Aggregating Sets of Judgments: An Impossibility Result

2009

Collective decision-making is a familiar feature of our social, political, and economic lives. It ranges from the relatively trivial (e.g. the choice of the next family car) to the globally significant (e.g. whether or not a country should go to war). Yet, whether trivial or globally significant, such decisions involve a number of challenging problems. These problems arise in the standard social choice setting, where individuals differ in their preferences. They also arise in the standard decision-making setting, where individuals share the same preferences, but differ in their decisional capabilities. The distinctive feature of Collective Preference and Choice is that it looks at classical aggregation problems that arise in three closely related areas: social choice theory, voting theory, and group decision-making under uncertainty. Using a series of exercises and examples, the book explains these problems with reference to a number of important contributions to the study of collec...

Autonomous Agents and Multi-Agent Systems, 2011

Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow's theorem and Condorcet's paradox-which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation-including classical preference aggregation. As a case study we present a logical study of, including a formal proof of, the neutrality lemma, the main ingredient in a well-known proof of Arrow's theorem.

Lecture Notes in Computer Science, 2014

Similar to Arrow's impossibility theorem for preference aggregation, judgment aggregation has also an intrinsic impossibility for generating consistent group judgment from individual judgments. Removing some of the pre-assumed conditions would mitigate the problem but may still lead to too restrictive solutions. It was proved that if completeness is removed but other plausible conditions are kept, the only possible aggregation functions are oligarchic, which means that the group judgment is purely determined by a certain subset of participating judges. Instead of further challenging the other conditions, this paper investigates how the judgment from each individual judge affects the group judgment in an oligarchic environment. We explore a set of intuitively demanded conditions under abstentions and design a feasible judgment aggregation rule based on the agents' hierarchy. We show this proposed aggregation rule satisfies the desirable conditions. More importantly, this rule is oligarchic with respect to a subset of agenda instead of the whole agenda due to its literal-based characteristics.

Journal of Mathematical Psychology, 1985

He is a specialist in issues of representation and reapportionment and has written on a variety of topics dealing with collective decision-making. His most recent co-edited books are Information Pooling and Group Decision-Making, 1985, and Electoral Laws and Their Political Consequences,

Group Decision and Negotiation, 2005

Group decisions are of longstanding interest to researchers from a wide spectrum of disciplines. Group Decision Support Systems (GDSS) can play a vital role in situations where multiple persons are involved, each having their own private perceptions of the context and the decision problem to be tackled. In such an environment the conflict between the members of the planning group is not an unusual situation. Multiple criteria decision aid (MCDA) methods may be a useful tool in coping with such interpersonal conflicts where the aim is to achieve consensus between the group members. This paper combines two well-known multicriteria methods, based on the notion of aggregation of preferences, in order to construct a consensus seeking methodology for collective decision-making.

Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems - AAMAS '07, 2007

Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The recently emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation of such rules. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow's theorem and Condorcet's paradox -which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation -including classical preference aggregation.

Social Choice and Welfare, 2008

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinstein's group identi…cation problem.

Lecture Notes in Computer Science, 2015

2012

Author's abstract. In a situation of decision under uncertainty, a decision maker wishes to choose according to the maxmin expected utility rule, and he can observe the preferences of a set of experts who all share his utility function and all use the maxmin EU rule. This paper considers rules for aggregating the experts’ sets of priors into a set that the decision maker can use. It is shown that, in a multi profile setting, among the rules that allow the decision maker’s evaluation of an act to depend only on the experts'evaluations of that act, the only rule satisfying the standard unanimity or Pareto condition on preferences is the “set of weights” aggregation rule, according to which the decision maker’s set of priors is the set of weighted averages of the priors in the experts’ sets, with the weights taken from a set of probability vectors over the experts. An analogous characterisation is obtained for variational preferences.

Synthese, 2004

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.