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.     

The Act of Understanding

Summary Attention is drawn to experience that much social work has beenundertaken on a base of common sense, personal intuition andfeelings that the client's situation has been 'understood'.Medical, sociological and psychological sources for theory whichhave given status to objectivity and implicitly deterministicways of thinking are criticized as sometimes contrary to usefulunderstanding and not supportive to constructive change. Itis claimed that a goal of social work theory should be to disciplinethe 'understanding' which exists in the practice and it is suggestedthat this might be based upon the investigation of the meaningwhich people give to the world in which they live and to theirself within that world. This phenomenologkal approach placessubjectivity and the role of the self-concept at the focus ofattention. As a working philosophy it incorporates the formativeinfluences of past experiences whilst also asserting the primacyof the person who may impose a new significance upon both thosepast events and his immediate situation. It is claimed thatsuch an approach might be the source of a discipline which couldrelate more intimately than hitherto the theory and good practiceof all concerned with social welfare and social change.     

Cooperation between politicians and econometricians and the search for optimal economic policy

The problem of formalizing the cooperation between politicians and econometricians as an iterative—or trial and error—process, such that the weights in a preference function can be obtained in a finite number of steps, is considered. The approach is based on the pioneering suggestions made by Tinbergen and Frisch within the framework of a formal theory of economic policy. Using the implicit information given by a possible and implementable dialogue between a policy maker and a model builder it is shown that it is possible to take into account explicitly the interdependence between targets, instruments, and weights.     

Double deception: Two against one in three-person games

This article examines deception possibilities for two players in simple three-person voting games. An example of one game vulnerable to (tacit) deception by two players is given and its implications discussed. The most unexpected findings of this study is that in those games vulnerable to deception by two players, the optimal strategy of one of them is always to announce his (true) preference order. Moreover, since the player whose optimal announcement is his true one is unable to induce a better outcome for himself by misrepresenting his preference, while his partner can, this player will find that possessing a monopoly of information will not give him any special advantage. In fact, this analysis demonstrates that he may have incentives to share his information selectively with one or another of his opponents should he alone possess complete information at the outset.     

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.     

A graph-theoretical approach to the aggregation of individual preference orderings

A graph-theoretical approach as first outlined by Harary et al. is developed for solving the problem of aggregating a given set of individual preference orderings provided that no individual preferential judgment is inconsistent with the collective preference ordering obtained by aggregation.

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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.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).     

On the existence of fair matching algorithms

We analyze the Gale-Shapley matching problem within the context of Rawlsian justice. Defining a fair matching algorithm by a set of 4 axioms (Gender Indifference, Peer Indifference, Maximin Optimality, and Stability), we show that not all preference profiles admit a fair matching algorithm, the reason being that even this set of minimal axioms is too strong in a sense. Because of conflict between Stability and Maximin Optimality, even the algorithm which generates the mutual agreement match, paradoxically, has no chance to be fair.We then relax the definition of fairness (by giving preference to Stability over Maximin Optimality) and again find that some preference profiles admit a fair matching algorithm, while others still do not, but the mutual agreement algorithm now is fair under this definition.The paper then develops a test, which determines, for a given preference profile, whether a fair algorithm exists or not.     

Combining strength and uncertainty for preferences in the graph model for conflict resolution with multiple decision makers

A hybrid preference framework is proposed for strategic conflict analysis to integrate preference strength and preference uncertainty into the paradigm of the graph model for conflict resolution (GMCR) under multiple decision makers. This structure offers decision makers a more flexible mechanism for preference expression, which can include strong or mild preference of one state or scenario over another, as well as equal preference. In addition, preference between two states can be uncertain. The result is a preference framework that is more general than existing models which consider preference strength and preference uncertainty separately. Within the hybrid preference structure, four kinds of stability are defined as solution concepts and a post-stability analysis, called status quo analysis, which can be used to track the evolution of a given conflict. Algorithms are provided for implementing the key inputs of stability analysis and status quo analysis within the extended preference structure. The new stability concepts under the hybrid preference structure can be used to model complex strategic conflicts arising in practical applications, and can provide new insights for the conflicts. The method is illustrated using the conflict over proposed bulk water exports from Lake Gisborne in Newfoundland, Canada.     

A model to support the assessment of subjective probabilities

Conclusions The purpose of this article has been to provide a new method for the determination of subjective probabilities. The kind of support which is suggested for the assessment process has several interesting properties. Constructed in accordance with the Savage theory the procedure considers a preference structure on the space of the possible actions A or on a suitably chosen space A ^* as its basis. This structure will be attained by pairwise comparison of the elements a _i A (or a _i ^*A ^*). It may be incomplete and inconsistent. A linear programming approach will use this information to derive numerical probabilities which are in accordance with the stated preference judgments. Therefore, it avoids the difficulties which arise when the judge has to assign the numerical values to the possible states of nature. On the other hand, this method presents the advantage of taking into consideration all relevant a priori information because of its systematic judgment process. Another important element is given by the use of coefficients which express the judge's confidence in the stated comparisons. Further relevance is added to the procedure by the relatively small consumption of time and work to report the judgments even in complex situations, and by the renunciation of any methodological knowledge on the part of the judges.This study was financed by a research grant of the Deutsche Forschungsgemeinschaft (DFG).     

Discriminating Between Preference Functionals: A Monte Carlo Study

This paper reports on the results of a Monte Carlo investigation into the power of commonly employed procedures for identifying the ‘correct’ preference functional of individuals, and hence for discriminating between the large number of preference functionals now advocated in the theoretical literature. The paper also asks which of two commonly employed experimental procedures might be the most efficient in this respect. The results show that several of the ‘newer’ preference functionals are difficult to distinguish empirically-at least on the basis of conventional experimental tests-and that the Complete ranking experimental design might be better than the Pairwise Choice design. The conclusion of the paper is that more thought should therefore be given to the question of the experimental design.     

Ordinal preference representations

Ordinal preferences have several advantages over the traditional cardinal expressions of preference. Three different representations of ordinal preferences useful in multi-participant modelling are presented, and their features compared. One approach is thepayoff representation that is based on an ordinal normal form game. A second representation of ordinal preferences is thepreference vector, based on the option form of the game. The option form consists of a list of players, with each player followed by the options under its control. The third representation of ordinal preferences is thepreference tree. A preference tree is an implied binary tree that captures the information of preference vector in a more compact manner by making use of its lexicographic structure. The preference tree offers considerable compactness and computational efficiency over the other two approaches.     

An axiomatic system for multidimensional preferences

An axiomatic system TP is developed which allows the amalgamation of linear preferences (preferences in respect to different criteria) according to the weights of those criteria. Section 1 deals with linear preferences. In Section 2 an axiomatic system for the ordering of classes of criteria is formulated. Section 3 explains the development of system TP. Two TP-systems are distinguished, based on two different linear preference systems. The preference relation of TP is shown to be nontransitive, while the linear preference relation is transitive. In 3.7 an alternative-system TP′ is given. Section 3.8 deals with formulas concerning the disjunction of alternatives, which are not valid in TP, even though they are valid in a linear preference system. In Section 4 types of preference relations and types of alternatives are distinguished to get the opportunity to express preferences between preferences.     

The risk aversion measure without the independence axiom

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.Helpful comments by I. Gilboa and suggestions by the referee are gratefully acknowledged.     

The impossibility of the Paretian liberal and its relevance to welfare economics

The purpose of this study is to analyse the implications of Sen's impossibility result, the liberal paradox, for orthodox welfare economics. Because the rather special format of social choice theory makes it a little difficult to be sure of the relevance of this result, the whole dilemma is posed here in terms of a rather informal analysis of information al patterns.On the one hand, it is argued that the traditional approach to welfare economics, including both utilitarianism and Paretian ordinalism, contains severe informational constraints eliminating the use of all kinds of independent non-utility information in the social evaluation process. This property, called welfarism, is also present in the weak Pareto principle, which conflicts with even minimal requirements of personal liberty according to Sen's result.On the other hand, it is argued that there is in fact little to be resolved in this problem in spite of several attempts to circumvent the conflict. These studies are argued to be mainly ad hoc solutions to the formal problem and relevant only to the extent they indicate how severe restrictions are needed to avoid the paradox. The analogy with the prisoner's dilemma does not work either. Since liberal values are intrinsically non-welfaristic, the liberal paradox can be interpreted as only one, but a rather powerful, example of the informational deficiency of the orthodox approach.Finally, it is argued that the liberal paradox has striking implications for both the concept of preference and social optimum as well as empirical research on social welfare. This means that if the impossibility is to be taken seriously we need to revalue both the status of utility information and the role of the Pareto principle in social welfare analysis.The author is grateful to Professor Amartya Sen and Matti Tuomala for helpful comments and suggestions.     

A geometric approach to revealed preference via Hamiltonian cycles

It is shown that a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference (WARP) implies the strong axiom of revealed preference (SARP), can be reduced to a Hamiltonian cycle problem: A set of bundles allows a preference cycle of irreducible length if and only if the convex monotonic hull of these bundles admits a Hamiltonian cycle. This leads to a new proof to show that preference cycles can be of arbitrary length for more than two but not for two commodities. For this, it is shown that a set of bundles satisfying the given condition exists if and only if the dimension of the commodity space is at least three. Preference cycles can be constructed by embedding a cyclic $(L-1)$ -polytope into a facet of a convex monotonic hull in $L$ -space, because cyclic polytopes always admit Hamiltonian cycles. An immediate corollary is that WARP only implies SARP for two commodities. The proof is intuitively appealing as this gives a geometric interpretation of preference cycles.     

Dynamic inconsistency and choice

In this paper, we analyze an intra-personal game where a decision-maker is summarized by a succession of selves. Selves may (or may not) have conflicting interests, and earlier selves may have imperfect knowledge of the preferences of future selves. At date 1, self-1 chooses a menu, at date 2, the preferences of self-2 realize and self-2 chooses an item from the menu. We show that equilibrium choice is consistent with either a preference for flexibility, a preference for betweenness or a preference for systematic restriction. Overall, the analysis reconciles the decision–theoretic approach of choice over time with the game–theoretic multiple-selves approach.     

Morality and welfare

The economic theory of utility does not take into account moral behaviour, that is behaviour in which an individual faced with two courses of action may choose the one which is less profitable to him. It is shown in this paper that the economic approach can be broadened to take such behaviour into account. The individual is considered to maximize a higher welfare function whose arguments are (a) his utility and (b) higher goods or moral value. The latter are goals which he ought to achieve. The amount of higher goods achieved depends upon his moral preference. The ranking of different acts is complete - an assumption commonly made in economics with regard to bundles of goods. In an extension of the economic model of optimization over time, the individual is considered to maximize his higher welfare over his lifetime. This may require raising his moral preference.     

Self‐control,peer preference,and loneliness in Chinese children: A three‐year longitudinal study

The purpose of this study was to explore the longitudinal links among Chinese children's self‐control, social experiences, and loneliness, largely from a developmental cascades perspective (which postulates mechanisms about how effects within a particular domain of functioning can impact across additional domains over time). Participants were N = 1,066 primary school students in Shanghai, P. R. China, who were followed over three years from Grade 3 to Grade 5. Measures of children's behavioral self‐control, peer preference, and loneliness were obtained each year from peer nominations and child self‐reports. Results indicated that as compared with the unidirectional and bidirectional models, the developmental cascade model represented the best fit for the data. Within this model, a number of significant direct and indirect pathways were identified among variables and over time. For example, self‐control was found to indirectly contribute to later decreases in loneliness via a pathway through peer preference. As well, peer preference both directly and indirectly contributed to later increases in self‐control. Finally, loneliness directly led to decreases in self‐control from Grade 3 to Grade 4, but not from Grade 4 to Grade 5. Results are discussed in terms of the implications of self‐control for Chinese children's social and emotional functioning over time.