Possibility Theorems, Incentives, and Progress in Economic Theory

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.

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].

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.

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.

All prescriptions of behavior for individuals require enforcement. Usually the obligation to behave in a prescribed way is entered into voluntarily by explicit or implicit contract. For example, I promise to teach certain classes with designated frequency and to discuss matters which I, and possibly others, believe are relevant to the course titles. By negotiation, and in the event of its failure, by legal action, I and my employer seek to enforce the contract of employment against large departures from the promised behavior. Performance of some kinds of behavior is difficult or impossible to enforce-such as promises to be creative, noble, or steadfast in crisis-and as a result such contractual promises are either not made or enforced only when there is an uncontroversially flagrant violation. The influence upon contract, and upon economic organization generally, of the costs of enforcing various kinds of contracts has received virtually no study by economists, despite its immense potential explanatory power. When the prescribed behavior is fixed unilaterally rather than by individual agreement, we have the regulation or law, and enforcement of these unilateral rules is the subject of the present essay. Departures of actual from prescribed behavior are crimes or violations, although one could wish for a less formidable description than "criminal" to describe many of the trifling offenses or the offenses against unjust laws. My primary purpose is to construct a theory of rational enforcement, a theory which owes much to Gary Becker's major article on the subject (1968). In the conclusion the problem of explanation, as distinguished from prescription, will be commented upon.

Book Review: An Anatomy of Values: Problems of Personal and Social Choice

AINYONE CONCERNED to think clearly and well has to make at least two principle decisions. The first, and often most neglected, involves separating the from the trivial: making distinctions between things that we need to know now, from those that we might want to know at a later, less perilous time, from those that we really can do without now and forever. This is not necessarily easy as the advocates of pure science often remind us. Still, I suspect that the choice is not as difficult as it may seem, especially in a world of widening extremes between the winners and the losers, that is busily engaged in pulling the ecological rug out from under its own feet while developing more clever and more probable ways to incinerate itself. The second decision involves making distinctions within the category of important between the true, the less true, and the false, knowing all the while that reality is more complex than our ability to think about it. Moreover, the characteristics of relativity, randomness, indeterminacy, and incompleteness found in the physical/mathematical universe doubtless have analogs in the social world, which are compounded by problems of irony, values, ethics, and social choice. Moreover, disciplinary paradigms, research methods, and the institutional context of research and thought can seriously distort what we think to be true. The two decisions, of course, are related. The choice of what is depends to some extent on what is thought to be true and vice versa. In both decisions a healthy concern for good thought en-

The problem of social choice when individual orderings are lexicographic is discussed in this paper. In addition to assuming that individual orderings over the social alternatives are lexicographic, the assumptions are made that the set of criteria which constitute the basis of each individual's ordering are identical for all individuals and also that the ranking of these criteria, in terms of priority, is the same for all individuals. Given these assumptions, the author establishes conditions for the existence of a socially best alternative under binary, decisive, neutral and nonnegatively responsive group decision rules, which constitute a very wide class of group decision rules. The result established turns out to be more general than the results proved earlier by Encarnación (1969) and Taylor (1970).

The assignment problem, a well-known variant of which is the problem of optimal staff distribution, has formerly been treated as a problem of linear optimization starting with numerical evaluations of the assignments. In this paper the assignment problem is regarded as a problem of social choice, where preference orders are used as the sole initial information. A formal definition of an assignment function is given and various conditions for such functions are suggested. Some conditions are found to be incompatible. Several reasonable assignment functions are examined with respect to which conditions they fulfil. Finally, some problems, related to the assignment problem, are informally discussed.

Samuelson's statement about Marx being a minor post-Ricardian from the viewpoint of pure economic theory leaves one perplexed. Is theoretical welfare economics a part of pure economic theory? This paper shows that theoretical welfare economics is the foundation of the theory of economic policy and intervention. Marx also sets forth a theory of intervention based on economic arguments. Therefore, Marx should be studied within the broader scope of theoretical welfare economics. Moreover, Marx's system contains not only analytical devices but also important value judgments and the theory of welfare economics aims to study the interrelations between value judgments and economic interventions. Methodological contradictions are shown in, and removed from, the theory of welfare economics in order to show the framework of Marx's entire system. The system as a whole is far beyond the realm of Ricardian economics, for it can be consistent and scientific although it contains value judgments. Samuelson's narrow approach and fuzziness in the methodological foundations of welfare economics hindered an appreciation of the full value of Marx's approach.

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in +k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of +c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on 2 then choice on a compact and convex constraint exhibits a version of the median voter theorem.