Complexity of Judgment Aggregation

Abstract

We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can influence the outcome of a judgment aggregation procedure in her favour by reporting insincere judgments (the strategic manipulation problem); and (3) deciding whether a given judgment aggregation scenario is guaranteed to result in a logically consistent outcome, independently from what the judgments supplied by the individuals are (the problem of the safety of the agenda). We provide results both for specific aggregation procedures (the quota rules, the premise-based procedure, and a distance-based procedure) and for classes of aggregation procedures characterised in terms of fundamental axioms

Ontology merging as social choice: judgment aggregation under the open world assumption

“…Now a simple reformulation of a known result due to Nehring and Puppe shows that an agenda Φ is safe for the majority rule if and only if it satisfies the median property (Nehring and Puppe, 2007;List and Puppe, 2009;Endriss et al, 2010). This result can be refined if we put restrictions on the range of profiles on Φ that we consider.…”

“…We have seen that the majority rule can produce inconsistent collective ontologies. Following Endriss et al (2010), we call an agenda Φ safe for a given aggregator F if F (O) is consistent for any profile O ∈ On(Φ) N . We will now identify necessary and sufficient conditions for the safety of Φ under the majority rule.…”

Applications of Logic in Social Choice Theory

“…Over the past decade or so, this paradox of JA has given rise to a fast moving area of research, spanning Legal Theory, Philosophy, Economic Theory and AI. For instance, we have recently begun to analyse the computational complexity of a number of problems that naturally arise in JA [2,3].…”

Ontology Merging as Social Choice

“…This is an appropriate assumption in JA, but not here. 4 We also propose three axioms that are specific to ontology aggregation and that do not have a counterpart in standard SCT or JA. The first is groundedness: a formula should only occur in the collective ontology if it is included in at least one of the individual ontologies, i.e., if it is an element of O 1 ∪ · · · ∪ O n , the support of a given profile (O 1 , .…”

“…Following Endriss et al [4], we call L safe for a given aggregator F if F (O) is satisfiable for any profile O ∈ On(L) N . We will now identify necessary and sufficient conditions for the safety of L under the majority rule.…”