Arrow's Theorem, Indeterminacy, and Multiplicity Reconsidered

The problem ofjustifying democracy arises when a society sees the need for cooperative, collective action. Collective action may be needed to solve coordination problems, public goods problems, Prisoner's Dilemmas, and other structural problems of human interaction or perhaps to realize common ideals of justice in concrete political institutions. Once the need for collective action is established, one must ask how the goals of a cooperative endeavor are to be secured. Cooperation requires that individuals, at least over a specified range of activity, pursue a joint strategy. To assure voluntary compliance in a joint venture, cooperation must be to each person's advantage. However, each person may have an incentive to induce others to cooperate and to defect from the joint strategy in the hope of enjoying the fruits of cooperation without incurring the opportunity costs of compliance. This is the essence of the Prisoner's Dilemma, and considerations of this sort suggest that if collective action is to succeed, policies or strategies formed collectively must be coercively enforceable. Coercion, however, requires justification. Because solving the problem of rational noncompliance requires that collective decisions be coercively enforceable, the rules by which collective decisions are reached require justification. Douglas Rae neatly puts the problem of justification that emerges when political solutions are coercively enforceable as follows: "Once a political community has decided which of its members are to participate directly in the making of collective policy, an important question remains: 'How many of them must agree before a policy is imposed on the community?"" This is essentially the question to which the principle of democratic rule provides an answer: by what process are collective decisions to be made? Answering Rae's question requires a normative framework. We could say that a procedure for making collective decisions is justified if and

This new textbook addresses the neglect of practical research methods in cultural studies. It provides readers with clearly written overviews of research methods in cultural studies, along with guidelines on how to put these methods into operation. It advocates a multi-method approach, with students drawing from a pool of techniques and approaches suitable for their own topics of investigation. The book covers the following main areas: Drawing on experience, and studying how narratives make sense of experience. Investigating production processes in the cultural industries, and the consumption and assimilation of cultural products by audiences and fans. Taking both quantitative and qualitative approaches to the study of cultural life. Analysing visual images and both spoken and written forms of discourse. Exploring cultural memory and historical representation. Key Features A unique guide to research methods in Cultural Studies Explores key methods of research, with examples of how to pursue (or not to pursue) a particular method. Expert contributors include Martin Barker, Aeron Davis, David Deacon, Emily Keightley, Steph Lawler, Anneke Meyer, Virginia Nightingale and Sarah Pink.

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].

Extension of Arrow's theorem to symmetric sets of tournaments

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.

Social choice and telling the truth

Computational social choice is a new interdisciplinary research area at the intersection of social choice theory and computer science that studies computational aspects of social choice mechanisms as well as applications of social choice ideas in the design and analysis of computing systems. Research on computational social choice now encompasses a variety of topics, including the algorithmic aspects of voting rules, computational barriers to strategic behaviour, collective decision-making in multi-agent systems, preference elicitation and communication issues in voting, fair division, computational aspects of weighted voting games, collective decision-making in combinatorial domains, logic-based formalisms for social choice problems, belief and judgement aggregation, and social software. The aim of this special issue is to build on the spectacular success of two workshops on computational social choice (COMSOC’06, Amsterdam and COMSOC’08, Liverpool), and provide a forum for reseachers in the area to present their ideas on a variety of topics. This special issue is composed of nine research articles and a short note that span a fairly wide range of issues, including voting, mechanism design, judgment aggregation, and fairness, and a similarly wide range of techniques, such as logic, experiments, and computational complexity. The paper by Agotnes, van der Hoek and Wooldridge defines a new modal logic for representing and reasoning about preference and judgment aggregation. The authors show that their logic can be used to express a variety of concepts and results from the social choice domain, including majority voting, the discursive paradox, Arrow’s theorem and Condorcet’s paradox. Baharad, Goldberger, Koppel and Nitzan propose a maximum likelihood estimation-based approach for aggregating voters’ opinions over several issues. They argue that their method is vastly superior to simple majority voting, and use both analytical and empirical evaluation to support their argument. Balan, Richards and Luke deal with the problem of selecting a sequence of actions when the actions have multiple beneficiaries. Their goal is to select the actions according to a lexicographic fairness criterion. The explore the complexity of this problem for both finite and infinite sequences of actions and present both algorithms and intractability results for this problem. The paper by Caminada and Pigozzi applies argumentation theory to judgment aggregation, and proposes aggregation operators satisfying not only the usual property of collective rationality, but another highly desirable property,

In his recent book William Riker (1982) argues that the results of socialchoice theory require that we lower our traditional aspirations for the role of elections in democratic theory. In particular, he concludes that socialchoice theory sounds the death knell for the 'populist' interpretation of voting, the view that election outcomes embody the 'collective will' of the electorate. Other, more modest claims about elections also fall to the theorems of social-choice, until we remain with the extremely modest claim embodied in Riker's 'liberal' interpretation of voting. On this view elections provide the possibility of rejecting unpopular leaders, and the major value of elections is as a weapon against government tyranny. Riker's argument proceeds in several stages. First, he reviews the analysis of social-choice in n-dimensional issue spaces. This literature conclusively demonstrates that the necessary and sufficient conditions for a majority voting equilibrium are so strict that one cannot expect them to ever be met.' Thus Riker consigns the notion of a 'majority will' to the same 'dust bin' to which earlier liberals (from Madison to Bentley) had consigned Rousseau's general will. Having discovered at the most abstract level that the search for a collective will of the electorate is pointless, for no such will exists, Riker proceeds to review the conclusions of social-choice theory about elections with a finite number of candidates. Here Arrow's theorem (1963) and its progeny take over. Not only do majority cycles still occur, but different systems with strong axiomatic credentials may select different candidates. Thus, even at this level the prospect is slight that we can endow election outcomes with great normative significance. But the prospects disappear almost entirely, in Riker's judgment, when we consider the manipulability theorems of

Kenneth J. Arrow was one of the most important intellectuals of the twentieth century, and his “impossibility theorem” is arguably the starting point of modern, axiomatic social choice theory. In this review, we begin with a brief discussion of Arrow's theorem and subsequent work that extended the result. We then discuss its implications for voting and constitutional systems, including a number of seminal results—both positive and negative—that characterize what such systems can accomplish and why. We then depart from this narrow interpretation of the result to consider more varied institutional design questions such as apportionment and geographical districting. Following this, we address the theorem's implications for measurement of concepts of fundamental interest to political science such as justice and inequality. Finally, we address current work applying social choice concepts and the axiomatic method to data analysis more generally.

By classic results in social choice theory, any reasonable preferential voting method sometimes gives individuals an incentive to report an insincere preference. The extent to which different voting methods are more or less resistant to such strategic manipulation has become a key consideration for comparing voting methods. Here we measure resistance to manipulation by whether neural networks of varying sizes can learn to profitably manipulate a given voting method in expectation, given different types of limited information about how other voters will vote. We trained over 100,000 neural networks of 26 sizes to manipulate against 8 different voting methods, under 6 types of limited information, in committee-sized elections with 5-21 voters and 3-6 candidates. We find that some voting methods, such as Borda, are highly manipulable by networks with limited information, while others, such as Instant Runoff, are not, despite being quite profitably manipulated by an ideal manipulator with full information. For the three probability models for elections that we use, the overall least manipulable of the 8 methods we study are Condorcet methods, namely Minimax and Split Cycle.

Competition results are traditionally decided based on referees' rating scores. However, due to the inconsistent evaluation standard and the different subjective wishes of referees, the evaluation results will be affected to some extent. Thus, the objectivity of competition results will be reduced. We propose a method by using referees' preference ordering for players instead of traditional scores to evaluate the competition result. We constructs a relationship of referees' preferences for players and calculates players' overall ranking based on social choice theory and group decision. By constructing a directed graph and calculating the value of strongest path, we can improve the objectivity of result evaluation and the difficulty of manipulating competition results. We proved that the proposed method satisfies the criteria of fairness, consistency, maneuverability and monotonicity. Competition results will not be changed due to individual referees' preferences, so we can obtain a more reasonable result. We make a theoretical analysis and a comprehensive experimental study using real datasets to verify the rationality and effectiveness of the proposed method.

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Politics. Please check back later for the full article.Narrowly understood, social choice theory is a specialized branch of applied logic and mathematics that analyzes abstract objects called preference aggregation functions, social welfare functions, and social choice functions. But more broadly, social choice theory identifies, analyzes, and evaluates rules that may be used to make collective decisions. So understood, social choice is a subfield of the social sciences that examines what may be called “voting rules” of various sorts. While social choice theory typically assumes a finite set of alternatives over which voter preferences are unrestricted, the spatial model of social choice assumes that policy alternatives can be represented by points in a space of one or more dimensions, and that voters have preferences that are plausibly shaped by this spatial structure.Social choice theory has considerable relevance for the study of legislative (as well as electoral) institutions. The concepts and tools of social choice theory make possible formal descriptions of legislative institutions such as bicameralism, parliamentary voting procedures, effects of decision rules (e.g., supramajority vs. simple majority rule and executive veto rules), sincere vs. strategic voting by legislators, agenda control, and other parliamentary maneuvers. Spatial models of social choice further enrich this analysis and raise additional questions regarding policy stability and change. Spatial models are used increasingly to guide empirical research on legislative institutions and processes.

This paper presents a proof of the Gibbard-Satterthwaite theorem. The importance of the theorem itself has been widely recognized, and there already exist several alternative proofs. The present one is offered because it is technically simple, direct, and emphasizes the role of pivotal individuals in the manipulation of voting schemes. Some indirect proofs of the theorem rely on the connections between voting schemes and social welfare functions, and one interesting corollary of such proofs is to provide new arguments in behalf of Arrow's condition of independence of irrelevant alternatives. Yet, a direct proof has the advantage of showing the complete independence of the manipulation phenomena from any requirement of consistency in choice under varying sets of alternatives. Also, our proof emphasizes the role of pivotal individuals in manipulation. A classical approach in social choice theory is to concentrate on the global distribution of power among groups of agenits, i.e., on the ability of coalitions to impose outcomes regardless of the preferences of agents not in the coalition. For example, standard proofs of Arrow's Theorem concentrate on decisive coalitions, to end up proving that there must exist one such coalition containing a single individual, the dictator. Our approach concentrates on the local distribution of power among single agents, i.e., on the ability of each single individual to determine the social outcome, given the preferences of all other members of society.' This approach may be more intuitive for some readers; moreover, it closely parallels the work on incentives in economic environments with public goods and suggests a possible way to unify results which have been obtained from different initial specifications. We do not attempt such a rejoinder here, but let us observe the following. For certain environments with public goods, where transfer payments make perfect sense, nontrivial strategy-proof mechanisms can be obtained by appropriately taxing pivotal agents so that they take into account the costs that their actions impose upon others.2 One explanation for the nonexistence of equivalent mechanisms in our case is that the environment, as specified, does not allow for a corresponding procedure whereby different costs to individuals could be attached to different acts of preference revelation.

Journal Article Algorithms for Social Choice Functions Get access Donald E. Campbell Donald E. Campbell Ontario Economic Council and University of Toronto Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 47, Issue 3, April 1980, Pages 617–627, https://doi.org/10.2307/2297312 Published: 01 April 1980 Article history Received: 01 June 1977 Accepted: 01 April 1979 Published: 01 April 1980

This paper shows that different institutional structures for aggregation of preferences under majority rule may generate social choices that are quite similar, so that the actual social choice may be rather insensitive to the choice of institutional rules. Specifically, in a multidimensional setting, where all voters have strictly quasi-concave preferences, it is shown that the "uncovered set" contains the outcomes that would arise from equilibrium behavior under three different institutional settings. The three institutional settings are two-candidate competition in a large electorate, cooperative behavior in small committees, and sophisticated voting behavior in a legislative environment where the agenda is determined endogenously. Because of its apparent institution-free properties, the uncovered set may provide a useful generalization of the core when a core does not exist. A general existence theorem for the uncovered set is proven, and for the Downsian case, bounds for the uncovered set are computed. These bounds show that the uncovered set is centered around a generalized median set whose size is a measure of the degree of symmetry of the voter ideal points.