On the logic of preference and judgment aggregation
- 162 Downloads
Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow’s theorem and Condorcet’s paradox—which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation—including classical preference aggregation. As a case study we present a logical study of, including a formal proof of, the neutrality lemma, the main ingredient in a well-known proof of Arrow’s theorem.
KeywordsJudgment aggregation Preference aggregation Modal logic Complexity Completeness
Unable to display preview. Download preview PDF.
- 2.Ågotnes, T., Wooldridge, M., & van der Hoek, W. (2007) Reasoning about judgment and preference aggregation. In M. Huhns, O. Shehory, (Eds.), Proceedings of the sixth international conference on autonomous agents and multiagent systems (AAMAS 2007) (pp. 554–561). IFAMAAS.Google Scholar
- 4.Arrow, K. J., Sen, A. K., & Suzumura, K., (Eds). (2002). Handbook of social choice and welfare, vol 1. North-Holland.Google Scholar
- 6.Clarke E. M., Grumberg O., Peled D. A. (2000) Model checking. The MIT Press, Cambridge, MAGoogle Scholar
- 10.Geanakoplos, J. (2001). Three brief proofs of Arrow’s impossibility theorem. Cowles foundation discussion papers 1123R3, Cowles Foundation, Yale University.Google Scholar
- 12.Grandi, U., & Endriss, U. (2009). First-order formalisation of Arrow’s Theorem. Presentation given at a seminar at the University of Amsterdam, http://www.illc.uva.nl/lgc/seminar/docs/Arrow.pdf.
- 14.Lafage, C., & Lang, J. (2000). Logical representation of preferences for group decision making. In A. G. Cohn, F. Giunchiglia, & B. Selman (Eds.), Proceedings of the conference on principles of knowledge representation and reasoning (KR-00) (pp. 457–470). Morgan Kaufman.Google Scholar
- 15.Lang, J. (2002). From preference representation to combinatorial vote. In D. Fensel, F. Giunchiglia, D. L. McGuinness, M. -A. Williams (Eds.), Proceedings of the conference on principles and knowledge representation and reasoning (KR-02), April 22–25, 2002 (pp. 277–290). Morgan Kaufmann.Google Scholar
- 17.Lin, F., & Tang, P. (2008). Computer-aided proofs of Arrow’s and other impossibility theorems. Proceedings of the 23rd AAAI conference on artificial intelligence (pp. 114–119).Google Scholar
- 18.List, C. (2009). Judgment aggregation. A bibliography on the discursive dilemma, doctrinal paradox and decisions on multiple propositions. Website, see http://personal.lse.ac.uk/LIST/DOCTRINALPARADOX.HTM.
- 19.List C., Pettit P. (2005) Aggregating sets of judgments: An impossibility result. Economics and Philosophy 18: 89–110Google Scholar
- 21.Pauly, M. (2006). Axiomatizing collective judgment sets in a minimal logical language. Manuscript.Google Scholar
- 22.van der Hoek, W., & Pauly, M. (2006). Modal logic for games and information. In P. Blackburn, J. van Benthem, & F. Wolter, (Eds)., Handbook of modal logic (pp. 1077–1148). Amsterdam, The Netherlands: Elsevier Science Publishers B.V.Google Scholar
- 23.Venema, Y. (1996). A crash course in arrow logic. In M. Marx, M. Masuch, & L. Polos, Arrow logic and multi-modal logic (pp. 3–34). Stanford: CSLI Publications.Google Scholar