Succinctness of Languages for Judgment Aggregation

We review several different languages for collective decision making problems, in which agents express their judgments, opinions, or beliefs over elements of a logically structured domain. Several such languages have been proposed in the literature to compactly represent the questions on which the agents are asked to give their views. In particular, the framework of judgment aggregation allows agents to vote directly on complex, logically related formulas, whereas the setting of binary aggregation asks agents to vote on propositional variables, over which dependencies are expressed by means of an integrity constraint. We compare these two languages and some of their variants according to their relative succinctness and according to the computational complexity of aggregating several individual views expressed in such languages into a collective judgment. Our main finding is that the formula-based language of judgment aggregation is more succinct than the constraint-based language of binary aggregation. In many (but not all) practically relevant situations, this increase in succinctness does not entail an increase in complexity of the corresponding problem of computing the outcome of an aggregation rule.

[1]  Marco Cadoli,et al.  A Survey on Knowledge Compilation , 1997, AI Commun..

[2]  Umberto Grandi,et al.  Binary Aggregation with Integrity Constraints , 2011, IJCAI.

[3]  Sébastien Konieczny,et al.  Logic Based Merging , 2011, J. Philos. Log..

[4]  Gabriella Pigozzi,et al.  Judgment Aggregation: A Primer , 2014, Judgment Aggregation.

[5]  Klaus Nehring,et al.  Abstract Arrowian aggregation , 2010, J. Econ. Theory.

[6]  Marija Slavkovik,et al.  How Hard is it to Compute Majority-Preserving Judgment Aggregation Rules? , 2014, ECAI.

[7]  Ulrich Endriss,et al.  Complexity of the Winner Determination Problem in Judgment Aggregation: Kemeny, Slater, Tideman, Young , 2015, AAMAS.

[8]  Lawrence G. Sager,et al.  The One and the Many: Adjudication in Collegial Courts , 1993 .

[9]  Ulrich Endriss,et al.  Lifting integrity constraints in binary aggregation , 2013, Artif. Intell..

[10]  Marcus Pivato,et al.  The Condorcet set: Majority voting over interconnected propositions , 2014, J. Econ. Theory.

[11]  Ulrich Endriss,et al.  Judgment Aggregation under Issue Dependencies , 2016, AAAI.

[12]  Marc Pauly,et al.  Logical Constraints on Judgement Aggregation , 2006, J. Philos. Log..

[13]  Pierre Marquis,et al.  A Knowledge Compilation Map , 2002, J. Artif. Intell. Res..

[14]  Franz Dietrich.  Scoring rules for judgment aggregation , 2014, Soc. Choice Welf..

[15]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[16]  Ron Holzman,et al.  Aggregation of binary evaluations for truth-functional agendas , 2009, Soc. Choice Welf..

[17]  Franz Dietrich,et al.  A generalised model of judgment aggregation , 2007, Soc. Choice Welf..

[18]  F. Dietrich,et al.  Judgment Aggregation By Quota Rules , 2007 .

[19]  Jérôme Lang,et al.  Expressive Power and Succinctness of Propositional Languages for Preference Representation , 2004, KR.

[20]  Lirong Xia,et al.  Voting in Combinatorial Domains , 2016, Handbook of Computational Social Choice.

[21]  Gabriella Pigozzi,et al.  Judgment aggregation rules based on minimization , 2011, TARK XIII.

[22]  Christian List,et al.  Arrow’s theorem in judgment aggregation , 2005, Soc. Choice Welf..

[23]  Christian List,et al.  Judgment aggregation without full rationality , 2008, Soc. Choice Welf..

[24]  Vincent Conitzer,et al.  Handbook of Computational Social Choice , 2016 .

[25]  Yann Chevaleyre,et al.  Representing Utility Functions via Weighted Goals , 2009, Math. Log. Q..

[26]  Ulrich Pferschy,et al.  The traveling group problem , 2007, Soc. Choice Welf..

[27]  Daniel N. Osherson,et al.  Methods for distance-based judgment aggregation , 2009, Soc. Choice Welf..

[28]  Ulrich Endriss,et al.  Complexity of Judgment Aggregation , 2012, J. Artif. Intell. Res..

[29]  Wiebe van der Hoek,et al.  On the succinctness of some modal logics , 2013, Artif. Intell..

[30]  C. List,et al.  Aggregating Sets of Judgments: An Impossibility Result , 2002, Economics and Philosophy.

[31]  Ron Holzman,et al.  Aggregation of binary evaluations , 2010, J. Econ. Theory.

[32]  P. Mongin.  Judgment Aggregation , 2011 .

[33]  Ulrich Endriss,et al.  Binary Aggregation by Selection of the Most Representative Voters , 2014, AAAI.

[34]  Bernhard Nebel,et al.  On the Compilability and Expressive Power of Propositional Planning Formalisms , 2000, J. Artif. Intell. Res..

[35]  Sébastien Konieczny,et al.  Propositional Merging and Judgment Aggregation: Two Compatible Approaches? , 2014, ECAI.

[36]  Sébastien Konieczny,et al.  Belief Merging versus Judgment Aggregation , 2015, AAMAS.

[37]  P. Gärdenfors.  A REPRESENTATION THEOREM FOR VOTING WITH LOGICAL CONSEQUENCES , 2006, Economics and Philosophy.

[38]  Bart Selman,et al.  The Comparative Linguistics of Knowledge Representation , 1995, IJCAI.

[39]  Francesco M. Donini,et al.  Space Efficiency of Propositional Knowledge Representation Formalisms , 2000, J. Artif. Intell. Res..