Fuzzy preferences and Arrow-type problems in social choice

The theory of fuzzy sets has been applied to social choice primarily in the contexts where one is given a set of individual fuzzy preference relations and the aim is to find a non-fuzzy choice set of winners or best alternatives. We discuss the problem of composing multi-member deliberative bodies starting from a set of individual fuzzy preference relations. We outline methods of aggregating these relations into a measure of how well each candidate represents each voter in terms of the latter's preferences. Our main goal is to show how the considerations discussed in the context of individual non-fuzzy complete and transitive preference relations can be extended into the domain of fuzzy preference relations.

Classically, formal individual choice and rationality are framed using precise preference relations. Individual preference relations and hence the rationality of choice are generally not precise, and this necessitates the studies under vague preference relations. Until now, available studies of rationality have done by utilizing R-implications in the associated continuous t-norm. However, another class of fuzzy implications, S-implications, which represent behavioral interpretation, i.e., lenience and non-satisfaction of criteria, have not yet been used for the study of rationality. We validated that S-implication with continuous t-norm fails to satisfy key results of rationality and hence introduced an operator called S-residuum, which logically represents the minimal degree required to hold the implication against the joint degree of satisfaction of the implication by t-norm. This happens due to failure of adjointness, the law of identity and the boundary condition in this structure. This S-implication-S-residuum framework is introduced and exploited to develop logical interrelations among rationality axioms built on vague preferences for the first time. This leads to reformulating the well-known rationality axioms, namely S-FWARP, S-FSARP, S-FWCA and S-FSCA. The main results are : S-FSCA ⇒ S-FWCA ⇒ S-FSWCA and S-FSARP ⇒ S-FWARP ⇒ S-SFWARP. Furthermore, we characterize regularity of fuzzy preference relation in terms of FCF satisfying S-FWCA and hence S-SFCA. We concluded that this new framework offers a behaviorally meaningful and logically coherent foundation for the study of vague choice and rationality.

The theory of social choice is concerned with the problem of aggregating the preferences of several persons into a single preference order. The problem has the same structure as that of aggregating the preference orders of a single person with regard to several aspects of alternatives into a single preference order on the set of alternatives. For this reason, the theory of social choice comes logically not under the topic of collective action (as one might suppose) but rather under the topic of multi-objective decisions. This is so because the several participants in a situation defined as a problem of social choice do not really select courses of action among several available ones when they present their preference orders on a set of alternatives. Therefore they are not ‘actors’, as actors are defined in the theory of decision. To be sure, there are situations where voters do choose strategies, that is, choose among different preference orders to present, which may or may not represent their true preferences but which they believe will be more likely to lead to a more desirable aggregated order. These situations can be considered as n-person decision problems, and we will examine them in the context of non-cooperative n-person games (see Chapter 14. At this time, however, we will suppose that each voter acts ‘sincerely’, that is, presents his ‘actual’ preference order. The decision is now up to some agency, which must combine all these submitted preference orders into a ‘social’ preference order.KeywordsSocial ChoicePreference OrderPareto OptimalitySocial Welfare FunctionSimple MajorityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions.

We present how fuzzy logic with linguistic quantifiers, mainly its calculi of linguistically quantified propositions, can be used in group decision making. Basically, the fuzzy linguistic quantifiers (exemplified bymost, almost all,...) are employed to represent a fuzzy majority which is in many cases closer to a real human perception of the very essence of majority. Fuzzy logic provides here means for a formal handling of such a fuzzy majority which was not possible by using traditional formal apparata. Using a fuzzy majority, and assuming fuzzy individual and social preference relations, we redefine solution concepts in group decision making, and present new «soft» degrees of consensus.

Preference modelling is a fundamental part of several applied fields but at the same time it has its own interesting theoretical problems. There exists a well-known axiomatic approach to fuzzy preference structures. In this axiomatic framework, general constructions of strict preference, indifference and incomparability relations associated with a fuzzy large preference relation are established via solving a system of functional equations. In the present paper we overview the solutions of two key functional equations (one of Frank, and one of Alsina) under different conditions on the underlying logic operations. Consequences of these general results to fuzzy preference modelling are also summarized

In this paper we first consider the problem of extending fuzzy (weak and strict) preference relations, represented by fuzzy preorders on a set to a fuzzy preferences on subsets, and we characterise different possibilities. Based on their properties, we then semantically define and axiomatize several two-tiered graded modal logics to reason about the corresponding different notions of fuzzy preferences.

Crowdsourcing is an important tool, e.g., in computational linguistics and computer vision, to efficiently label large amounts of data using nonexpert annotators. The individual annotations collected need to be aggregated into a single collective annotation. The hope is that the quality of this collective annotation will be comparable to that of a traditionally sourced expert annotation. In practice, most scientists working with crowdsourcing methods use simple majority voting to aggregate their data, although some have also used probabilistic models and treated aggregation as a problem of maximum likelihood estimation. The observation that the aggregation step in a collective annotation exercise may be considered a problem of social choice has only been made very recently. Following up on this observation, we show that the axiomatic method, as practiced in social choice theory, can make a contribution to this important domain and we develop an axiomatic framework for collective annotation, focusing amongst other things on the notion of an annotator's bias. We complement our theoretical study with a discussion of a crowdsourcing experiment using data from dialogue modelling in computational linguistics.

Collective decision making is the process of mapping the individual views of several individual agents into a joint decision. The need for collective decision making mechanisms is abundant, not just in the realm of politics, but also for a wide range of scientific and technological applications. These include, for instance, logistics, grid computing, and recommender systems. The alternatives to be decided upon often have a combinatorial structure: an alternative is characterised by a tuple of variables, each ranging over a finite domain. Classical approaches to collective decision making, developed in social choice theory, do not take the computational limitations induced by the combinatorial nature of the problem into account. For instance, if we are asked to elect a committee consisting of k representatives, choosing from a pool of n candidates, then we are in fact faced with a social choice problem with $(^n_k)$ alternatives. Asking each voter for their full preferences over these $(^n_k)$ alternatives may not be feasible in practice. Similarly, if we have to choose between accepting or rejecting each of n different propositions, then we are dealing with a social choice problem with 2 n possible outcomes. In this talk I will report on a number of recent developments in the area of collective decision making in combinatorial domains. This includes the study of languages for the compact representation of preferences [2,4], the design of voting rules for multi-issue elections [1], and the analysis of the computational complexity of decision problems arising in judgment aggregation [3].

One of the most exciting areas in economic theory is the study of incentives. Within this rubric a broad range of problems are currently being addressed. They range from the problems of social choice, to the economics of planning, to the character of labor contracts. The central theme of the research agenda concerns how the incentive structure affects the quality of information agents are willing to communicate. The principal question concerns how truthful revelation of information can be induced.1 Such questions are important whenever agents can profit from distorting information. The problem thus arises in all sorts of guises in economics. The purpose of this essay is to illustrate this point by focusing on three distinct areas, social choice, public finance, and economic planning.

Social choice problems with fixed sets of alternatives

We consider a social choice problem where individual rationality is required. The status quo belongs to the outcome space, and the selected alternative must be weakly better than the status quo for everybody. If the mechanism designer has no knowledge of the alternatives, we obtain a negative result: any individually rational (IR) and strategy-proof (SP) mechanism can choose at most one alternative (besides the status quo), regardless of the preferences. To overcome this negative result, we consider a domain where the alternatives have a known structure, i.e., an agent is indifferent between the status quo and a subset of the outcomes. This set is exogenously given and public information. This assumption is natural if the social choice involves the participation of agents. For example, consider a group of people organizing a trip where participation is voluntary. We can assume each agent is indifferent between the trip plans in which she does not participate and the status quo (i.e., no trip). In this setting, we obtain more positive results: we develop a class of mechanisms called Approve and Choose mechanisms, which are IR and SP, and can choose multiple alternatives as well as the status quo.

The extraordinary technological progress we have witnessed in recent years has made it possible to generate and exchange multimedia content at an unprecedented rate. As a consequence, massive collections of multimedia objects are now widely available to a large population of users. As the task of browsing such large collections could be daunting, Recommender Systems are being developed to assist users in finding items that match their needs and preferences. In this article, we present a novel approach to recommendation in multimedia browsing systems, based on modeling recommendation as a social choice problem. In social choice theory, a set of voters is called to rank a set of alternatives, and individual rankings are aggregated into a global ranking. In our formulation, the set of voters and the set of alternatives both coincide with the set of objects in the data collection. We first define what constitutes a choice in the browsing domain and then define a mechanism to aggregate individual choices into a global ranking. The result is a framework for computing customized recommendations by originally combining intrinsic features of multimedia objects, past behavior of individual users, and overall behavior of the entire community of users. Recommendations are ranked using an importance ranking algorithm that resembles the well-known PageRank strategy. Experiments conducted on a prototype of the proposed system confirm the effectiveness and efficiency of our approach.

Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness

A parametric model for determining consensus priority vectors from fuzzy comparison matrices