# Peter P. Wakker's research while affiliated with Erasmus University Rotterdam and other places

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## Publications (207)

Chung, Glimcher, and Tymula (2019) observed both consumers’ choices over commodity bundles and choices under risk. They assumed a cardinal riskless utility function V representing consumer choices and a cardinal risky utility function U.The two were inconsistent. This note shows that the two functions can be reconciled if we assume that V is ordina...

Whereas original prospect theory was introduced over 40 years ago, its formula for multi-outcome prospects has never yet been published, resulting in many misunderstandings. This note provides that formula.

We introduce belief hedges, i.e., sets of events whose uncertain subjective beliefs neutralize each other. Belief hedges allow us to measure ambiguity attitudes without knowing those subjective beliefs. They lead to improved ambiguity indexes that are valid under all popular ambiguity theories. Our indexes can be applied to real-world problems and...

This paper analyzes concave and convex utility and probability distortion functions for decision under risk (law-invariant functionals). We characterize concave utility for virtually all existing models, and concave/convex probability distortion functions for rank-dependent utility and prospect theory in complete generality, through an appealing an...

This paper introduces the Prince incentive system for measuring preferences. Prince combines the tractability of direct matching, allowing for the precise and direct elicitation of indifference values, with the clarity and validity of choice lists. It makes incentive compatibility completely transparent to subjects, avoiding the opaqueness of the B...

Complementary symmetry was derived before under particular theories, and used to test those. Progressively general results were published. This paper proves the condition in full generality, providing a one-line proof, and shedding new light on its empirical implications.

This paper examines the difference between strategic ambiguity as in game theory and ambiguity arising in individual decisions. We identify a new, non-strategic component underlying all strategic ambiguities, called social ambiguity. We recommend controlling for it to better identify strategic causes. Thus, we shed new light on Bohnet and Zeckhause...

Savage's foundation of expected utility is considered to be the most convincing justification of Bayesian expected utility and the crowning glory of decision theory. It combines exceptionally appealing axioms with deep mathematics. Despite the wide influence and deep respect that Savage received in economics and statistics, virtually no one touched...

We present a theoretical model of Rabin’s famous calibration paradox that resolves confusions in the literature and that makes it possible to identify the causes of the paradox. Using suitable experimental stimuli, we show that the paradox truly violates expected utility and that it is caused by reference dependence. Rabin already showed that utili...

This paper shows that convexity of preference has stronger implications for weighted utility models than had been known hitherto, both for utility and for weighting functions. Our main theorem derives concave utility from convexity of preference on the two-dimensional comonotonic cone, without presupposing continuity. We then show that this, seemin...

Measurements of ambiguity attitudes have so far focused on artificial events, where (subjective) beliefs can be derived from symmetry of events and can be then controlled for. For natural events as relevant in applications, such a symmetry and corresponding control are usually absent, precluding traditional measurement methods. This paper introduce...

Decisions to trust in strategic situations involve ambiguity (unknown probabilities). Despite many theoretical studies on ambiguity in game theory, empirical studies have lagged behind due to a lack of measurement methods, where separating ambiguity attitudes from beliefs is crucial. Baillon et al. (Econometrica, 2018b) introduced a method that all...

The Anscombe-Aumann (AA) model, originally introduced to give a normative basis to expected utility, is nowadays mostly used for another purpose: to analyze deviations from expected utility due to ambiguity (unknown probabilities). The AA model makes two ancillary assumptions that do not refer to ambiguity: expected utility for risk and backward in...

Ellsberg and others suggested that decision under ambiguity is a rich empirical domain with many phenomena to be investigated beyond the Ellsberg urns. We provide a systematic empirical investigation of this richness by varying the uncertain events, the outcomes, and combinations of both. Although ambiguity aversion is prevailing, we also find syst...

This paper investigates how letting people predict others’ choices under risk affects subsequent own choices. We find an improvement of strong rationality (risk neutrality) for losses in own choices, but no such improvement for gains. There is no improvement of weak rationality (avoiding preference reversals). Overall, risk aversion in own choices...

Nash is famous for many inventions, but it is less known that he, simultaneously with Marschak, also was the first to axiomatize expected utility for risk. In particular, these authors were the first to state the independence condition, a condition that should have been but was not stated by von Neumann and Morgenstern. Marschak’s paper resulted fr...

We introduce a new method to measure the temporal discounting of money. Unlike preceding methods, our method requires neither knowledge nor measurement of utility. It is easier to implement, clearer to subjects, and requires fewer measurements than existing methods.

This paper investigates the rationality of group decisions versus individual decisions under risk. We study two group decision rules, majority and unanimity, in stochastic dominance and Allais paradox tasks. We distinguish communication effects (the effects of group discussions and interactions) from aggregation effects (mere impact of the voting p...

Case-based decision theory (CBDT) provided a new way of revealing preferences, with decisions under uncertainty determined by similarities with cases in memory. This paper introduces a method to measure CBDT that requires no commitment to parametric families and that relates directly to decisions. Thus, CBDT becomes directly observable and can be u...

We introduce a new type of preference condition for intertemporal choice, which requires present values to be independent of various other variables. The new conditions are more concise and more transparent than traditional ones. They are directly related to applications because present values are widely used tools in intertemporal choice. Our cond...

Using a theorem showing that matching probabilities of ambiguous events can capture ambiguity attitudes, we introduce a tractable method for measuring ambiguity attitudes and apply it in a large representative sample. In addition to ambiguity aversion, we confirm an ambiguity component recently found in laboratory studies: a-insensitivity, the tend...

This paper presents the metric-frequency calculator (MF Calculator), an online application to analyze similarity. The MF Calculator implements a metric-frequency similarity algorithm for the quantitative assessment of similarity in ill-structured data sets. It is widely applicable as it can be used with nominal, ordinal, or interval data when there...

This paper presents the Metric-Frequency Calculator (MF Calculator), an online ap-plication to analyze similarity. The MF Calculator implements the MF similarity al-gorithm developed by Sales and Wakker (2009) for the quantitative assessment of sim-ilarity in ill-structured data sets. It is widely applicable as it can be used when there is little p...

In their famous 1982 paper in this Journal, Loomes and Sugden introduced regret theory. Now, more than 30 years later, the case for the historical importance of this contribution can be made.

Uncertainty pervades most aspects of life. From selecting a new technology to choosing a career, decision makers rarely know in advance the exact outcomes of their decisions. Whereas the consequences of decisions in standard decision theory are explicitly described (the decision from description (DFD) paradigm), the consequences of decisions in the...

A central question in many debates on paternalism is whether a decision analyst can ever go against the stated preference of a client, even if merely intending to improve the decisions for the client. Using four gedanken-experiments, this paper shows that this central question, so cleverly and aptly avoided by libertarian paternalism (nudge), canno...

Prospect theory is the most popular theory for predicting decisions under risk. This paper investigates its predictive power for decisions under ambiguity, using its specification through the source method. We find that it outperforms its most popular alternatives, including subjective expected utility, Choquet expected utility, and three multiple...

This paper provides necessary and sufficient preference conditions for average utility maximization over sequences of variable length. We obtain full generality by using a new algebraic technique that exploits the richness structure naturally provided by the variable length of the sequences. Thus we generalize many preceding results in the literatu...

Doyle’s (2013) theoretical survey of discount functions criticizes two parametric families abbreviated as CRDI and CADI families. We show that Doyle’s criticisms are based on a mathematical mistake and are incorrect.

Published as "Beware of black swans: Taking stock of the description–experience gap in decision under uncertainty," Marketing Letters, Springer, vol. 25(3), pages 269-280, September.
Uncertainty pervades most aspects of life. From selecting a new technology to choosing a career, decision makers often ignore the outcomes of their decisions. In the...

Behavioral conditions such as compound invariance for risky choice and constant decreasing relative impatience for intertemporal choice have surprising implications for the underlying decision model. They imply a multiplicative separability of outcomes and either probability or time. Hence the underlying model must be prospect theory or discounted...

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 finit...

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...

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...

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...

In a large representative sample, we measure ambiguity attitudes and investigate their relation with stock market participation. Our tractable measurement of the general population’s ambiguity attitudes is made possible by a simplification of the recently introduced source method. In addition to ambiguity aversion, the results from our representati...

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...

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...

Several contributions in this book present axiomatizations of decision models, and of special forms thereof. This chapter explains the general usefulness of such axiomatizations, and reviews the basic axiomatizations for static individual decisions under uncertainty. It will demonstrate that David Schmeidler’s contributions to this field were cruci...

This paper discusses Jean-Yves Jaffray’s ideas on ambiguity and the views underlying his ideas. His models, developed 20years
ago, provide the most tractable separation of risk attitudes, ambiguity attitudes, and ambiguity beliefs available in the
literature today.
KeywordsAmbiguity–Total absence of information–Belief functions

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 S...

Preference foundations give necessary and sufficient conditions for a decision model, stated directly in terms of the empirical
primitive: the preference relation. For the most popular descriptive model for decision making under risk and uncertainty
today, prospect theory, preference foundations have as yet been provided only for prospects taking f...

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. Th...

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. I...

We investigate the effect of ambiguity attitudes on households’ portfolio decisions. We elicit ambiguity attitudes in a large representative survey using real rewards and the recently introduced source method. This method provides refined yet tractable measurements of not only the well known aversion to ambiguity, but also a new component, ambiguit...

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.

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...

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....

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 quant...

Prospect Theory: For Risk and Ambiguity provides the first comprehensive and accessible textbook treatment of the way decisions are made both when we have the statistical probabilities associated with uncertain future events (risk) and when we lack them (ambiguity). The book presents models, primarily prospect theory, that are both tractable and ps...

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 benefit...

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 decreas...

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 metr...

a b s t r a c t 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 field...

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 audie...

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 r...

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 mea...

a b s t r a c t 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 u...

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...

Machina (2004) introduced the notion of an 'almost objective' event in a continuous state space—high frequency events in a subjective setting such as 'the realization of the n th decimal place of a stock index.' Payoffs on such events intuitively appear as objective lotteries in the sense that decision makers should not prefer to place bets on any...

This article deals with individual decision making under uncertainty (unknown probabilities). Risk (known probabilities) is not treated as a separate case, but as a sub-case of uncertainty. Many results from risk naturally extend to uncertainty. The Allais paradox, commonly applied to risk, also reveals empirical deficiencies of expected utility fo...

In the United States, religious attendance rises sharply with education across individuals, but religious attendance declines sharply with education across denominations. This puzzle is explained if education both increases the returns to social connection and reduces the extent of religious belief, and if beliefs are closely linked to denomination...

This paper presents a field study into the effects of statistical information concerning risks on willingness to take insurance, with special attention being paid to the usefulness of these effects for the clients (the insured). Unlike many academic studies, we were able to use in-depth individual interviews of a large representative sample from th...

The introduction of the euro gave a unique opportunity to empirically disentangle two components of utility: intrinsic value,
a rational component central in economics, and the numerosity effect (going by numbers while ignoring units), a descriptive
and irrational component central in prospect theory and underlying the money illusion. We measured r...

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...

This paper extends de Finetti’s betting-odds method for assessing subjective beliefs to ambiguous events. Thus, a tractable
manner for measuring decision weights under ambiguity is obtained. De Finetti’s method is so transparent that decision makers
can evaluate the relevant tradeoffs in complex situations. The resulting data can easily be analyzed...

Abstract This paper proposes a new method,to measure the utility of life duration. A major advantage of our method, as opposed to existing methods for the measurement of the utility of life

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 pr...

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...