T00:00:00Z Preference reversals have usually been explained by weighted additive models, in which different tasks give rise to different importance weights for the stimulus attributes, resulting in contradictory trade-offs. This article presents a preference reversal of a more extreme nature. Let (10, 5 Migr) denote living 10 years with a migraine for 5 days per week. Many participants preferred (10, 5 Migr) to (20, 5 Migr). However, when asked to equate these two options with a shorter period of good health, they usually demanded more healthy life years for (20, 5 Migr) than for (10, 5 Migr). This preference reversal within a single dimension cannot be explained by different importance weights and suggests irrationalities at a more fundamental level. Most participants did not change their responses after being confronted with their inconsistencies. http://repub.eur.nl/res/pub/23088/ 1997-07-01T00:00:00Z Anscombe and Aumann showed that if one accepts the existence of a physical randomizing device such as a roulette wheel then Savage's derivation of subjective expected utility can be considerably simplified. They, however, invoked compound gambles to define their axioms. We demonstrate that the subjective expected utility derivation can be further simplified and need not invoke compound gambles. Our simplification is obtained by closely following the steps by which probabilities and utilities are elicited. http://repub.eur.nl/res/pub/11026/ 1997-01-01T00:00:00Z This paper shows that QALYs can be derived from more elementary conditions than thought hitherto in the literature: it suffices to impose risk neutrality for life years in every health state. This derivation of QALYs is appealing because it does not require knowledge of concepts from utility theory such as utility independence. Therefore our axiomatization greatly facilitates the assessment of the normative (non)validity of QALYs in medical decision making. Moreover, risk neutrality can easily be tested in experimental designs, which makes it straightforward to assess the descriptive (non)validity of QALYs. http://repub.eur.nl/res/pub/23011/ 1997-01-01T00:00:00Z Two core meanings of “utility” are distinguished. “Decision utility” is the weight of an outcome in a decision. “Experienced utility” is hedonic quality, as in Bentham’s usage. Experienced utility can be reported in real time (instant utility), or in retrospective evaluations of past episodes (remembered utility). Psychological research has documented systematic errors in retrospective evaluations, which can induce a preference for dominated options. We propose a formal normative theory of the total experienced utility of temporally extended outcomes. Measuring the experienced utility of outcomes permits tests of utility maximization and opens other lines of empirical research. http://repub.eur.nl/res/pub/23089/ 1997-01-01T00:00:00Z http://repub.eur.nl/res/pub/23101/ 1996-12-01T00:00:00Z A method is given to extend demand functions to new commodities under preservation of the cycle number, i.e. the minimal length of a preference cycle revealed by the demand function. Thus, Gale's (Economica, N.S., 1960, 27, 348–354) demand function that shows that the weak axiom of revealed preference does not imply the strong axiom of revealed preference for three commodities can be extended to more than three commodities. Also Shafer's (Journal of Economic Theory, 1977, 16, 293–309) result, that arbitrarily high cycle numbers exist for three commodities, can now be extended to any number of commodities larger than three. This completely settles a question raised by Samuelson (Economica, N.S., 1953, 20, 1–9). http://repub.eur.nl/res/pub/23106/ 1996-12-01T00:00:00Z This paper compares classical expected utility with the more general rank-dependent utility models. It shows that the difference between the sure-thing principle for preferences of expected utility and its comonotonic generalization in rank-dependent utility provides the exact demarcation between expected utility and rank-dependent models. http://repub.eur.nl/res/pub/23107/ 1996-12-01T00:00:00Z http://repub.eur.nl/res/pub/23096/ 1996-08-01T00:00:00Z This paper proposes a new method, the (gamble-)tradeoff method, for eliciting utilities in decision under risk or uncertainty. The elicitation of utilities, to be used in the expected utility criterion, turns out to be possible even if probabilities are ambiguous or unknown. A disadvantage of the tradeoff method is that a few more questions usually must be asked to clients. Also, the lotteries that are needed are somewhat more complex than in the certainty-equivalent method or in the probability-equivalent method. The major advantage of the tradeoff method is its robustness against probability distortions and misconceptions, which constitute a major cause of violations of expected utility and generate inconsistencies in utility elicitation. Thus the tradeoff method retains full validity under prospect theory, rank-dependent utility, and the combination of the two, i.e., cumulative prospect theory. The tradeoff method is tested for monetary outcomes and for outcomes describing life-duration. We find higher risk aversion for life duration, but the tradeoff method elicits similar curvature of utility. Apparently the higher risk aversion for life duration is due to more pronounced deviations from expected utility. http://repub.eur.nl/res/pub/23105/ 1996-08-01T00:00:00Z The following questions describe the scope of this paper. When decision trees are used to analyze optimal decisions, should end nodes be evaluated on the basis of QALYs or on the basis of healthy-years equivalents? Which measures should be used in communications with others, e.g., patients? Which of these measures incorporate nsk attitudes, and which do not? It is demonstrated that the healthy-years equivalent measure does not stand scrutiny. Key words: utility; QALY; quality of life; healthy-years equivalent; standard gamble. http://repub.eur.nl/res/pub/23103/ 1996-07-01T00:00:00Z Experimental investigations of non-expected utility have primarily concentrated on decision under risk (probability triangles). The literature suggests, however, that ambiguity is one of the main causes for deviations from expected utility (EU). This article investigates the descriptive performance of rank-dependent utility (RDU) in the context of choice under ambiguity. We use the axiomatic difference between RDU and EU to critically test RDU against EU. Surprisingly, the RDU model does not provide any descriptive improvement over EU. Our data suggest other framing factors that do provide descriptive improvements over EU. http://repub.eur.nl/res/pub/23099/ 1996-03-01T00:00:00Z Methods for determining the form of utilities are needed for the implementation of utility theory in specific decisions. An important step forward was achieved when utility theorists characterized useful parametric families of utilities and simplifying decompositions of multiattribute utilities. The standard development of these results is based on expected utility theory which is now known to be descriptively invalid. The empirical violations of expected utility impair the credibility of utility assessments. This paper shows, however, that parametric and multiattribute utility results are robust against the major violations of expected utility. They retain their validity under nonexpected utility theories that have been developed to account for actual choice behavior. To be precise, characterizations of parametric and multiattribute representations are extended to rank-dependent utility, state-dependent utility, Choquet-expected utility, and prospect theory. http://repub.eur.nl/res/pub/23010/ 1996-01-01T00:00:00Z Established firms can diversify into new markets in two distinct modes: through internal development or through conglomerate merger. Building on a dynamic three-stage bargaining model with variable threats, this paper shows that a lenient antitrust position toward horizontal mergers can induce established firms that would otherwise not have entered to enter via conglomerate merger. The vigor of antitrust enforcement toward horizontal mergers also affects the conglomerate acquisition price but it does not influence the choice of entry mode. Finally, the paper brings to light a heretofore neglected avenue through which conglomerate mergers can increase welfare. http://repub.eur.nl/res/pub/23097/ 1996-01-01T00:00:00Z This article identifies the common characterizing property, the comonotonic sure-thing principle, that underlies the rank-dependent direction in non-expected utility. This property restricts Savage's sure-thing principle to comonotonic acts, and is characterized in full generality by means of a new functional form—cumulative utility—that generalizes the Choquet integral. Thus, a common generalization of all existing rank-dependent forms is obtained, including rank-dependent expected utility, Choquet expected utility, and cumulative prospect theory. http://repub.eur.nl/res/pub/23118/ 1995-11-01T00:00:00Z To accommodate the observed pattern of risk-aversion and risk-seeking, as well as common violations of expected utility (e.g., the certainty effect), we introduce and characterize a weighting function according to which an event has greater impact when it turns impossibility into possibility, or possibility into certainty, that when it merely makes a possibility more or less likely. We show how to compare such weighting functions (of different individuals) with respect to the degree of departure from expected utility, and we present a method for comparing an individual's weighting functions for risk and for uncertainty. http://repub.eur.nl/res/pub/23110/ 1995-04-01T00:00:00Z An operational method is presented for deriving a linear ranking of alternatives from repeated paired comparisons of the alternatives. Intransitivities in the observed preferences are cleared away by the introduction of decision errors of varying importance. An observed preference between two alternatives that causes an intransitivity in the course of the procedure will be reversed if it is of lesser importance. The method is applicable in case one wants to take account of intensities of preference and assume these to be monotone with the probability that an observed choice coincides with a fixed underlying true choice. http://repub.eur.nl/res/pub/23108/ 1995-01-01T00:00:00Z This paper discusses the history and interrelations of three central ideas in preference theory: the independence condition in decision under risk, the sure-thing principle in decision under uncertainty, and conjoint independence for multiattribute decisions and consumer theory. Independence was recognized as an important component of decision under risk in the late 1940s by Jacob Marschak, John Nash, Herman Rubin, and Norman Dalkey, and first appeared in publication in Marschak (1950) and Nash (1950). The sure-thing principle can be credited to Savage (1953, 1954). Conjoint independence for consumer theory was introduced by Sono (1943) and Leontief (1947a, b); a form of it can also be recognized in Samuelson (1947), presented earlier in Samuelson (1940). Independence and the sure-thing principle are equivalent for decision under risk, but in a less elementary way than has sometimes been thought. The sure-thing principle for decision under uncertainty and conjoint independence are identical in a mathematical sense. The mathematics underlying our three preference conditions has an older history. The independence condition for decision under risk can be recognized in the characterization of "associative means," and conjoint independence for multiattribute decisions in solutions to the "generalized associativity functional equation." http://repub.eur.nl/res/pub/23113/ 1995-01-01T00:00:00Z The standard gamble (SG) method has been accepted as the gold standard for the elicitation of utility when risk or uncertainty is involved in decisions, and thus for the measurement of utility in medical decisions. Unfortunately, the SG method is distorted by a general dislike for gambles, the "gambling effect," leading to an overestimation of risk aversion and of utility of impaired health. This problem does not occur for visual analogue scales or the time tradeoff method. For risky decisions, however, the latter methods lack validity. This paper shows how "rank-dependent utility" theory, a newly developed theory in the decision science literature, can provide a new explanation for the gambling effect. Thus it provides a means to correct the SG method and to improve the assessments of quality adjusted life years for medical decisions in which there is uncertainty about outcomes. http://repub.eur.nl/res/pub/23116/ 1995-01-01T00:00:00Z The reduction of costs is becoming increasingly important in the medical field. The relevant topic of many clinical trials is not effectiveness per se, but rather cost-effectiveness ratios. Strangely enough, no statistical tools for analyzing cost-effectiveness ratios have been provided in the medical literature yet. This paper explains the gap in the literature, and provides a first technique for obtaining confidence intervals for cost-effectiveness ratios. The technique does not use sophisticated tools to achieve maximal optimality criteria, but seeks for tractability and ease of application, while still satisfying all formal statistical requirements. http://repub.eur.nl/res/pub/23426/ 1995-01-01T00:00:00Z Het 'preference reversal'-verschijnsel is van fundamenteel belang voor de economie, en geeft ook inzicht in de sociologie van de wetenschap. De ontdekkingsgeschiedenis leest als een sprookje of, zo u wilt, een wild-west verhaal. http://repub.eur.nl/res/pub/23121/ 1994-12-01T00:00:00Z This article compares classical expected utility (EU) with the more general rank-dependent utility (RDU) models. The difference between the independence condition for preferences of EU and its comonotonic generalization in RDU provides the exact demarcation between EU and rank-dependent models. Other axiomatic differences are not essential. An experimental design is described that tests this difference between independence and comonotonic independence in its most basic form and is robust against violations of other assumptions that may confound the results, in particular the reduction principle and transitivity. It is well known that in the classical counterexamples to EU, comonotonic independence performs better than full-force independence. For our more general choice pairs, however, we find that comonotonic independence does not perform better. This is contrary to our prior expectation and suggests that rank-dependent models, in full generality, do not provide a descriptive improvement over EU. For rank-dependent models to have a future, submodels and choice situations need to be identified for which rank-dependence does contribute descriptively. http://repub.eur.nl/res/pub/23186/ 1994-12-01T00:00:00Z Quiggin (J. Econ. Behav. Organization3 (1982), 323-345) introduced anticipated ("rank-dependent") utility theory into decision making under risk. Questions have been raised about mathematical aspects of Quiggin′s analysis. This paper settles these questions and shows that a minor modification of Quiggin′s axioms leads to a useful and correct result, with features not found in other recent axiomatizations. http://repub.eur.nl/res/pub/23189/ 1994-12-01T00:00:00Z http://repub.eur.nl/res/pub/23190/ 1994-12-01T00:00:00Z John Quiggin is a professor at the centre for Economic Policy Research of the Australian National University. His research interests are decision making under risk and agricultural economics. The reviewer is an associate professor at the Medical Decision Making Unit, University of Leiden, Leiden, The Netherlands. His main interest is decision making under uncertainty. http://repub.eur.nl/res/pub/23123/ 1994-05-01T00:00:00Z This note demonstrates that two minimal requirements of decision tree analysis, the folding back procedure and the interchangeability of consecutive event nodes, imply independence http://repub.eur.nl/res/pub/23125/ 1994-05-01T00:00:00Z http://repub.eur.nl/res/pub/23184/ 1994-03-01T00:00:00Z This paper shows that the role of risky alternatives can be greatly reduced in the clicitation procedures of multiattribute utility. This reduction can be achieved by invoking methods from additive conjoint measurement; it is desirable because risky choices involve more cognitive problems, thus more biases and unreliability, than riskless ones. Existing results of multiattribute utility are generalized to obtain a complete axiomatization of the new clicitation procedure. The approach has been developed in a medical decision analysis project to advise on the choice between surgery and radiotherapy for laryngeal cancer. http://repub.eur.nl/res/pub/23119/ 1994-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/23188/ 1994-01-01T00:00:00Z This paper provides a simple approach for deriving cumulative prospect theory. The key axiom is a cumulative dominance axiom which requires that a prospect be judged more attractive if in it greater gains are more likely and greater losses are less likely. In the presence of this cumulative dominance, once a model is satisfied on a "sufficiently rich" domain, then it holds everywhere. This leads to highly transparent results. http://repub.eur.nl/res/pub/23209/ 1993-12-01T00:00:00Z This article studies situations in which information is ambiguous and only part of it can be probabilized. It is shown that the information can be modeled through belief functions if and only if the nonprobabilizable information is subject to the principles of complete ignorance. Next the representability of decisions by belief functions on outcomes is justified by means of a neutrality axiom. The natural weakening of Savage's sure-thing principle to unambiguous events is examined and its implications for decision making are identified. http://repub.eur.nl/res/pub/23205/ 1993-10-01T00:00:00Z This paper presents a method for axiomatizing a variety of models for decision making under uncertainty, including Expected Utility and Cumulative Prospect Theory. This method identifies, for each model, the situations that permit consistent inferences about the ordering of value differences. Examples of rankdependent and sign-dependent preference patterns are used to motivate the models and the tradeoff consistency axioms that characterize them. The major properties of the value function in Cumulative Prospect Theory—diminishing sensitivity and loss aversion—are contrasted with the principle of diminishing marginal utility that is commonly assumed in Expected Utility. http://repub.eur.nl/res/pub/23200/ 1993-05-01T00:00:00Z A general procedure for extending finite-dimensional "additive-like" representations for binary relations to infinite-dimensional "integral-like" representations is developed by means of a condition called truncation-continuity. The restriction of boundedness of utility, met throughout the literature, can now be dispensed with, and for instance normal distributions, or any other distribution with finite first moment, can be incorporated. Classical representation results of expected utility, such as Savage (1954), von Neumann and Morgenstern (1944), Anscombe and Aumann (1963), de Finetti (1937), and many others, can now be extended. The results are generalized to Schmeidler's (1989) approach with nonadditive measures and Choquet integrals, and Quiggin's (1982) rank-dependent utility. The different approaches have been brought together in this paper to bring to the fore the unity in the extension process. http://repub.eur.nl/res/pub/23201/ 1993-04-01T00:00:00Z Contrary to common belief, Savage's axioms do not imply strict stochastic dominance. Instead, they usually involve violation of that. Violations occur as soon as the range of the utility function is rich enough, e.g. contains an interval, and the probability measure is, loosely speaking, "constructive". An example is given where all of Savage's axioms are satisfied, but still strict statewise monotonicity is violated: An agent is willing to exchange an act for another act that with certainty yields a strictly worse outcome. Thus book can be made against the agent. Weak stochastic dominance and weak statewise monotonicity are always satisfied, as well as strict stochastic dominance and strict statewise monotonicity when restricted to acts with finitely many outcomes. http://repub.eur.nl/res/pub/23195/ 1993-02-01T00:00:00Z Chew and Epstein attempted to provide a unifying axiomatic framework for a number of generalizations of expected utility theory. Wakker pointed out that Theorem A, on which the central unifying proposition is based, is false. In this note, we apply Segal′s result to prove that Theorem 2 is nevertheless valid with minor modifications. http://repub.eur.nl/res/pub/23198/ 1993-02-01T00:00:00Z This note discusses some mathematical misunderstandings about Savage (1954). It is shown that in his model the probability measure cannot be countably additive, that the set of events must be a σ-algebra and not just an algebra, that Savage did not characterize all atomless finitely additive probability measures, and that the state space in his model, while infinite, does not have to be uncountable. http://repub.eur.nl/res/pub/23192/ 1993-01-01T00:00:00Z Additive representation theory on subsets of Cartesian products has characteristics different from additive representation theory on full Cartesian products. This paper describes the difficulties that can arise on subsets. These difficulties have been underestimated in the literature. For the special case of rank-ordered subsets of Cartesian products the paper obtains characterizations of additive representations. These results can be applied in the modern rank-dependent approaches to decision making under risk/uncertainty, and to generalizations of the Gini index in the measurement of inequality. http://repub.eur.nl/res/pub/23193/ 1993-01-01T00:00:00Z This article discusses relations between several notions of continuity in rank-dependent utility, and in the generalized version of rank-dependent utility as initiated by Segal. Primarily, examples are given to show logical independencies between these notions of continuity. This also leads to counterexamples to Segal's (1989) characterizing theorem 1. This article is a rewritten version of Wakker (1990a). Puppe (1990) independently discovered that Segal's (1989) theorem 1 is not correct. This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. http://repub.eur.nl/res/pub/23206/ 1993-01-01T00:00:00Z This paper studies continuous additive representations of transitive preferences on connected subdomains of product sets. Contrary to what has sometimes been thought, local additive representability does not imply global additive representability. It is shown that the result can nevertheless be established under some additional connectedness conditions. This generalizes previous results on additive representations on (subsets of) product sets. http://repub.eur.nl/res/pub/23212/ 1992-12-01T00:00:00Z http://repub.eur.nl/res/pub/23210/ 1992-11-01T00:00:00Z This paper provides an extension of Savage's subjective expected utility theory for decisions under uncertainty. It includes in the set of events both unambiguous events for which probabilities are additive and ambiguous events for which probabilities are permitted to be nonadditive. The main axiom is cumulative dominance, which adapts stochastic dominance to decision making under uncertainty. We derive a Choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation. The relationship of our approach with that of Schmeidler, who uses a two-stage formulation to derive Choquet expected utility, is also explored. Our work may be viewed as a unification of Schmeidler (1989) and Gilboa (1987). http://repub.eur.nl/res/pub/23215/ 1991-12-01T00:00:00Z This paper considers additive conjoint measurement on subsets of Cartesian products containing “rank-ordered‘ n-tuples. Contrary to what has often been thought, additive conjoint measurement on subsets of Cartesian products has characteristics different from additive conjoint measurement on full Cartesian products. http://repub.eur.nl/res/pub/23214/ 1991-11-15T00:00:00Z http://repub.eur.nl/res/pub/23217/ 1991-11-01T00:00:00Z http://repub.eur.nl/res/pub/23213/ 1991-06-01T00:00:00Z It is shown that additive conjoint measurement theory can be considerably generalized and simplified in the equally spaced case. http://repub.eur.nl/res/pub/23218/ 1991-01-01T00:00:00Z In Wakker (1989, "Additive Representations of Preferences, A New Foundation of Decision Analysis"), a new foundation of decision analysis was given. The main tool was a way to derive comparisons of "tradeoffs" from ordinal preferences, with comparisons of tradeoffs revealing orderings of utility differences. These comparisons of tradeoffs underly the construction of standard sequences in conjoint measurement theory. The restrictive structural assumption (every approach has its restrictive structural assumption) was of a topological nature, requiring continuity. This paper adapts the main results of Wakker (1989) to the algebraic approach, where a solvability condition is required which is less restrictive than continuity. http://repub.eur.nl/res/pub/23220/ 1990-12-01T00:00:00Z Pessimism-independence is introduced to characterize pessimistic risk attitudes for the nonlinear-probability models of Schmeidler, Quiggin, and Yaari directly in terms of comonotonicity, rather than through additional conditions such as convexity of preferences. Pessimism-independence requires the mixture of an arbitrary good and a fixed act to be preferred to the mixture of a comonotonic bad act and the fixed act. Thus, more general than full-force independence, it does not exclude the additional (pessimistic) appreciation of the hedging involved in the mixture of a noncomonotonic bad and fixed act. More restrictive than comonotonic independence, it does exclude (optimistic) aversion of hedging. http://repub.eur.nl/res/pub/23225/ 1990-11-01T00:00:00Z http://repub.eur.nl/res/pub/23219/ 1990-09-28T00:00:00Z In Savage [41] a ‘behavioral foundation’ was given for subjective probabilities, to be used in the maximization of expected utility. This paper analogously gives a behavioral foundation for fuzzy measures, to be used in the maximization of ‘Choquet-expected utility’. This opens the way to empirical verification or falsification of fuzzy measures, and frees them of their ‘ad hoc’ character. http://repub.eur.nl/res/pub/23222/ 1990-09-01T00:00:00Z The aim of this paper is to convince the reader that Choquet-expected utility, as initiated by Schmeidler (1982, 1989) for decision making under uncertainty, when formulated for decision making under risk naturally leads to anticipated utility, as initiated by Quiggin/Yaari. Thus the two generalizations of expected utility in fact are one. http://repub.eur.nl/res/pub/23229/ 1989-12-01T00:00:00Z It is shown that assumptions about risk aversion, usually studied under the pre-supposition of expected utility maximization, have a surprising extra merit at an earlier stage of the measurement work: together with the sure-thing principle, these assumptions imply subjective expected utility maximization for monotonic continuous weak orders. http://repub.eur.nl/res/pub/23232/ 1989-09-01T00:00:00Z http://repub.eur.nl/res/pub/23226/ 1989-01-01T00:00:00Z A well-known theorem of Debreu about additive representations of preferences is applied in a non-additive context, to characterize continuous subjective expected utility maximization for the case where the probability measures may be non-additive. The approach of this paper does not need the assumption that lotteries with known (objective) probability distributions over consequences are available. http://repub.eur.nl/res/pub/23228/ 1989-01-01T00:00:00Z One of the issues in the impossibility theorem of Arrow is the difference between choice behaviour, as considered by Arrow in most of the illustrations for the conditions in his theorem, and binary relations as dealt with in Arrow's theorem. The relations between choice behaviour and binary relations are studied in revealed preference theory, a theory which originates from consumer demand theory. This paper presents a graph-theoretic approach to revealed preference theory. This is done by considering alternatives as vertices, and choice situations as arcs. By means of this method alternative proofs are obtained for some known results. In particular it is shown that many results from literature can be derived from what may be the main result from revealed preference theory, a theorem of Richter (1966). Next a duality approach is sketched, where vertices and arcs are interchanged as done in dual graph theory. Finally some results are given for non-transitive binary relations. For these there is an increasing interest because of Arrow's theorem. http://repub.eur.nl/res/pub/23231/ 1989-01-01T00:00:00Z The idea of expected utility, to transform payments into their utilities before calculating expectation, traces back at least to Bernoulli (1738). It is a very natural idea to transform, analogously, probabilities. This paper gives heuristic visual arguments to show that the, at first sight, natural way to do this, at second thought seems questionable. At second thought a sound and natural way is the way indicated by Quiggin (1982) and Yaari (1987a). http://repub.eur.nl/res/pub/23236/ 1988-12-01T00:00:00Z It is proved that, under a nontriviality assumption, an additive function on a Cartesian product of connected topological spaces is continuous, whenever the preference relation, represented by this function, is continuous. The result is used to generalize a theorem of Debreu ((1960). Mathematical methods in the social sciences (pp. 16–26). Stanford: Stanford Univ. Press) on additive representations and to argue that the algebraic approach of KLST to additive conjoint measurement is preferable to the more customary topological approach. Applications to the representation of strength of preference relations and to the characterization of subjective expected utility maximization are given. http://repub.eur.nl/res/pub/23238/ 1988-12-01T00:00:00Z way is indicated to derive, from a preference relation on a Cartesian product, strength of preference relations on the coordinate sets. These strengths of preference relations are then used to reformulate several well-known properties of preference relations, and make their meaning more transparent. A new result for dynamic contexts is given. http://repub.eur.nl/res/pub/23234/ 1988-02-01T00:00:00Z In Debreu (1954, 1959) some classical results were provided for consumer theory. Necessary and sufficient conditions were given for the existence of (con- tinuous) utility functions to represent preference relations of consumers. Further results are given in Bowen (1968), Jaffray (1975), Richter (1980), and Chateauneuf (1985). http://repub.eur.nl/res/pub/23241/ 1987-12-01T00:00:00Z For the expected utility model with state dependent utilities, Karni, Schmeidler and Vind (1983) have shown how to recover uniquely the involved subjective probabilities if the preferences, contingent on a hypothetical probability distribution over the state space, are known. This they do for consequence spaces, consisting of lotteries on sets of prizes. We adapt their work to consequence spaces that are connected topological spaces, without using lotteries on them. E.g. our consequences may be money, or commodity bundles. http://repub.eur.nl/res/pub/23240/ 1987-01-01T00:00:00Z It is shown that a convex function, defined on an arbitrary, possibly finite, subset of a linear space, can be extended to the whole space. An application to decision making under risk is given. http://repub.eur.nl/res/pub/23244/ 1987-01-01T00:00:00Z http://repub.eur.nl/res/pub/23246/ 1986-01-01T00:00:00Z Building on previous work of A. Camacho, we give necessary and sufficient conditions for the existence of a cardinal utility function to represent, through summation, a preference relation on sequences of alternatives. http://repub.eur.nl/res/pub/23260/ 1984-03-01T00:00:00Z A representation theorem for binary relations on n is derived. It is interpreted in the context of decision making under uncertainty. There we consider the existence of a subjective expected utility model to represent a preference relation of a person on the set of bets for money on a finite state space. The theorem shows that, for this model to exist, it is not only necessary (as has often been observed), but it also is sufficient, that the appreciation for money of the person has a cardinal character, independent of the state of nature. This condition of cardinal appreciation is simple and thus easily testable in experiments. Also it may be of help in relating the neo-classical economic interpretation of cardinal utility to the von Neumann-Morgenstern interpretation. http://repub.eur.nl/res/pub/11817/ 1983-01-01T00:00:00Z Bargaining solutions are considered which have the following four properties: individual rationality, Pareto optimality, independence of equivalent utility representations, and independence of irrelevant alternatives. A main result of this paper is a simple proof of the fact that all such bargaining solutions are risk sensitive. Further a description is given of all bargaining solutions satisfying the four mentioned properties. Finally, a continuous bargaining solution, satisfying the first three properties, is given which is not risk sensitive. http://repub.eur.nl/res/pub/23261/ 1981-05-01T00:00:00Z It is proved that fine and tight comparative probability structures (where the set of events is assumed to be an algebra, not necessarily a σ-algebra) have agreeing probability measures. Although this was often claimed in the literature, all proofs the author encountered are not valid for the general case, but only for σ-algebras. Here the proof of Niiniluoto (1972) is supplemented. Furthermore an example is presented that reveals many misunderstandings in the literature. At the end a necessary and sufficient condition is given for comparative probability structures to have an almost agreeing probability measure