On the Representation of Beliefs by Probabilities

This paper explores two axiomatic structures of subjective expected utility assuming a finite state-space and state-dependent, connected, topological outcome-spaces. Building on the work of Karni and Schmeidler (1981) the analytical framework includes, in addition to the preference relation on acts, introspective preferences on hypothetical lotteries that are linked to the preference relation on acts by consistency axioms. The two models accommodate state-dependent preferences and yield subjective probabilities that correctly represent the decision-maker's beliefs. State-independent preferences are a special case.     

Preference aggregation and statistical estimation

This paper deals with various connections that are found to exist between statistical estimation methods for decision-making and rules of group choice in the social choice area. Initially the aggregation of individual opinions is formulated as a pattern recognition problem; firstly it is shown that individual preferences lead to a natural representation in terms of binary patterns. Then we proceed to show how the search for a group preference pattern can be conducted by classifying the input preference patterns into various pattern classes and using the resulting classification boundaries to define the area of mutual agreement over some of the available alternatives. This leads to a decision-theoretic problem which consists in defining a decision rule (for classification) that is least likely to lead to misrecognition of arbitrary preference patterns. A maximum likelihood solution is obtained and compared with some well-known rules of group decision-making. Other solutions are also possible, on the basis of different optimality criteria, and their social choice interpretation is suggested. Finally, a method using Coleman's linear model for attributes is applied to yield group decision rules by feature weighting of election issues.     

A characterization of the maximin rule in the context of voting

In a voting context, when the preferences of voters are described by linear orderings over a finite set of alternatives, the Maximin rule orders the alternatives according to their minimal rank in the voters’ preferences. It is equivalent to the Fallback bargaining process described by Brams and Kilgour (Group Decision and Negotiation 10:287–316, 2001). This article proposes a characterization of the Maximin rule as a social welfare function (SWF) based upon five conditions: Neutrality, Duplication, Unanimity, Top Invariance, and Weak Separability. In a similar way, we obtain a characterization for the Maximax SWF by using Bottom Invariance instead of Top Invariance. Then, these results are compared to the axiomatic characterizations of two famous scoring rules, the Plurality rule and the Antiplurality rule.     

Explaining robust additive utility models by sequences of preference swaps

As decision-aiding tools become more popular everyday—but at the same time more sophisticated—it is of utmost importance to develop their explanatory capabilities. Some decisions require careful explanations, which can be challenging to provide when the underlying mathematical model is complex. This is the case when recommendations are based on incomplete expression of preferences, as the decision-aiding tool has to infer despite this scarcity of information. This step is key in the process but hardly intelligible for the user. The robust additive utility model is a necessary preference relation which makes minimal assumptions, at the price of handling a collection of compatible utility functions, virtually impossible to exhibit to the user. This strength for the model is a challenge for the explanation. In this paper, we come up with an explanation engine based on sequences of preference swaps, that is, pairwise comparison of alternatives. The intuition is to confront the decision maker with “elementary” comparisons, thus building incremental explanations. Elementary here means that alternatives compared may only differ on two criteria. Technically, our explanation engine exploits some properties of the necessary preference relation that we unveil in the paper. Equipped with this, we explore the issues of the existence and length of the resulting sequences. We show in particular that in the general case, no bound can be given on the length of explanations, but that in binary domains, the sequences remain short.     

Indecisiveness aversion and preference for commitment

We present an axiomatic model of preferences over menus that is motivated by three assumptions. First, the decision maker is uncertain ex ante (i.e., at the time of choosing a menu) about her ex post (i.e., at the time of choosing an option within her chosen menu) preferences over options, and she anticipates that this subjective uncertainty will not resolve before the ex post stage. Second, she is averse to ex post indecisiveness (i.e., to having to choose between options that she cannot rank with certainty). Third, when evaluating a menu she discards options that are dominated (i.e., inferior to another option whatever her ex post preferences may be) and restricts attention to the undominated ones. Under these assumptions, the decision maker has a preference for commitment in the sense of preferring menus with fewer undominated alternatives. We derive a representation in which the decision maker’s uncertainty about her ex post preferences is captured by means of a subjective state space, which in turn determines which options are undominated in a given menu, and in which the decision maker fears, whenever indecisive, to choose an option that will turn out to be the worst (undominated) one according to the realization of her ex post preferences.     

A new axiomatic foundation of partial comparability

The paper presents some results obtained in searching for a new axiomatic foundation for partial comparability (PC) in the frame of non-conventional preference modeling. The basic idea is to define an extended preference structure able to represent lack of information, uncertainty, ambiguity, multidimensional and conflicting preferences, using formal logic as the basic formalism.A four-valued paraconsistent logic is therefore described in the paper as a more suitable language for the purposes of the research. The concepts of partition, general binary relations properties, fundamental relational system of preferences (f.r.s.p.), maximal f.r.s.p. and well founded f.r.s.p. are then introduced and some theorems are demonstrated in order to provide the axiomatic foundation of PC. The main result obtained is a preference structure that is a maximal well founded f.r.s.p. This preference structure facilitates a more flexible, reliable and robust preference modeling. Moreover it can be viewed as a generalization of the conventional approach, so that all the results obtained until now can be used under it.Two examples are provided at the end of the paper in order to give an account of the operational potentialities of the new theory, mainly in the area of multicriteria decision aid and social choice theory. Further research directions conclude the paper.This research has been done while the first author was in the Université Libre de Bruxelles under the Research in Brussels actions.     

The logic of preference reconsidered

Preferences are an important object of study in economic theory. Their logico-mathematical study has become prominent with the raise of modern decision theory and with the new conceptions of utility-functions and personalistic probabilities. The ‘basic logic’ at the foundation of the more advanced theories of preference, however, has been relatively little investigated. The pioneer work is Hallden's The Logic of ‘Better’ of the year 1957, followed by von Wright's The Logic of Preference in 1963. The topic has turned out unexpectedly problematic and there is as yet little consensus among logicians about the basic laws of preferring. A reason for this is apparently that there exist several concepts of preference which must be disentangled and kept apart in a logical theory. How this is to be done is discussed in the introductory sections (1–5) of the present paper. In Section 8 is sketched a logic for an asymmetric and connected preference-relation which holds between ‘possible worlds’ within a subject's ‘preference-horizon’ (Section 7). Preferences between states of affairs generally are called holistic when they hold ceteris paribus, i.e., when there is a corresponding preference-relation between any pairs of possible worlds which differ only in those two states and in no others. (Section 6.) Holistic preferences between states are asymmetrical and transitive but they do not form a linear preference order. (Section 9.) Failure to notice this, the author maintains, is responsible for much confusion in the traditional treatment of the subject. In the concluding sections (10–12) the author discusses the mutual relations of the value-absolutes, the good and the bad. Can they be defined in the terms of the relative notion of betterness or preference (and logical constants) alone? The definition which says that the goodness of a state is the holistic preference of it over its contradictory, answers the question affirmatively but conflicts with several deep-rooted axiological intuitions of ours. A more satisfying definition of the value-absolutes requires the additional notion of a value-less state (world) with which all the other states may be compared. In this connection some ideas of G. E. Moore about the notion of an ‘empty’ world can be interestingly exploited.     

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.      

Even-chance lotteries in social choice theory

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.     

Stochastic evolution of rationality

Following up on previous results by Falmagne, this paper investigates possible mechanisms explaining how preference relations are created and how they evolve over time. We postulate a preference relation which is initially empty and becomes increasingly intricate under the influence of a random environment delivering discrete tokens of information concerning the alternatives. The framework is that of a class of real-time stochastic processes having interlinked Markov and Poisson components. Specifically, the occurence of the tokens is governed by a Poisson process, while the succession of preference relations is a Markov process. In an example case, the preference relations are the various possible semiorders on the set of alternatives. Asymptotic results are obtained in the form of the limit probabilities of any semiorder. The arguments extend to a much more general situation including interval orders, biorders and partial orders. The results provide (up to a small number of parameters) complete quantitative predictions for panel data of a standard type, in which the same sample of subjects has been asked to compare the alternatives a number of times.     

Preference reversals and probabilistic decisions

Preference reversals occur when different (but formally equivalent) elicitation methods reveal conflicting preferences over two alternatives. This paper shows that when people have fuzzy preferences, i.e. when they decide in a probabilistic manner, their observed decisions can generate systematic preference reversals. A simple model of probabilistic choice and valuation can account for a higher incidence of standard (nonstandard) preference reversals for certainty (probability) equivalents and it can also rationalize the existence of strong reversals. An important methodological contribution of the paper is a new definition of a probabilistic certainty/probability equivalent of a risky lottery.     

The Deep Side of Preference Theory

An individual is said to have a taste for a particular menu, i.e. a subset of available commodities, if he is indifferent between all commodity bundles that contain the same quantity for each commodity which actually is in the menu, whatever the rest of the bundle. Then, a taste is directly defined as a deep property of preferences. As a first result, it is shown that a complete and transitive preference relation over the commodity bundles is equivalent to regular tastes where regularity means that tastes can be derived from a pure qualitative relation between the different commodities. Besides, a preference family based on preference relations corresponding to each particular commodity is said to be rationalizable if there exists a metapreference over commodity bundles which consistently summarizes the preference family and then allows to decide. As a second result, it is shown that if a preference family is rationalizable, then the tastes are organized thanks to a reflexive and transitive qualitative relation between the different commodities.     

Schur convexity,quasi-convexity and preference for early resolution of uncertainty

This paper deals with decision makers who choose among information systems. It shows that the properties of Schur convexity and of quasi-convexity are equivalent, even when general preferences are considered. Since Schur convexity is closely related to having a willingness to accept information and since quasi-convexity is closely related to having a preference for early resolution of the uncertainty about which information system prevails, then it follows that the equivalence implies that decision makers prefer more information to less if, and only if, they prefer early resolution of uncertainty to later resolution.     

The independence condition in the theory of social choice

Arrow's theorem is really a theorem about the independence condition. In order to show the very crucial role that this condition plays, the theorem is proved in a refined version, where the use of the Pareto condition is almost avoided.A distinction is made between group preference functions and group decision functions, yielding respectively preference relations and optimal subsets as values. Arrow's theorem is about the first kind, but some ambiguities and mistakes in his book are explained if we assume that he was really thinking of decision functions. The trouble then is that it is not clear how to formulate the independence condition for decision functions. Therefore the next step is to analyse Arrow's argument for accepting the independence condition.The most frequent ambiguity depends on an interpretation of A as the set of all conceivable alternatives, while the variable subset B is the set of all feasible or available alternatives. He then argues that preferences between alternatives that are not feasible shall not influence the choice from the set of available alternatives. But even if this principle is accepted, it only forces us to require independence with respect to some specific set B and not to every B simultaneously. Therefore the independence condition cannot be accepted on these grounds.Another argument is about an election where one of the candidates dies. On one interpretation this argument can be taken to support an independence requirement which leads to a contradiction. On another interpretation it is a condition about connexions between choices from different sets.The so-called problem of binary choice is found to be different from the independence problem and it plays no essential role in Arrow's impossibility result. Other impossibility results by Sen, Batra and Pattanaik and by Schwartz are of a different character.In the last section, several weaker independence conditions are presented. Their relations to Arrow's condition are stated and the arguments supporting them are discussed.     

Stochastic dominance in multicriterion analysis under risk

Traditionally, in the literature on the modelling of decision aids one notes the propensity to treat expected utility models and outranking relation models as rivals. It may be possible, however, to benefit from the use of both approaches in a risky decision context. Stochastic dominance conditions can be used to establish, for each criterion, the preferences of a decision maker and to characterise them by a concave or convex utility function.Two levels of complexity in preference elicitation, designated as clear and unclear, are distinguished. Only in the case of unclear preferences is it potentially interesting to attempt to estimate the value function of the decision maker, thus obtaining his (her) preferences with a reduced number of questions. The number of questions that must be asked of the decision maker depends upon the level of the concordance threshold that he(she) requires in the construction of the outranking relations using the ELECTRE method.     

Ratio-Scale Measurement with Intransitivity or Incompleteness: The Homogeneous Case

In the homogeneous case of one-dimensional objects, we show that any relation that is positive and homothetic can be represented by a ratio-scale and a unique and constant biasing factor. This factor may favor or disfavor the preference for an object over another. In the first case, preferences are complete but not transitive and an object may be preferred even when its value is lower. In the second case, preferences are asymmetric and transitive but not negatively transitive and it may not be sufficient for an object to have a greater value to be preferred. In this manner, the biasing factor reflects the extent to which preferences may depart from a maximization process.     

A short step between democracy and dictatorship

When preferences are defined over two alternatives and societies are variable, the group formed by the relative majority rule, the unanimity rule, the dictatorial rules, and the strongly dictatorial rules is characterized in terms of five axioms: unanimity, reducibility, substitutability, exchangeability, and parity. This result is used to provide characterizations of each of these rules by postulating separating axioms, that is, an axiom and its negation. Such axioms identify traits specifically differentiating a type of rule from the other types. For instance, majority differs from strong dictatorship in the existence of a society for which collective indifference should be a less likely outcome than the strict preference of one alternative over the other. As a second example, the difference between majority and strong dictatorship can be traced back to the requirement that the likelihood of collective indifference diminishes with the size of society.     

Metapreferences and the reasons for stability in social choice: Thoughts on broadening and clarifying the debate

The standard model of collective choice looks at aggregation procedures which take individual preferences as existing for a specified set of alternatives. In this paper we propose that actors also have preferences for rules of choice or characteristics of choice processes (e.g., the perceived fairness of procedures or the popularity of outcomes) rather than simply for alternatives (outcomes) themselves. We argue that the positing of the existence of meta-preferences can illuminate a number of areas of choice theory. Here we focus on one such area: the problem of “too much” stability in majority rule decision making - a stability which belies the standard theoretical results on the generic instability of majority rule processes. We also show that discussion of the prevalence of stability in collective decision making needs to be clarified because there are at least six distinct types of stability which are sometimes confounded in the literature.     

Subjective probabilities and utility with event-dependent preferences

This article generalizes Savage's theory to include event-dependent preferences. The state space is partitioned into finitely many events. The induced preferences over consequences are assumed independent of the underlying states within, but not across, these events. This results in an additively separable representation of preferences over acts. The dependence of the preference relation over consequences on the events is represented by event-dependent mappings of the set of consequences onto itself. Given these mappings, the preferences on acts are represented by the expectation of event-dependent utilities on the consequences with respect to unique subjective probabilities on the states.Helpful discussions with David Schmeidler are gratefully acknowledged.