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.

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.

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].

Perhaps the most influential result in social choice theory is Arrow’s impossibility theorem, which states that a seemingly modest set of desiderata cannot be satisfied when aggregating preferences [1]. While Arrow’s theorem might appear rather negative, it can also be interpreted in a positive way by identifying what can be achieved in preference aggregation.In this talk, I present a number of variations of Arrow’s theorem–such as those due to Mas-Colell and Sonnenschein [8] and Blau and Deb [2]–in their choice-theoretic version. The critical condition in all these theorems is the assumption of a rationalizing binary relation or equivalent notions of choice-consistency. The bulk of my presentation contains three escape routes from these results. The first one is to ignore consistency with respect to a variable set of alternatives altogether and require consistency with respect to a variable electorate instead. As Smith [12] and Young [14] have famously shown, this essentially characterizes the class of scoring rules, which contains plurality and Borda’s rule. For the second escape route, we factorize choice-consistency into two parts, contraction-consistency and expansions-consistency [11]. While even the mildest dose of the former has severe consequences on the possibility of choice, varying degrees of the latter characterize a number of appealing social choice functions, namely the top cycle, the uncovered set, and the Banks set [3,9,4]. Finally, I suggest to redefine choice-consistency by making reference to the set of chosen alternatives rather than individual chosen alternatives [6]. It turns out that the resulting condition is a weakening of transitive rationalizability and can be used to characterize the minimal covering set and the bipartisan set. Based on a two decades-old conjecture due to Schwartz [10], the tournament equilibrium set can be characterized by the same condition or, alternatively, by a weak expansion-consistency condition from the second escape route. Whether Schwartz’s conjecture actually holds remains a challenging combinatorial problem as well as one of the enigmatic open problems of social choice theory.Throughout the presentation I will discuss the algorithmic aspects of all considered social choice functions. While some of the mentioned functions can be easily computed, other ones do not admit an efficient algorithm unless P equals NP [13,5,7].

Previous articleNext article No AccessArrow's Theorem, Indeterminacy, and Multiplicity Reconsidered*Mathias RisseMathias Risse Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Ethics Volume 111, Number 4July 2001 Article DOIhttps://doi.org/10.1086/233570 Views: 53Total views on this site Citations: 18Citations are reported from Crossref © 2001 by The University of Chicago. All rights reserved.PDF download Crossref reports the following articles citing this article:Henrik D. Kugelberg Social choice problems with public reason proceduralism, Economics and Philosophy 38, no.11 (Feb 2021): 51–70.https://doi.org/10.1017/S0266267120000498Wesley H. Holliday, Eric Pacuit Axioms for defeat in democratic elections, Journal of Theoretical Politics 33, no.44 (Nov 2021): 475–524.https://doi.org/10.1177/09516298211043236Hannu Nurmi The Incidence of Some Voting Paradoxes Under Domain Restrictions, Group Decision and Negotiation 29, no.66 (Aug 2020): 1107–1120.https://doi.org/10.1007/s10726-020-09697-9Adiel Teixeira de Almeida, Danielle Costa Morais, Hannu Nurmi More Than Two Alternatives, (Oct 2019): 21–30.https://doi.org/10.1007/978-3-030-30955-8_4Eerik Lagerspetz Interpretations, (Jan 2016): 247–340.https://doi.org/10.1007/978-3-319-23261-4_5MARK A. BURGMAN, HELEN M. REGAN, LYNN A. MAGUIRE, MARK COLYVAN, JAMES JUSTUS, TARA G. MARTIN, KRIS ROTHLEY Voting Systems for Environmental Decisions, Conservation Biology 28, no.22 (Jan 2014): 322–332.https://doi.org/10.1111/cobi.12209Rosa Camps, Xavier Mora, Laia Saumell A continuous rating method for preferential voting. The incomplete case, Social Choice and Welfare 40, no.44 (Apr 2012): 1111–1142.https://doi.org/10.1007/s00355-012-0663-5Aki Lehtinen A welfarist critique of social choice theory, Journal of Theoretical Politics 23, no.33 (Jul 2011): 359–381.https://doi.org/10.1177/0951629811411753Mathias Risse On the Philosophy of Group Decision Methods I: The Nonobviousness of Majority Rule, Philosophy Compass 4, no.55 (Sep 2009): 793–802.https://doi.org/10.1111/j.1747-9991.2009.00224.xMathias Risse On the Philosophy of Group Decision Methods II: Alternatives to Majority Rule, Philosophy Compass 4, no.55 (Sep 2009): 803–812.https://doi.org/10.1111/j.1747-9991.2009.00225.xArnold Shepperson An economy of impossibility: a preliminary study for an ordinal approach to research methods in cultural studies, Critical Arts 22, no.22 (Nov 2008): 333–357.https://doi.org/10.1080/02560040802472492Joel Parker Peter Emerson, ed., Designing an All-Inclusive Democracy: Consensual Voting Procedures For Use in Parliaments, Councils and Committees, Public Choice 135, no.3-43-4 (Nov 2007): 493–496.https://doi.org/10.1007/s11127-007-9261-yMathias Risse On the Philosophy of Group Decision Methods I: The Non-Obviousness of Majority Rule, SSRN Electronic Journal (Jan 2008).https://doi.org/10.2139/ssrn.1296638MARC FLEURBAEY SOCIAL CHOICE AND JUST INSTITUTIONS: NEW PERSPECTIVES, Economics and Philosophy 23, no.11 (Mar 2007): 15–43.https://doi.org/10.1017/S0266267107001204Donald G. Saari Which is better: the Condorcet or Borda winner?, Social Choice and Welfare 26, no.11 (Jan 2006): 107–129.https://doi.org/10.1007/s00355-005-0046-2Mathias Risse Why the count de Borda cannot beat the Marquis de Condorcet, Social Choice and Welfare 25, no.11 (Nov 2005): 95–113.https://doi.org/10.1007/s00355-005-0045-3Gerry Mackie Democracy Defended, 20 (Sep 2009).https://doi.org/10.1017/CBO9780511490293 by Donald G. Saari Capturing the “Will of the People” Saari, Ethics 113, no.22 (Jul 2015): 333–349.https://doi.org/10.1086/342857

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.

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.

Extension of Arrow's theorem to symmetric sets of tournaments

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.

We explore the implications of “nosy” preferences—when individuals have rankings over the specific actions of others—using recent theoretical results in the behavioral economics literature. Our model jointly captures preference interdependence over utilities (benevolence) and actions (nosiness). We apply the model to two well‐known environments. The first is a classic social choice problem; the second is a model of relative consumption concerns. For the former we characterize the existence of the impossibility once the social choice problem has been modeled as a behavioral game. For the latter we characterize when the negative externality arising from relative consumption concerns can be overcome without a policy intervention.

Many practical scenarios involve solving a social choice problem: a group of self-interested agents have to agree on an outcome that best fits their combined preferences. We assume that each outcome presents a certain utility to an agent and that the best outcome is the one that maximizes the sum of these utilities. We call a mechanism for solving social choice problems incentive-compatible if for each agent, the behavior that maximizes its own utility is also the one that maximizes the group’s utility.One way to achieve incentive-compatibility is the Vickrey-Clarke-Groves (VCG) tax ([5]) mechanism. However, it produces a surplus of taxes that cannot be redistributed to the agents and can severely reduce agents’ utilities. Game theory has shown that it is not possible to have a general scheme that is incentive-compatible, budget-balanced and guarantees a Pareto-efficient solution.We present a scheme that sacrifices Pareto-efficiency to achieve budget balance while being both incentive-compatible and individually rational. On randomly generated social choice problems, the scheme results in significantly better overall agent utility than the VCG tax mechanism.KeywordsSocial ChoiceConstraint Satisfaction ProblemVickrey AuctionConstraint Optimization ProblemConsistent AssignmentThese 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.

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.

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 and telling the truth

Chapter Eighteen - Topological Theories of Social Choice