Social Choice Theory

A standard formulation of luck-egalitarianism says that ‘it is [in itself] bad – unjust and unfair – for some to be worse off than others [through no fault or choice of their own]’, where ‘fault or choice’ means substantive responsibility-generating fault or choice. This formulation is ambiguous: one ambiguity concerns the possible existence of a gap between what is true of each worse-off individual and what is true of the group of worse-off individuals, fault or choice-wise, the other concerns the (...) notion of fault. I show that certain ways of resolving these ambiguities lead to counterintuitive results; and that the most plausible way of resolving them leads to a theory of distributive justice in which responsibility plays a role significantly different from that in standard luck-egalitarian thinking. My main conclusion here is that luck-egalitarianism is best formulated as the view that it is [in itself] bad – unjust and unfair – for an individual to be worse off than others if, and only if, her being worse off does not fit the degree to which she is at fault in a not purely prudential sense. (shrink)

There are various ways to reach a group decision on a factual yes–no question. One way is to vote and decide what the majority votes for. This procedure receives some epistemological support from the Condorcet Jury Theorem. Alternatively, the group members may prefer to deliberate and will eventually reach a decision that everybody endorses—a consensus. While the latter procedure has the advantage that it makes everybody happy, it has the disadvantage that it is difficult to implement, especially for larger groups. (...) Besides, the resulting consensus may be far away from the truth. And so we ask: Is deliberation truth-conducive in the sense that majority voting is? To address this question, we construct a highly idealized model of a particular deliberation process, inspired by the movie Twelve Angry Men, and show that the answer is ‘yes’. Deliberation procedures can be truth-conducive just as the voting procedure is. We then explore, again on the basis of our model and using agent-based simulations, under which conditions it is better epistemically to deliberate than to vote. Our analysis shows that there are contexts in which deliberation is epistemically preferable and we will provide reasons for why this is so. (shrink)

The term “rationality” is applied to many different things, from beliefs and preferences to decisions and choices, actions and behaviors, people, collectivities, andinstitutions. Therefore this paper will limit its considerations only to social preferences and choices in order to clarify the role of rationality in social explanation. The paper will focus on degrees of rationality, calling upon the concept of transitivity for help.

Professor Kenneth J. Arrow is one of the most distinguished economic theorists. He has played a major role in shaping the subject and is honoured by the publication of three volumes of essays on economic theory. Each volume deals with a different area of economic theory. The books include contributions by some of the best economic theorists from the United States, Japan, Israel and Europe.

Antonio Quesada (Public Choice 130:395–400, 2007) argues that a dictator has no more than two to three times the ‘average power’ of a non-dictatorial voter. If Quesada is correct, then his argument has major consequences for social choice theory; for instance, it warrants reconsidering the significance of Arrow’s Theorem. If Quesada is incorrect, however, then his position is dangerously misleading. This paper argues that Quesada is wrong. His argument depends on his own formal account of power, an account that is (...) implausible because it disregards a basic insight common to the standard characterisations of voting power: the idea that one has power over an outcome to the extent that one is able to change that outcome. Claims about power have a counterfactual component; to assert that an individual actually has determined an outcome is also to make an assertion about what would have been the case had that individual acted differently. We can employ David Lewis’s influential account of counterfactuals to show, contra Quesada, that in a dictatorship, nondictatorial individuals and groups cannot possibly determine a social preference. In short, Quesada is fundamentally mistaken about power, and thus also about the distribution of power in a dictatorship. (shrink)

Combines a theoretical interest in the mathematics of voting procedures with practical interest in the circumstances in which votes are cast. The most important results in the theory of voting are surveyed, and the differences between the principal types of voting procedures are explained.

Loren King | : Liberal democracies encourage citizen participation and protect our freedoms, yet these regimes elect politicians and decide important issues with electoral and legislative systems that are less inclusive than other arrangements. Some citizens inevitably have more influence than others. Is this a problem? Yes, because similarly just but more inclusive systems are possible. Political theorists and philosophers should be arguing for particular institutional forms, with particular geographies, consistent with justice. | : Les démocraties libérales encouragent la participation (...) citoyenne et protègent nos libertés. Pourtant, ces régimes élisent des politiciens et décident de problèmes importants via les systèmes électoral et législatif, qui sont moins inclusifs que d’autres arrangements. Certains citoyens ont inévitablement plus d’influence que d’autres. Est-ce un problème? Oui, parce que des systèmes similairement justes, mais plus inclusifs, sont possibles. Les théoriciens et philosophes politiques devraient ainsi plaider en faveur de formes institutionnelles particulières, aux géographies précises, en cohérence avec la justice. (shrink)

This paper argues that standard theories of judgment aggregation cannot apply to scientific practice, since science is a temporally extended process that involves both different individuals and different hypotheses during that process. Thus, for example, we seem to have no idea how to determine the judgments of dead scientists about theoretical alternatives that were proposed after their death. The paper then proposes an algorithm for judgment aggregation to try to address some of these challenges.

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they should be competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justified depends on the notion of uncertainty or probability employed; no such notion renders both premises simultaneously justified. Especially the independence assumption should be weakened.

In this paper, we show that Arrow’s well-known impossibility theorem is instrumental in bringing the ongoing discussion about verisimilitude to a more general level of abstraction. After some preparatory technical steps, we show that Arrow’s requirements for voting procedures in social choice are also natural desiderata for a general verisimilitude definition that places content and likeness considerations on the same footing. Our main result states that no qualitative unifying procedure of a functional form can simultaneously satisfy the requirements of Unanimity, (...) Independence of irrelevant alternatives and Non-dictatorship at the level of sentence variables. By giving a formal account of the incompatibility of the considerations of content and likeness, our impossibility result makes it possible to systematize the discussion about verisimilitude, and to understand it in more general terms. (shrink)

In his 2010 paper “Philosophical Naturalism and Intuitional Methodology”, Alvin I. Goldman invokes the Condorcet Jury Theorem in order to defend the reliability of intuitions. The present note argues that the original conditions of the theorem are all unrealistic when analysed in connection to the case of intuitions. Alternative conditions are discussed.

The Condorcet jury theorem (CJT) is based on the assumption of homogeneous voters who imperfectly know the correct policy. We reassess the validity of the CJT when voters are homogeneous and each knows the correct decision with an average probability of more than a half.

Utilitarianism and other aggregationist moral theories view the public interest or the general welfare as an aggregate of individual goods. But critics of these theories question whether there is adequate justification for employing the concept of an aggregate social good. How are we supposed to sum up individual interests? Is it even possible to compare the utilities of different people or to assign values to individual utilities that can be added or subtracted? If not, how is the general good to (...) be aggregated? Critics have also raised concerns about the aggregative approach in ethics - concerns about its implications for distributive justice, individual liberty and democratic institutions. The essays in this volume explore these issues and address related questions. Some of them examine specific objections to aggregation, others analyze the very idea of a social good or social welfare. Other essays discuss the application of aggregative principles to particular problems. (shrink)

In a recent article I compared the problem of theory choice, in which scientists must choose between competing theories, with the problem of social choice, in which society must choose between competing social alternatives. I argued that the formal machinery of social choice theory can be used to shed light on the problem of theory choice in science, an argument that has been criticized by Michael Morreau and Jacob Stegenga. This article replies to Morreau’s and Stegenga’s criticisms.

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Arrow's Theorem implies that the problem of cyclical majorities is endemic any non-dictatorial system of preference aggregation that meets certain minimal conditions of fairness. The present study attempts to access the significance of this result to the theory and practice of democratic government. ;The analysis begins with an empirical examination of individual and social level preferences for candidates in several recent American presidential elections. It is argued that clear Condorcet winners existed in each contest in question. Further, it was found (...) that individual preferences tended, on whole, to be both transitive and single peaked along an ideological dimension. The significance of these findings to the American electoral process in particular and the economic analysis of politics in general are discussed. ;Attention then turns to an appraisal of several tradition models of representative democracy fare given the Pandora's Box of Arrow's Theorem and related impossibility results. It is maintained that the only theory to remain theoretically coherent is the highly minimalist notion of democracy associated with contemporary liberalism. However, it is maintained this theory might form a foundation upon which more robust visions of democracy might be built. ;Moving from theories of representation to those of direct participation, it is argued that while the General Will can be construed in such a way that it survives the challenge posed by the Arrow problem, such can be done only by via the importation of assumptions that are empirically dubious, normatively troubling, or otherwise so heroic as to make tax credulity. ;The analysis concludes that no acceptable theory of democracy can be based upon the assignment of particular properties to the results of majority rule, as the previous results indicate. Accordingly, an alternative model based in the inherent value of participation itself, as opposed to the outcomes produced via participation, is proposed. It is argued that this conception of democracy remains theoretically coherent in the face of the aggregation problem. The implications of this result to the real world of democratic government are discussed. (shrink)

In the final part, Arrow's own formal treatment of the social choice problem is examined and its relation to the preference aggregation model is analyzed. Particular attention is paid to Arrow's discussion of independence of irrelevant alternatives, and, with the aid of some additional formal work, much of what has been found problematic in this is resolved. Finally, this analysis is used as the basis of a critique of the "received view" in the literature about independence of irrelevant alternatives. ;In (...) the second part, several formal results from the literature are adapted to the present formal framework and interpreted in preference aggregation terms. These include both characterization theorems for specific classes of decision rules, and impossibility results, such as the Arrow theorem itself. Most of these results utilize the requirement of context-independence, and its role in the proofs is analyzed. Some discussion as well is offered about the characterization of rules that do not satisfy the requirement of context--independence. ;The treatment of these matters is divided into three parts. In the first part, the basic formal representation of preference aggregation rules is laid out. A variety of the formal conditions from the literature are stated and discussed in preference aggregation terms. Several of the conditions usually called "rational choice" or "revealed preference" conditions are included. One of these, here called the "Weak Axiom of Revealed Preference", is singled out as the formal embodiment of the requirement of context-independence. Various examples of preference aggregation rules are stated and assessed vis-a-vis the formal conditions presented. ;A subsidiary theme, which is dealt with throughout much of the discussion, is the nature, import, and desirability of independence-or-irrelevant-alternatives conditions. A general informal requirement on preference aggregation rules, which is called "the requirement of context-independence", is characterized. This requirement, it is argued, is a basis for the proper understanding of the several formal conditions that have been called "independence of irrelevant alternatives" in the literature. A variety of formal results, including a new version of Arrow's "General Possibility Result" for social choice rules, are offered in support of this contention. ;The purpose of this work is to provide an analysis of the interrelationships among formal models for choice under conditions of uncertainty, social choice, and multicriterial decision-making. A simplified version of the usual formalism for the first named of these problems is adopted. It is argued that this can be used, with appropriate interpretations, for all three problems. Decision rules for these problems are construed in the abstract form of preference aggregation rules: rules for the selection of alternatives which are "best", in some sense, in the face of conflicting evaluations. The multicriterial interpretation is taken as a touchstone for the understanding of the proposed model. (shrink)

This book elucidates the theoretical structure and scope of interpersonal and intra-personal aggregation--a trade-off between benefits to a group of individuals and losses to another group of individuals--and defends a form of aggregation -- formal aggregation -- that resolves a variety of outstanding problems arising from the conventional understanding of aggregation, including the Number Problem concerning the moral relevance of the number of individuals.

In axiomatic approaches to expert opinion aggregation, so-called independence conditions have been ubiquitous. Such conditions dictate that the group value assigned to each decision variable should depend only on the values assigned by individuals to that variable, taking no account of values that they assign to other variables. This radically anti-holistic stricture on the synthesis of expert opinion severely limits the set of allowable aggregation methods. As we show, the limitations are particularly acute in the case of three or more (...) variables which must be assigned nonnegative real values summing to a fixed positive real number s. For if the subset V of [0,s] comprising the allowable values of the variables satisfies the closure conditions 0 is an element of V ; if x is an element of V, then s-x is an element of V ; and if x and y are elements of V and x+y is an element of [0,s], then x+y is an element of V, then, if V is finite, which is always the case in practice, subjecting the aggregation of such s-allocations to an independence condition allows only for dictatorial or imposed aggregation. (shrink)

Selinger provides a new take on what is being referred to in the computational literature as ‘structured argumentation’. In this commentary the differences and similarities with existing work are highlighted as a way of demonstrating how philosophical and computational approaches to argumentation are increasingly coming together and complementing one another.

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) this idea by introducing a 'premise-based' approach to probabilistic opinon pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

What is wrong with participating in a democratic decision-making process, and then doing something other than the outcome of the decision? It is often thought that collective decision-making entails being prima facie bound to the outcome of that decision, although little analysis has been done on why that is the case. Conventional perspectives are inadequate to explain its wrongness. I offer a new and more robust analysis on the nature of voting: voting when you will accept the outcome only if (...) the decision goes your way is an act of bad faith: you are not taking part in a ‘process that decides what we will do’. This analysis sheds light on understanding the intrinsic nature of voting and what we are doing when we make decisions collectively. (shrink)

There is no consensus on the properties of voting power indices when there is a large number of voters in a weighted-voting body. On the one hand, in some real-world cases that have been studied the power indices have been found to be nearly proportional to the weights (e.g., the EUCM, US Electoral College); this is true for both the Penrose-Banzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (known subsequently (...) as the Penrose limit theorem, that has been shown to hold under certain conditions). On the other hand, we have the older literature from cooperative game theory, due to Shapley and his collaborators, showing that, where there is a finite number of voters whose weights remain constant in relative terms, and where the quota remains constant in relative terms, while the total number of voters increases without limit, the powers of the voters with finite weight tend to limiting values that are, in general, not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies (e.g., the IMF/WB boards, corporate shareholder meetings). This paper proposes a restatement of the Penrose limit theorem and shows that, in general, both the 'classical' power indices converge in the limit to proportionality with weights as the Laakso-Taagepera index of political fragmentation increases. This new version reconciles the different theoretical and empirical results that have been found for large voting bodies. (shrink)

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to …lling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) Our …rst theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the speci…c kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation. (shrink)