Declarative Approaches to Outcome Determination in Judgment Aggregation

Abstract

Judgment aggregation (JA) offers a generic formal framework for modeling various set- tings involving information aggregation by social choice mechanisms. For many judgment aggregation rules, computing collective judgments is computationally notoriously hard. The central outcome determination problem, in particular, is often complete for higher levels of the polynomial hierarchy. This complexity barrier makes it challenging to de- velop practical exact algorithms to outcome determination. Taking on this challenge, in this work we develop practical exact algorithms for outcome determination under a range of the most central JA rules—namely Kemeny, Slater, MaxHamming, Young, Dodgson, Reversal scoring, Condorcet, Ranked agenda, and LexiMax—by harnessing the declarative approach, in particular, Boolean satisfiability (SAT) and integer programming techniques. For the Kemeny, Slater, MaxHamming, Young, and Dodgson rules, we detail direct ap- proaches based on maximum satisfiability (MaxSAT) and integer programming. For the Reversal scoring, Condorcet, Ranked agenda, and LexiMax rules, we develop iterative al- gorithms, including algorithms based on the counterexample-guided abstraction refinement (CEGAR) paradigm, making use of recent advances in incremental MaxSAT solving and preferential SAT-based reasoning. We provide an open-source implementation of the algo- rithms, and empirically evaluate them using real-world preference data. We compare the performance of our implementation to a recent approach which makes use of declarative solver technology for answer set programming (ASP). The results demonstrate that our approaches scale significantly beyond the reach of the ASP-based algorithms for all of the judgment aggregation rules considered.