Aggregation theory and the relevance of some issues to others

Dietrich, Franz (2006): Aggregation theory and the relevance of some issues to others. Forthcoming in: Journal of Economic Theory

Abstract

I propose a relevance-based independence axiom on how to aggregate individual yes/no judgments on given propositions into collective judgments: the collective judgment on a proposition depends only on people's judgments on propositions which are relevant to that proposition. This axiom contrasts with the classical independence axiom: the collective judgment on a proposition depends only on people's judgments on the same proposition. I generalize the premise-based rule and the sequential-priority rule to an arbitrary priority order of the propositions, instead of a dichotomous premise/conclusion order resp. a linear priority order. I prove four impossibility theorems on relevance-based aggregation. One theorem simultaneously generalizes Arrow's Theorem (in its general and indifference-free versions) and the well-known Arrow-like theorem in judgment aggregation.

• Item Type: MPRA Paper
• Language: English
• Keywords: judgment aggregation, relevance connections vs. logical connections between issues, generalized Arrow theorem, generalized premise-based rule, sequential priority rule, general priority rule, priority graph, aggregation of non-binary evaluations
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D70 - General; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations

References

Balinski, M, Laraki, R (2010) Majority Judgment: Measuring, Ranking, and Electing, MIT Press

Bradley, R (2006) Taking advantage of difference in opinion, Episteme 3: 141-155

Chapman, B (2002) Rational Aggregation, Polit Philos Econ 1(3): 337-354

Claussen, C, Roisland, O (2010) A quantitative discursive dilemma, Social Choice and Welfare 35(1): 49-64

Dietrich, F (2006a) Judgment aggregation: (im)possibility theorems, J Econ Theory 126(1): 286-298

Dietrich, F (2006b) Aggregation theory and the relevance of some issues to others, first version, available at www.franzdietrich.net and as Meteor Research Memorandum RM/07/002, Maastricht University

Dietrich, F (2007) A generalised model of judgment aggregation, Social Choice and Welfare 28: 529-65

Dietrich, F (2014) Scoring rules for judgment aggregation, Social Choice and Welfare 42(4): 873-911

Dietrich, F, List, C (2007a) Judgment aggregation by quota rules, Journal of Theoretical Politics 19(4): 391-424

Dietrich, F, List, C (2007b) Arrow's theorem in judgment aggregation, Social Choice and Welfare 29: 19-33

Dietrich, F, List, C (2008) Judgment aggregation without full rationality, Social Choice and Welfare 31: 15-39

Dietrich, F, List, C (2010a) Judgment aggregation on restricted domains, Journal of Economic Theory 145(2): 512-543

Dietrich, F, List, C (2010b) The aggregation of propositional attitudes: towards a general theory, Oxford Studies in Epistemology 3: 215-34 (Corrigendum on the authors' homepages)

Dietrich, F, List, C (2013) Propositionwise judgment aggregation: the general case, Social Choice and Welfare 40: 1067-95

Dietrich, F, Mongin, P (2010) The premise-based approach to judgment aggregation, Journal of Economic Theory 145(2): 562-582

Dokow, E, Holzman, R (2010a) Aggregation of binary evaluations, Journal of Economic Theory 145(2): 495-511

Dokow, E, Holzman, R (2010b) Aggregation of non-binary evaluations, Advances in Applied Mathematics 45: 487-504

Dokow, E, Holzman, R (2010c) Aggregation of binary evaluations with abstentions, Journal of Economic Theory 145(2): 544-561

Duddy, C, Piggins, A (2013) Many-valued judgment aggregation: characterizing the possibility/impossibility boundary, Journal of Economic Theory 148(2):793-805

Duddy, C, Piggins, A, Zwicker, W (2014) Aggregation of binary evaluations: a Borda-like approach, working paper

Fenstad, JF (1980) General recursion theory: an axiomatic approach, Springer

Gaertner, W. (2006) A Primer in Social Choice Theory, Oxford University Press

Gärdenfors, P (2006) An Arrow-like theorem for voting with logical consequences, Economics and Philosophy 22(2): 181-190

Genest, C, Zidek, JV (1986) Combining Probability Distributions: A Critique and Annotated Bibliography, Statistical Science 1(1): 113-135

Hartmann, S, Pigozzi, G, Sprenger, J (2010) Reliable methods of judgment aggregation, Journal for Logic and Computation 20: 603--617

Konieczny, S, Pino-Perez, R (2002) Merging information under constraints: a logical framework, J Logic Comput 12(5): 773-808

Lang, J, Pigozzi, G, Slavkovik, M, van der Torre, L (2011) Judgment aggregation rules based on minimization. In Proceedings of the 13th Conference on the Theoretical Aspects of Rationality and Knowledge (TARK XIII), ACM, pp. 238-246

List, C (2003) A Possibility Theorem on Aggregation over Multiple Interconnected Propositions, Mathematical Social Sciences 45(1): 1-13 (Corrigendum in Mathematical Social Sciences 52:109-110)

List, C (2004) A model of path-dependence in decisions over multiple propositions, Am Polit Sci Rev 98(3): 495-513

List, C (2005) Group knowledge and group rationality: a judgment aggregation perspective, Episteme 2(1): 25-38

List, C, Pettit, P (2002) Aggregating sets of judgments: an impossibility result, Economics and Philosophy 18: 89-110

List, C, Pettit, P (2004) Aggregating sets of judgments: two impossibility results compared. Synthese 140(1-2): 207-235

Miller, M, Osherson, D (2008) Methods for distance-based judgment aggregation, Social Choice and Welfare 32(4): 575--601

Mongin, P (2005) Spurious Unanimity and the Pareto Principle, LSE Choice Group Working Papers 1 (5)

Mongin, P (2008) Factoring out the impossibility of logical aggregation, Journal of Economic Theory 141: 100-113

Nehring, K (2003) Arrow's theorem as a corollary. Econ Letters 80(3): 379-382

Nehring, K (2005) The (im)possibility of a Paretian rational, working paper, Univ California at Davies

Nehring, K, Pivato, M, Puppe, C (2011) Condorcet admissibility: indeterminacy and path-dependence under majority voting on interconnected decisions, working paper, University of California at Davis

Nehring, K, Puppe, C (2002) Strategy-proof social choice on single-peaked domains: possibility, impossibility and the space between, working paper, Univ California at Davies

Nehring, K, Puppe, C (2005) The structure of strategy-proof social choice, part II: non-dictatorship, anonymity and neutrality, working paper, Karlsruhe Univ

Nehring, K., Puppe, C (2008) Consistent judgment aggregation: the truth-functional Case, Social Choice and Welfare 31: 41-57

Pauly, M, van Hees, M (2006) Logical constraints on judgment aggregation, J Philos Logic 35: 569-585

Pigozzi, G (2006) Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation, Synthese 152(2): 285-298

Pivato, M (2009) The geometry of consistent majoritarian judgement aggregation, Social Choice and Welfare, 33: 559-74

Rubinstein, A, Fishburn, P (1986) Algebraic aggregation theory, J Econ Theory 38: 63-77

Wilson, R (1975) On the Theory of Aggregation, J Econ Theory 10: 89-99

Zwicker, W (2011) Towards a Borda count for judgment aggregation, extended abstract, Conference `Judgment Aggregation and Voting', Karlsruhe Institute of Technology