Aggregating Sets of Judgments: Two Impossibility Results Compared1

The ``doctrinal paradox'' or ``discursive dilemma'' shows that propositionwise majority voting over the judgments held by multiple individuals on some interconnected propositions can lead to inconsistent collective judgments on these propositions. List and Pettit (2002) have proved that this paradox illustrates a more general impossibility theorem showing that there exists no aggregation procedure that generally produces consistent collective judgments and satisfies certain minimal conditions. Although the paradox and the theorem concern the aggregation of judgments rather than preferences, they invite comparison with two established results on the aggregation of preferences: the Condorcet paradox and Arrow's impossibility theorem. We may ask whether the new impossibility theorem is a special case of Arrow's theorem, or whether there are interesting disanalogies between the two results. In this paper, we compare the two theorems, and show that they are not straightforward corollaries of each other. We further suggest that, while the framework of preference aggregation can be mapped into the framework of judgment aggregation, there exists no obvious reverse mapping. Finally, we address one particular minimal condition that is used in both theorems – an independence condition – and suggest that this condition points towards a unifying property underlying both impossibility results.     

Ranking judgments in Arrow’s setting

In this paper, I investigate the relationship between preference and judgment aggregation, using the notion of ranking judgment introduced in List and Pettit (Synthese 140(1–2):207–235, 2004). Ranking judgments were introduced in order to state the logical connections between the impossibility theorem of aggregating sets of judgments proved in List and Pettit (Economics and Philosophy 18:89–110, 2002) and Arrow’s theorem (Arrow, Social choice and individual values, 1963). I present a proof of the theorem concerning ranking judgments as a corollary of Arrow’s theorem, extending the translation between preferences and judgments defined in List and Pettit (Synthese 140(1–2):207–235, 2004) to the conditions on the aggregation procedure.     

Impossibility Results for Infinite-Electorate Abstract Aggregation Rules

Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.     

Arrovian Aggregation of Generalised Expected-Utility Preferences: (Im)possibility Results by Means of Model Theory

Cerreia-Vioglio et al. (Econ Theory 48(2–3):341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.     

Decision framing in judgment aggregation

Judgment aggregation problems are language dependent in that they may be framed in different yet equivalent ways. We formalize this dependence via the notion of translation invariance, adopted from the philosophy of science, and we argue for the normative desirability of translation invariance. We characterize the class of translation invariant aggregation functions in the canonical judgment aggregation model, which requires collective judgments to be complete. Since there are reasonable translation invariant aggregation functions, our result can be viewed as a possibility theorem. At the same time, we show that translation invariance does have certain normatively undesirable consequences (e.g. failure of anonymity). We present a way of circumventing them by moving to a more general model of judgment aggregation, one that allows for incomplete collective judgments.     

Reasons,Coherence, and Group Rationality

If groups can have beliefs and other attitudes of their own, what determines which such attitudes the group rationally ought to have? A widespread presupposition is that group‐level beliefs should be a function of the beliefs of the group's members, and similarly for other attitudes. But a host of impossibility theorems show that no such aggregation function can satisfy intuitively attractive constraints while ensuring coherent group‐level attitudes. I argue that this presupposition is false. Group‐level attitudes should be a function of group‐level reasons (evidence, in the epistemic case), not individual‐level attitudes. This allows for a theory of group rationality that (i) bypasses a host of pessimistic results in the literature on judgment aggregation and (ii) treats rational individual‐level attitudes and rational group‐level attitudes in parallel.     

Deflecting Arrow by Aggregating Rankings of Multiple Correlated Criteria According to Borda

This paper has two objectives. The first is to review and address concerns raised by Hazelrigg that Arrow's impossibility theorem prevents the selection of rational aggregation methods for use in engineering trade studies. In addressing these concerns, the work of Saari is cited to establish the fact that the Borda count is the only ‘non‐dictatorial’ positional voting method that satisfies the criteria for a rational decision procedure while using complete information. Hence, the resulting rank ordering of the alternatives is the most reliable outcome. Several previous studies that use other aggregation methods are critiqued, and Borda is applied to examples to illustrate the differences in the outcomes. The second objective is to extend the applicability of Borda to include attributes such as cost, schedule duration, and certain technical and performance measures that are generally more reasonably described as correlated random variables. Exact Probabilities by Simulation with Borda, a method introduced by Hulkower that improves a technique by Book for determining which candidate in a trade study is the probable lowest‐cost alternative, is generalized to include multiple correlated criteria, each of which is expressed as a random variable and thus incorporates probabilistic uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.     

Task bias and the accuracy of judgment: Setting a baseline for expected group performance

When a judgment task evokes unbiased estimates (i.e. the errors in individual judgments are distributed randomly around the true value), mathematical aggregation of individual estimates, even by a simple arithmetic mean, often will outperform all group members. However, when a task evokes biased estimates, mathematical aggregation does not perform so well. In this study, simulated data were accumulated to specify the expected' accuracy of mathematical aggregation relative to the accuracy of observed judgment of individual group members under varying conditions of task bias. Three types of judgment tasks were employed: (1) single-estimate, holistic tasks, (2) multiple-estimate, ranking tasks, and (3) multi-cue, decomposed tasks. Findings indicated across all task types that a large percentage of judgment-making group estimates formed strictly by computing the arithmetic mean of individual estimates performed better than their most capable members when a judgment task evoked little or no bias, a result particularly pronounced for ranking tasks. When the task was more greatly bias-evoking, a large percentage of parallel groups performed more poorly than average (or median) members, again a pattern more starkly evident for ranking tasks. These results suggest that the extent to which a judgment task evokes bias in a population of prospective group members is an important explanatory variable deserving much greater attention in the study of group performance. For example, an assertion about the efficacy of a particular group intervention based on a reliable demonstration of group performance as accurate as the most capable members may be unfounded when a task evokes no bias, since the baseline standard under such conditions should be much higher. By selecting tasks and populations that jointly produced highly biased estimates, researchers can lower the performance floor enough to detect (with reasonably small samples of groups) experimental effects should they occur.     

The Logic of Group Decisions: Judgment Aggregation

Judgment aggregation studies how individual opinions on a given set of propositions can be aggregated to form a consistent group judgment on the same propositions. Despite the simplicity of the problem, seemingly natural aggregation procedures fail to return consistent collective outcomes, leading to what is now known as the doctrinal paradox. The first occurrences of the paradox were discovered in the legal realm. However, the interest of judgment aggregation is much broader and extends to political philosophy, epistemology, social choice theory, and computer science. The aim of this paper is to provide a concise survey of the discipline and to outline some of the most pressing questions and future lines of research.     

Use of situational judgment tests to predict job performance: a clarification of the literature

Although situational judgment tests have a long history in the psychological assessment literature and continue to be frequently used in employment contexts, there has been virtually no summarization of this literature. The purpose of this article is to review the history of such tests and present the results of a meta-analysis on criterion-related and construct validity. On the basis of 102 coefficients and 10,640 people, situational judgment tests showed useful levels of validity (rho = .34) that were generalizable. A review of 79 correlations between situational judgment tests and general cognitive ability involving 16,984 people indicated that situational judgment tests typically evidence relationships with cognitive ability (rho = .46). On the basis of the literature review and meta-analytic findings, implications for the continued use of situational judgment tests are discussed, particularly in terms of recent investigations into tacit knowledge.     

Endless Future: A Persistent Thorn in the Kalām Cosmological Argument

Abstract

Wes Morriston contends that William Lane Craig's argument for the impossibility of a beginningless past results in an equally good argument for the impossibility of an endless future (or at least a pre-determined one). Craig disagrees. I show that Craig's reply reveals a commitment to an unmotivated position concerning the relationship between actuality and the actual infinite. I then assess alternative routes to the impossibility of a beginningless past that have been offered in the literature, and show that, contrary to initial appearances, these routes similarly seem to support the impossibility of an endless future.     

Aggregating with reason(s)

Judgment aggregation is naturally applied to the modeling of collective attitudes. In the individual case, we represent agents as having not just beliefs, but also as supporting them with reasons. Can the Judgment Aggregation help model a concept of collective reason? I argue that the resources of the standard judgment aggregation framework are insufficiently general. I develop a generalization of the framework that improves along this dimension. In the new framework, new aggregation rules become available, as well as a natural account of collective reasons.     

Which Worlds are Possible? A Judgment Aggregation Problem

Suppose the members of a group (e.g., committee, jury, expert panel) each form a judgment on which worlds in a given set are possible, subject to the constraint that at least one world is possible but not all are. The group seeks to aggregate these individual judgments into a collective judgment, subject to the same constraint. I show that no judgment aggregation rule can solve this problem in accordance with three conditions: “unanimity,” “independence” and “non-dictatorship,” Although the result is a variant of an existing theorem on “group identification” (Kasher and Rubinstein, Logique et Analyse 160:385–395, 1997), the aggregation of judgments on which worlds are possible (or permissible, desirable, etc.) appears not to have been studied yet. The result challenges us to take a stance on which of its conditions to relax.     

Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation

The aggregation of individual judgments on logically interconnected propositions into a collective decision on the same propositions is called judgment aggregation. Literature in social choice and political theory has claimed that judgment aggregation raises serious concerns. For example, consider a set of premises and a conclusion where the latter is logically equivalent to the former. When majority voting is applied to some propositions (the premises) it may give a different outcome than majority voting applied to another set of propositions (the conclusion). This problem is known as the discursive dilemma (or paradox). The discursive dilemma is a serious problem since it is not clear whether a collective outcome exists in these cases, and if it does, what it is like. Moreover, the two suggested escape-routes from the paradox—the so-called premise-based procedure and the conclusion-based procedure—are not, as I will show, satisfactory methods for group decision-making. In this paper I introduce a new aggregation procedure inspired by an operator defined in artificial intelligence in order to merge belief bases. The result is that we do not need to worry about paradoxical outcomes, since these arise only when inconsistent collective judgments are not ruled out from the set of possible solutions.     

A solution to the discursive dilemma

An impossibility result pertaining to the aggregation of individual judgements is thought by many to have significant implications for political theory, social epistemology and metaphysics. When members of a group hold a rational set of judgments on some interconnected questions, the theorem shows, it isn’t always (logically) possible for them to aggregate their judgements into a collective one in conformity with seemingly very plausible constraints. I reject one of the constraints which engender the dilemma. The analogy with the lottery paradox, I argue, shows that rational belief needn’t be consistent. So the alleged implications of the dilemma are dispelled.     

Skepticism Between Excessiveness and Idleness

Abstract: Skepticism seems to have excessive consequences: the impossibility of successful enquiry and differentiated judgment. Yet if skepticism could avoid these consequences, it would seem idle. I offer an account of moderate skepticism that avoids both problems. Moderate skepticism avoids excessiveness because skeptical reflection and ordinary enquiry are immune from one another: a skeptical hypothesis is out of place when raised with in an ordinary enquiry. Conversely, the result of an ordinary enquiry cannot be used to disprove skepticism. This 'immunity' can be explained by theories such as contextualism, or sensitive invariantism. Moderate skepticism avoids idleness, because it can eliminate dogmatic elements from our commitments. An analogy is used to illustrate this: Consider someone who is rootless—someone who doesn't have a home. She won't take this conclusion to undermine her judgment that she is flying home for the holidays—even if she is sleeping in the guest bedroom. Similarly, a skeptic won't take the skeptical conclusion to undermine ordinary claims to know. Yet concluding that one is rootless is significant: it can shape one's commitments; for instance it can check one's nationalism. Similarly, accepting the skeptical conclusion is significant; it can undermine dogmatic commitments and ultimately bring about intellectual catharsis.     

Classes of models for selected axiomatic theories of choice

Adding a reversibility axiom to the other axioms of Luce's (1959) probabilistic ranking theory results in an impossibility theorem—that all alternatives in an alternative set are equally likely to be chosen (i.e., that preferences are random). This impossibility theorem is generally avoided by removing the reversibility axiom. Using simple algebraic methods such a modified theory is shown to contain a theorem similiar to the impossibility result. These results are discussed within the framework of mathematical model theory (model theory deals with the relations between sets of sentences (theories) and the structures which satisfy these sentences (models)) to illustrate the applicability of model theory as an analytic tool in theory development.     

The genetics of gambling and behavioral addictions

Behavioral addictions are considered as the repetitive occurrence of impulsive behaviors without consideration of their potential negative consequences. These addictions represent an increasing cost to society and are an important new field of research in psychiatric genetics. There has been a growing body of evidence on the familial aggregation and genetic influences on the development of behavioral addictions and mainly on pathological gambling. The aim of this article is to critically review findings of family and molecular genetic studies on behavioral addictions, focusing on pathological gambling and commenting on other disorders where appropriate. This review provides a comprehensive approach to genetic studies on behavioral addiction and points out the necessity of expanding the genetic research in this field. Future directions for genetic studies in this field are also discussed.