Complexity Results for Aggregating Judgments using Scoring or Distance-Based Procedures

Judgment aggregation is an abstract framework for studying collective decision making by aggregating individual opinions on logically related issues. Important types of judgment aggregation methods are those of scoring and distance-based methods, many of which can be seen as generalizations of voting rules. An important question to investigate for judgment aggregation methods is how hard it is to find a collective decision by applying these methods. In this article we study the complexity of this "winner determination" problem for some scoring and distance-based judgment aggregation procedures. Such procedures aggregate judgments by assigning values to judgment sets. Our work fills in some of the last gaps in the complexity landscape for winner determination in judgment aggregation. Our results reaffirm that aggregating judgments is computationally hard and strongly point towards the necessity of analyzing approximation methods or parameterized algorithms in judgment aggregation

Egalitarian Judgment Aggregation

Egalitarian considerations play a central role in many areas of social choice theory. Applications of egalitarian principles range from ensuring everyone gets an equal share of a cake when deciding how to divide it, to guaranteeing balance with respect to gender or ethnicity in committee elections. Yet, the egalitarian approach has received little attention in judgment aggregation -- a powerful framework for aggregating logically interconnected issues. We make the first steps towards filling that gap. We introduce axioms capturing two classical interpretations of egalitarianism in judgment aggregation and situate these within the context of existing axioms in the pertinent framework of belief merging. We then explore the relationship between these axioms and several notions of strategyproofness from social choice theory at large. Finally, a novel egalitarian judgment aggregation rule stems from our analysis; we present complexity results concerning both outcome determination and strategic manipulation for that rule.Comment: Extended version of paper in proceedings of the 20th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 202

Hunting for Tractable Languages for Judgment Aggregation

Judgment aggregation is a general framework for collective decision making that can be used to model many different settings. Due to its general nature, the worst case complexity of essentially all relevant problems in this framework is very high. However, these intractability results are mainly due to the fact that the language to represent the aggregation domain is overly expressive. We initiate an investigation of representation languages for judgment aggregation that strike a balance between (1) being limited enough to yield computational tractability results and (2) being expressive enough to model relevant applications. In particular, we consider the languages of Krom formulas, (definite) Horn formulas, and Boolean circuits in decomposable negation normal form (DNNF). We illustrate the use of the positive complexity results that we obtain for these languages with a concrete application: voting on how to spend a budget (i.e., participatory budgeting).Comment: To appear in the Proceedings of the 16th International Conference on Principles of Knowledge Representation and Reasoning (KR 2018