Rank- and sign-dependent linear utility models for finite first-order gambles

Finite first-order gambles are axiomatized. The representation combines features of prospect and rank-dependent theories. What is novel are distinctions between gains and losses and the inclusion of a binary operation of joint receipt. In addition to many of the usual structural and rationality axioms, joint receipt forms an ordered concatenation structure with special features for gains and losses. Pfanzagl's (1959) consistency principle is assumed for gains and losses separately. The nonrational assumption is that a gamble of gains and losses is indifferent to the joint receipt of its gains pitted against the status quo and of its losses against the status quo.Reprints may be obtained from either author. Luce's work was supported, in part, by the National Science Foundation grant IRI-8996149 to the University of California, Irvine.     

Reconciliation with the Utility of Chance by Elaborated Outcomes Destroys the Axiomatic Basis of Expected Utility Theory

Expected utility theory does not directly deal with the utility of chance. It has been suggested in the literature (Samuelson, 1952, Markowitz, 1959) that this can be remedied by an approach which explicitly models the emotional consequences which give rise to the utility of chance. We refer to this as the elaborated outcomes approach. It is argued that the elaborated outcomes approach destroys the possibility of deriving a representation theorem based on the usual axioms of expected utility theory. This is shown with the help of an example due to Markowitz. It turns out that the space of conceivable lotteries over elaborated outcomes is too narrow to permit the application of the axioms. Moreover it is shown that a representation theorem does not hold for the example.     

Ranked-Weighted Utilities and Qualitative Convolution

For gambles—non-numerical consequences attached to uncertain chance events—analogues are proposed for the sum of independent random variables and their convolution. Joint receipt of gambles is the analogue of the sum of random variables. Because it has no unique expansion as a first-order gamble analogous to convolution, a definition of qualitative convolution is proposed. Assuming ranked, weighted-utility representations (RWU) over gains (and, separately, over losses, but not mixtures of both), conditions are given for the equivalence of joint receipt, qualitative convolution, and a utility expression like expected value. As background, some properties of RWU are developed.     

The Domain and Interpretation of Utility Functions: An Exploration

This paper proposes an exploration of the methodology of utility functions that distinguishes interpretation from representation. While representation univocally assigns numbers to the entities of the domain of utility functions, interpretation relates these entities with empirically observable objects of choice. This allows us to make explicit the standard interpretation of utility functions which assumes that two objects have the same utility if and only if the individual is indifferent among them. We explore the underlying assumptions of such an hypothesis and propose a non-standard interpretation according to which objects of choice have a well-defined utility although individuals may vary in the way they treat these objects in a specific context. We provide examples of such a methodological approach that may explain some reversal of preferences and suggest possible mathematical formulations for further research. This revised version was published online in June 2006 with corrections to the Cover Date.     

INCREASING INCREMENT GENERALIZATIONS OF RANK-DEPENDENT THEORIES

Empirical evidence from both utility and psychophysical experiments suggests that people respond quite differently—perhaps discontinuously—to stimulus pairs when one consequence or signal is set to `zero.' Such stimuli are called unitary. The author's earlier theories assumed otherwise. In particular, the key property of segregation relating gambles and joint receipts (or presentations) involves unitary stimuli. Also, the representation of unitary stimuli was assumed to be separable (i.e., multiplicative). The theories developed here do not invoke separability. Four general cases based on two distinctions are explored. The first distinction is between commutative joint receipts, which are relevant to utility, and the non-commutative ones, which are relevant to psychophysics. The second distinction concerns how stimuli of the form (x, C; y) and the operation of joint receipt are linked: by segregation, which mixes stimuli and unitary ones, and by distributivity, which does not involve any unitary stimuli. A class of representations more general than rank-dependent utility (RDU) is found in which monotonic functions of increments U(x)-U(y), where U is an order preseving representation of gambles, and joint receipt play a role. This form and its natural generalization to gambles with n > 2 consequences, which is also axiomatized, appear to encompass models of configural weights and decision affect. When joint receipts are not commutative, somewhat similar representations of stimuli arise, and joint receipts are shown to have a conjoint additive representation and in some cases a constant bias independent of signal intensity is predicted.     

Anxiety and Decision Making with Delayed Resolution of Uncertainty

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.     

Expected discounted utility

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

     

A note on deriving rank-dependent utility using additive joint receipts

Luce and Fishburn (1991) derived a general rank-dependent utility model using an operation ⊕ of joint receipt. Their argument rested on an empirically supported property (now) calledsegregation and on the assumption that utility is additive over ⊕. This note generalizes that conclusion to the case where utility need not be additive over ⊕, but rather is of a more general form, which they derived but did not use in their article. Tversky and Kahneman (1992), conjecturing that the joint receipt of two sums of money is simply their sum, criticized that original model because ⊕=+ together with additive utility implies the unacceptable conclusion that the utility of money is proportional to money. In the present generalized theory, if ⊕=+, utility is a negative exponential function of money rather than proportional. Similar results hold for losses. The case of mixed gains and losses is less well understood.     

Reanalysis of the Chechile-Cooke Experiment: Correcting for Mismatched Gambles

Chechile and Cooke (1997) experimentally tested a broad class of utility models subsumed under the Miyamoto (1988, 1992) generic utility theory. The Chechile and Cooke study required participants to match on each trial, a fully specified reference gamble to a partially specified comparison gamble by adjusting the probability of a win on the comparison gamble. The Chechile and Cooke experiment, however, contained a subset of trials which were intrinsically unmatchable. In such cases, the participants could only give an extreme probability (either 0 or 1). In this paper, those extreme trials were omitted and the results from the experiment reanalyzed. Despite the mismatch problem, the conclusions of the Chechile and Cooke experiment were again supported. For nine implementations of generic utility there is model failure due to the systematic variation of a parameter that should be a constant.     

The law of large numbers and the attractiveness of compound gambles

A compound of many independent replicas of a gamble may be chosen by a risk averter even if the single gamble would be rejected given any initial wealth level. Samuelson has shown that such choices are impossible if the expected utility theory of preference is adopted. But they are consistent with more general theories of preference. Consequently, the intuition suggested by the law of large numbers can be supported.     

Interpersonal comparisons of utility for 2 of 3 types of people

This article argues that there is a natural solution to carry out interpersonal comparisons of utility when the theory of gambles is supplemented with a group operation of joint receipts. If so, three types of people can exist, and the two types having multiplicative representations of joint receipt have, in contrast to most utility theories, absolute scales of utility. This makes possible, at least in principle, meaningful interpersonal comparisons of utility with desirable properties, thus resolving a long standing philosophical problem and having potentially important implications in economics. Two behavioral criteria are given for the three classes of people. At this point the relative class sizes are unknown.     

Evaluating Time Streams of Income: Discounting What?

For decisions whose consequences accrue over time, there are several possible techniques to compute total utility. One is to discount utilities of future consequences at some appropriate rate. The second is to discount per-period certainty equivalents. And the third is to compute net present values (NPVs) of various possible streams and to then apply the utility function to these net present values. We find that the best approach is to first compute NPVs of various possible income streams and then take the utility of such NPVs. We show the drawbacks of other alternative models of evaluating income streams. The article discusses the advantages of the power and logarithmic forms in the modeling of time preference. These are the only forms for which utility of income and utility of consumption are strategically equivalent. Further, these forms permit the flexibility in the choice of a time period (e.g., monthly or quarterly) without modifying the utility function, thus simplifying analysis.     

Utility optimization when the utility function is virtually unknown

In this paper a surrogate to traditional utility optimization is presented. This approach is based on multiple criteria decision making techniques through a theorem which connects utility function optimization with compromise programming. Apart from common assumptions in the literature the only specific assumption underlying the approach seems realistic, and is markedly related to traditional analysis.     

A tractable method to measure utility and loss aversion under prospect theory

This paper provides an efficient method to measure utility under prospect theory. Our method minimizes both the number of elicitations required to measure utility and the cognitive burden for subjects, being based on the elicitation of certainty equivalents for two-outcome prospects. We applied our method in an experiment and were able to replicate the main findings on prospect theory, suggesting that our method measures what it is intended to. Our data confirmed empirically that risk seeking and concave utility can coincide under prospect theory. Utility did not depend on the probability used in the elicitation, which offers support for the validity of prospect theory.

Testing the Intransitivity Explanation of the Allais Paradox

This paper uses a two-dimensional version of a standard common consequence experiment to test the intransitivity explanation of Allais-paradox-type violations of expected utility theory. We compare the common consequence effect of two choice problems differing only with respect to whether alternatives are statistically correlated or independent. We framed the experiment so that intransitive preferences could explain violating behavior when alternatives are independent, but not when they are correlated. We found the same pattern of violation in the two cases. This is evidence against intransitivity as an explanation of the Allais Paradox. The question whether violations of expected utility are mainly due to intransitivity or to violation of independence is important since it is exactly on this issue the main new decision theories differ.     

Rank-dependent,subjective expected-utility representations

Gambles are recursively generated from pure payoffs, events, and other gambles, and a preference order over them is assumed. Weighted average utility representations are studied that are strictly increasing in each payoff and for which the weights depend both on the events underlying the gamble and the preference ranking over the several component payoffs. Basically two results are derived: a characterization of monotonicity in terms of the weights, and an axiomatization of the representation. The latter rests on two important conditions: a decomposition of gambles into binary ones and a necessary commutativity condition on events in a particular class of binary gambles. A number of unsolved problems are cited.     

Is choice the correct primitive? On using certainty equivalents and reference levels to predict choices among gambles

Choice is viewed as a derived, not a primitive, concept. Individual gambles are assigned subjective certainty equivalents (CE₁); the choice setX has an associated reference level [RL(X)] based on the CE₁S of its members; the outcomes of each gamble are recoded as deviations from the RL(X); and new CE₂S are constructed. The gamble having the largest CE₂ is chosen. The CEs are described by the rank-and sign-dependent theory of Luce (1992b). The concept of RL is studied axiomatically. The model predicts many behavioral anomalies and is tested with data sets of Mellers, Chang, Birnbaum, and Ordóñez (1992).     

Utility of Gambling when Events are Valued: an Application of Inset Entropy

The present theory leads to a set of subjective weights such that the utility of an uncertain alternative (gamble) is partitioned into three terms involving those weights—a conventional subjectively weighted utility function over pure consequences, a subjectively weighted value function over events, and a subjectively weighted function of the subjective weights. Under several assumptions, this becomes one of several standard utility representations, plus a weighted value function over events, plus an entropy term of the weights. In the finitely additive case, the latter is the Shannon entropy; in all other cases it is entropy of degree not 1. The primary mathematical tool is the theory of inset entropy. The work of Luce and Marley was supported in part by National Science Foundation grant SES-0452756 to the University of California, Irvine, and by the Natural Sciences and Engineering Research Council (NSERC) of Canada Discovery Grant 8124 to the University of Victoria for Marley. That of Ng was supported in part by the NSERC of Canada Discovery Grant 8212. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.