Signed orders in linear and nonlinear utility theory

Self-reflecting signed orders on a set A and its anti-set A ^* were introduced previously as a way to account for negative as well as positive feelings about the inclusion of items in A in potential subsets of choice. The present paper extends the notion of signed orders to lotteries on A A ^*, describes reflection axioms for the lottery context, and shows how these axioms simplify utility representations for preference between lotteries. The simplified representations are then used to guide procedures for extending preferences from A A ^* and its lotteries to preferences between subsets of items.     

Axiomatization of a Preference for Most Probable Winner

In binary choice between discrete outcome lotteries, an individual may prefer lottery L₁ to lottery L₂ when the probability that L₁ delivers a better outcome than L₂ is higher than the probability that L₂ delivers a better outcome than L₁. Such a preference can be rationalized by three standard axioms (solvability, convexity and symmetry) and one less standard axiom (a fanning-in). A preference for the most probable winner can be represented by a skew-symmetric bilinear utility function. Such a utility function has the structure of a regret theory when lottery outcomes are perceived as ordinal and the assumption of regret aversion is replaced with a preference for a win. The empirical evidence supporting the proposed system of axioms is discussed.     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

Order extensions,budget correspondences,and rational choice

This article is concerned with extensions of a continuous ordering R on a set X to a subset P(X) of the power set of X. The underlying topology will be the Hausdorff metric topology. We will see that continuous extensions of R do not require that P(X) contain every nonempty finite subset of X. Therefore, the analysis can be applied to consumer theory and inverse choice functions. In analogy to these functions budget correspondences are established which relate alternatives x with certain subsets of X, according to the extended ordering.     

Partial information on decisional competences and the desirability of the expert rule in uncertain dichotomous choice situations

The paper analyzes the expert resolution problem by employing extended versions of the uncertain dichotomous choice model. The main purpose of this study is to illustrate how the analysis of optimal decision rules can be carried out while dispensing with the common restrictive assumption of full information regarding individual decisional competences. In contrast to most previous studies in this field we here evaluate the expert rule under alternative assumptions regarding the available partial information on judgmental competences rather than compare it in an ad hoc manner to some common alternative rules, such as simple majority rule. A fuller optimality analysis allowing the evaluation of all relevant decision rules, and not merely the expert rule, is attempted for a five-member panel of experts assuming a uniform distribution of individual competences. For three-member groups the optimality issue is resolved by assuming no information on individual competences and interpreting the expert rule as an even-chance lottery on skills.     

Independent social choice correspondences

A fixed agenda social choice correspondence on outcome set X maps each profile of individual preferences into a nonempty subset of X. If satisfies an analogue of Arrow's independence of irrelevant alternatives condition, then either the range of contains exactly two alternatives, or else there is at most one individual whose preferences have any bearing on . This is the case even if is not defined for any proper subset of X.     

Changes in preference

A basic framework for studies of changes in preference is introduced, and four types of changes in preference are identified.Revision by a sentence such as A is better thanB means that a preference forA overB is acquired. The result ofcontraction by A is better thanB is that the subject no longer holdsA to be better thanB. Inaddition andsubtraction, an alternative is added to, or subtracted from, respectively, the set of alternatives that are under consideration. Formal models of these four types of change are introduced, and they are shown to satisfy plausible postulates for rational changes in preferences.     

Lottery Decisions and Probability Weighting Function

This article analyzes the implications of basic lottery tests for the probability weighting function w(p). We first show that the w(p) function with one argument cannot accommodate three basic tests of lottery choice. We also discuss in detail the links between the w(p) function with one argument and the preference reversal paradox. In the last section we propose an evaluation function that accommodates the restrictions imposed by the tests and we provide an example of such function.     

Mixed risk aversion and preference for risk disaggregation: a story of moments

In a recent article entitled “Putting Risk in its Proper Place,” Eeckhoudt and Schlesinger (2006) established a theorem linking the sign of the n-th derivative of an agent’s utility function to her preferences among pairs of simple lotteries. We characterize these lotteries and show that, in a given pair, they only differ by their moments of order greater than or equal to n. When the n-th derivative of the utility function is positive (negative) and n is odd (even), the agent prefers a lottery with higher (lower) n + 2p-th moments for p belonging to the set of positive integers. This result links the preference for disaggregation of risks across states of nature to the complete structure of moments preferred by mixed risk averse agents. It can be viewed as a generalization of a proposition appearing in Ekern (1980) which focused only on the differences in the n-th moments.     

Lottery qualities

The aim of this paper is to propose a model of decision-making for lotteries. Lottery qualities are the key concepts of the theory. Qualities allow the derivation of optimal decision-making processes and are taken explicitly into account for lottery evaluation. Our contribution explains the major violations of the expected utility theory for decisions on two-point lotteries and shows the necessity of giving explicit consideration to lottery qualities. Judged certainty equivalent and choice certainty equivalent concepts are discussed in detail along with the comparison of lotteries. Examples are provided by using different test results in the literature. JEL Classification D81     

The Budget-Voting Paradox

The budget-voting paradox states that, when social alternatives are proper subsets of a finite set of decisions, choosing decision-wise according to the majority rule may select an alternative that is covered in the majority tournament among alternatives. Individual preferences are defined on single decisions, and are extended to preferences over the alternative set by means of a preference extension rule. We prove the existence of the paradox for any rank-based, monotone, and independent extension rule.      

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.     

Are Event-Splitting Effects Actually Boundary Effects?

An event-splitting effect (ESE) occurs when an event, which yields a positive outcome in one lottery but zero under another, is separated into two sub-events and this increases the relative attractiveness of the former lottery. ESEs are attributed to either (1) a preference for more positive outcomes in lotteries, or (2) an aversion to more zero outcomes. An experiment is reported which replicates ESEs whilst controlling for explanation (2). This favours explanation (1), but a lack of generality in replication suggests previous ESEs to be partially due to (2). Implications are discussed for a related choice phenomena, decision theory, and real-world decision-making.     

Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

Echelons in Incomplete Relations

An efficient method of value assessment of a set of exchangeable alternatives A = {a ₁,a ₂, ,a _n} is presented. It particularly applies to situations where certain preferences may be easily evaluated or are already known, while other binary comparisons may not at once be available. Further applications are to ranking partial tournaments and the emergence and the characterisation of organisational hierarchy. By sequentially performing transitively efficient assessments of uncompared pairs, an initial weakly acyclical preference structure in A is transformed into an ordering of A in echelons. We call these nicely surveyable preference structures echelon orders. Theoretical properties of echelon orders are investigated, including a characterisation and a numerical representation.     

Revealed desirability: a novel instrument for social welfare

The note puts forward the idea of revealed desirability, a novel instrument, which like revealed preference is observable from choice and important for individual and social welfare. We provide the axiomatic underlying individual’s choice model, preliminary experimental results that support the idea, and an appealing allocation rule that uses the revealed desirability information along with the revealed-preference information.

     

How can an expert system help in choosing the optimal decision?

We introduce a rationality principle for a preference relation on an arbitrary set of lotteries. Such a principle is a necessary and sufficient condition for the existence of an expected utility agreeing with . The same principle also guarantees a rational extension of the preference relation to any larger set of lotteries. When the extended relation is unique with respect to the alternatives under consideration, the decision maker does not need a numerical evaluation in order to make a choice. Such a rationality condition needs little information in order to be applied, and its verification amounts to solving a linear system.The present research is supported by the Research Contract of CNR (Research National Council) 1989 and 1990 Decision Models under uncertainty and risk, for expert systems with incomplete and revisable information.     

Preference extension rules for ranking sets of alternatives with a fixed cardinality

This paper analyzes the problem of deriving a ranking of fixed-cardinality subsets of a universal set from a given ranking of the elements of this universal set. Only subsets with a given number of elements are being ranked, which is where the approach in this paper differs from the literature on extension rules that establish preference relations on the power set of the universal set. Common examples for areas where such preferences on subsets with a fixed cardinality are needed are elections of committees of a given size, many-to-one matchings, and decision problems under ignorance. The main result of the paper is a characterization of a class oflexicographic rank-ordered rules by means of two axioms, namely, aresponsiveness condition used in the matching literature and a well-knownneutrality requirement which ensures that the names of the alternatives are irrelevant for the ranking of the sets.     

Choosers as extension axioms

We consider the extension of a (strict) preference over a set to its power set. Elements of the power set are non-resolute outcomes. The final outcome is determined by an “(external) chooser” which is a resolute choice function. The individual whose preference is under consideration confronts a set of resolute choice functions which reflects the possible behaviors of the chooser. Every such set naturally induces an extension axiom (i.e., a rule that determines how an individual with a given preference over alternatives is required to rank certain sets). Our model allows to revisit various extension axioms of the literature. Interestingly, the Gärdenfors (1976) and Kelly (1977) principles are singled-out as the only two extension axioms compatible with the non-resolute outcome interpretation.