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The Logic of Group Decisions: Judgment Aggregation

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Abstract

Judgment aggregation studies how individual opinions on a given set of propositions can be aggregated to form a consistent group judgment on the same propositions. Despite the simplicity of the problem, seemingly natural aggregation procedures fail to return consistent collective outcomes, leading to what is now known as the doctrinal paradox. The first occurrences of the paradox were discovered in the legal realm. However, the interest of judgment aggregation is much broader and extends to political philosophy, epistemology, social choice theory, and computer science. The aim of this paper is to provide a concise survey of the discipline and to outline some of the most pressing questions and future lines of research.

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Notes

  1. To appreciate the difference between the doctrinal paradox and the discursive dilemma, and the key role played by the discursive dilemma in the development of the discipline, see the analysis by Mongin [51].

  2. Comprehensive surveys can be found in [30, 45, 48].

  3. Suppose there are three individuals (V 1,V 2,V 3) and three alternatives (x,y,z). Suppose now that V 1 prefers x to y and y to z (denoted by x≻1 y≻1 z), while V 2={y≻2 z≻2 x} and V 3={z≻3 x≻3 y}. Each of the alternatives is compared in pairs. For each pair, the winner is determined by majority voting, and the final outcome is obtained by a combination of all partial results. So we obtain that the collective preference ordering is x≻y≻z≻x, which is not an acceptable social outcome because it is a cycle.

  4. On the relation between judgment and preference aggregation, see also the work by Grossi [29].

  5. The propositions in the agenda are sentences in classical propositional logic, though more expressive logics like modal, predicate, conditional and deontic can also be used [15].

  6. Unanimity guarantees that, if all individuals agree on accepting proposition p, then so should society.

  7. Similar criticisms have been moved to the corresponding independence of irrelevant alternatives condition in preference aggregation.

  8. Monotonicity requires that, if a proposition is collectively accepted and an individual swaps from rejecting it to accepting it, the group should continue to accept the proposition.

  9. An important development in judgment aggregation is the study of manipulability of an aggregation problem. For an overview on these results see, for example, Section 4 in [30].

  10. If we assume that the decision has a definite correct answer, one way to choose the procedure is to consider their truth-tracking reliabilities, as examined in [6, 32, 33, 43].

  11. There is a clear analogy between the Paretian dilemma and floating conclusions, i.e. statements that are supported in each extension of a theory but by different arguments. We recommend [35] for a discussion on the problem of floating conclusions.

  12. As emphasized by authors like Bratman, Cohen and Tuomela, acceptances and beliefs are different mental attitudes. Herzig et al. [34] introduced a logic to model individual and collective acceptances. One of the examples of applications of their acceptance logic is the doctrinal paradox: their logic can model how the collective acceptance of a certain fact is derived from the individual acceptances of that fact.

  13. A more cautious position is that of Bradley [7], who illustrates a tension between aggregation theory and deliberation models.

  14. Unidimensional alignment transposes a classical result in preference aggregation by Black, namely single-peaked preferences [4].

References

  1. Arrow, K. (1963). Social choice and individual values, 2nd edn. New York: Wiley.

    Google Scholar 

  2. Baumeister, D., Erdélyi, G., Erdélyi, O.J., & Rothe, J. (2012). Bribery and control in judgment aggregation. In Proceedings of COMSOC 2012 (pp. 37–48).

  3. Baumeister, D., Erdélyi, , G., & Rothe, J. (2011). How hard is it to bribe the judges? A study of the complexity of bribery in judgment aggregation. In Proceedings of ADT’11 (pp. 1–15). Berlin Heidelberg: Springer.

  4. Black, D. (1948). On the rationale of group decision making. The Journal of Political Economy, 56, 23–34.

    Article  Google Scholar 

  5. de Borda, J.-C. (1781). Mémoire sur les élections au scrutin. In Mémoires de l’Académie Royale des Sciences (pp. 657–665). Paris: l’Imprimerie Royale.

  6. Bovens, L., & Rabinowicz, W. (2006). Democratic answers to complex questions. An epistemic perspective. Synthese, 150, 131–153.

    Article  Google Scholar 

  7. Bradley, R. (2007). Consensus by aggregation and deliberation. In T. Rönnow-Rasmussen, B. Petersson, J. Josefsson, & D. Egonsson (Eds.), Hommage à Wlodek: philosophical papers dedicated to Wlodek Rabinowicz.

  8. Caminada, M., & Pigozzi, G. (2011). On judgment aggregation in abstract argumentation. Autonomous Agents and Multi-Agent Systems, 22(1), 64–102.

    Article  Google Scholar 

  9. Caminada, M., Pigozzi, G., & Podlaszewski, M. (2011). Manipulation in group argument evaluation. In Proceedings of IJCAI 2011 (pp. 121–126).

  10. Chapman, B. (1998). More easily done than said: rules, reason and rational social choice. Oxford Journal of Legal Studies, 18, 293–329.

    Article  Google Scholar 

  11. Chapman, B. (2002). Rational aggregation. Politics, Philosophy & Economics, 1(3), 337–354.

    Article  Google Scholar 

  12. Chevaleyre, Y., Endriss, U., Lang, J., & Maudet, N. (2007). A short introduction to computational social choice. In Proceedings SOFSEM 2007: theory and practice of computer science, Lecture Notes in Computer Science, (Vol. 4362 pp. 51–69). Berlin Heidelberg: Springer.

  13. de Condorcet, M.J.A. (1785). Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralità des Voix. Paris: l’Imprimerie Royale.

  14. Dietrich, F. (2006). Judgment aggregation: (im)possibility theorems. Economic Theory, 126, 286–298.

    Article  Google Scholar 

  15. Dietrich, F. (2007). A generalized model of judgment aggregation. Social Choice and Welfare, 28(4), 529–565.

    Article  Google Scholar 

  16. Dietrich, F. (2012). Scoring rules for judgment aggregation. Working paper.

  17. Dietrich, F., & List, C. (2007). Arrow’s theorem in judgment aggregation. Social Choice and Welfare, 29, 19–33.

    Article  Google Scholar 

  18. Dietrich, F., & List, C. (2007). Judgment aggregation by quota rules: majority voting generalized. Journal of Theoretical Politics, 19, 391–424.

    Article  Google Scholar 

  19. Dietrich, F., & List, C. (2007). Strategy-proof judgment aggregation. Economics and Philosophy, 23, 269–300.

    Article  Google Scholar 

  20. Dietrich, F., & Mongin, P. (2010). The premiss-based approach to judgment aggregation. Journal of Economic Theory, 145(2), 562–582.

    Article  Google Scholar 

  21. Dokow, E., & Holzman, R. (2010). Aggregation of binary evaluations. Journal of Economic Theory, 145(2), 495–511.

    Article  Google Scholar 

  22. Dowding, K., & van Hees, M. (2007). In praise of manipulation. British Journal of Political Science, 38, 1–15.

    Google Scholar 

  23. Dryzek, J.S., & List, C. (2003). Social choice theory and deliberative democracy: a reconciliation. British Journal of Political Science, 33, 1–28.

    Article  Google Scholar 

  24. Duddy, C., & Piggins, A. (2012). A measure of distance between judgment sets. Social Choice and Welfare, 39, 855–867.

    Article  Google Scholar 

  25. Dung, P.M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77, 321–357.

    Article  Google Scholar 

  26. Elster, J. (2010). Décisions individuelles et décisions collectives. Social Science Information, 49(1), 11–28.

    Article  Google Scholar 

  27. Endriss, U., Grandi, U., & Porello, D. (2012). Complexity of judgment aggregation. Journal of Artificial Intelligence Research, 45, 481–514.

    Google Scholar 

  28. Goldman, A. (2010). Social epistemology. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Summer 2010 edition.

  29. Grossi, D. (2009). Unifying preference and judgment aggregation. In Proceedings of AAMAS 2009 (pp. 217–224). ACM Press.

  30. Grossi, D., & Pigozzi, G. (2012). Introduction to judgment aggregation. In N. Bezhanishvili, & V. Goranko (Eds.), ESSLLI 2010/2011 Lectures. LNCS 7388 (pp. 160–209). Berlin Heidelberg: Springer.

  31. Grossi, D., Pigozzi, G., & Slavkovik, M. (2009). White manipulation in judgment aggregation. In Proceedings of BNAIC 2009.

  32. Hartmann, S., Pigozzi, G., & Sprenger, J. (2010). Reliable methods of judgement aggregation. Journal of Logic and Computation, 20(2), 603–617.

    Article  Google Scholar 

  33. Hartmann, S., & Sprenger, J. (2012). Judgment aggregation and the problem of tracking the truth. Synthese, 187, 209–221.

    Article  Google Scholar 

  34. Herzig, A., de Lima, T., & Lorini, E. On the dynamics of institutional agreements. Synthese, 171, 321–355.

  35. Horty, J.F. (2012). Reasons as defaults. New York: Oxford University Press.

  36. Konieczny, S., & Pino-Pérez, R. (1999). Merging with integrity constraints. Proceedings of ECSQARU’99, 7, 233–244.

    Google Scholar 

  37. Kornhauser, L.A. (1992). Modeling collegial courts. II. Legal doctrine. Journal of Law, Economics, and Organization, 8, 441.

    Google Scholar 

  38. Kornhauser, L.A., & Sager, L.G. (1986). Unpacking the court. Yale Law Journal, 96, 82–117.

    Article  Google Scholar 

  39. Kornhauser, L.A., & Sager, L.G. (1993). The one and the many: adjudication in collegial courts. California Law Review, 81, 1–51.

    Article  Google Scholar 

  40. Lang, J., Pigozzi, G., Slavkovik, M., & van der Torre, L. (2011). Judgment aggregation rules based on minimization. In Proceedings of TARK XIII (pp. 238–246). New York: ACM.

  41. List, C. (2002). A possibility theorem on aggregation over multiple interconnected propositions. Mathematical Social Sciences, 45(1), 1–13.

    Article  Google Scholar 

  42. List C. (2004). A model of path-dependence in decisions over multiple propositions. American Political Science Review, 98(3), 495–513.

    Article  Google Scholar 

  43. List, C. (2005). The probability of inconsistencies in complex collective decisions. Social Choice and Welfare, 24(1), 3–32.

    Article  Google Scholar 

  44. List, C. (2011). Group deliberation and the revision of judgments: an impossibility result. Journal of Political Philosophy, 19(1), 1–27.

    Article  Google Scholar 

  45. List, C. (2012). The theory of judgment aggregation: an introductory review. Synthese, 187(1), 179–207.

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  48. List, C., & Puppe, C. (2009). Judgment aggregation: a survey. In P. Anand, C. Puppe, & P. Pattanaik (Eds.), The handbook of rational and social choice. New York: Oxford University Press.

  49. Miller, M.K., & Osherson, D. (2009). Methods for distance-based judgment aggregation. Social Choice and Welfare, 32(4), 575–601.

    Article  Google Scholar 

  50. Mongin, P. (2008). Factoring out the impossibility of logical aggregation. Journal of Economic Theory, 141(1), 100–113.

    Article  Google Scholar 

  51. Mongin, P. (2012). The doctrinal paradox, the discursive dilemma, and logical aggregation theory. Theory and Decision, 73, 315–355.

    Article  Google Scholar 

  52. Nehring, K. (2005). The (im)possibility of a Paretian rational. Economics working papers, Institute for Advanced Study, School of Social Science.

  53. 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.

  54. Nehring, K., & Puppe, C. (2010). Justiable group choice. Journal of Economic Theory, 145, 583–602.

    Article  Google Scholar 

  55. Ottonelli, V. (2005). Errore deliberativo e legittimità democratica. Networks, 5, 32–51.

    Google Scholar 

  56. Pauly, M., & van Hees, M. (2006). Logical constraints on judgment aggregation. Journal of Philosophical Logic, 35, 569–585.

    Article  Google Scholar 

  57. Pettit, P. (2001). Deliberative democracy and the discursive dilemma. Philosophical Issues, 11, 268–299.

    Article  Google Scholar 

  58. Pettit, P. (2003). Groups with minds of their own. In F. Schmitt (Ed.), Socializing metaphysics: the nature of social reality. (pp. 167–93). New York: Rowman and Littlefield.

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

    Article  Google Scholar 

  60. Rahwan, I., & Larson, K. (2008). Welfare properties of argumentation-based semantics. In Proceedings of the 2nd international workshop on computational social choice (COMSOC).

  61. Rahwan, I., & Tohmé, F. (2010). Collective argument evaluation as judgment aggregation. In Proceedings of 9th AAMAS.

  62. Roth, A.S. (2011). Shared agency. In E.N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Spring 2011 edition.

  63. Searle, J. (1990). Collective intentions and actions. In P. Cohen, J. Morgan, & M. Pollack (Eds.), Intentions in communication. (pp. 401–415). Cambridge: MIT Press.

  64. Sen, A.K. (1970). Collective Choice and Social Welfare. Holden Day.

  65. Spector, H. (2009). The right to a constitutional jury. Legisprudence, 3(1), 111–123.

    Article  Google Scholar 

  66. Sunstein, C.R. (2002). Conformity and dissent. U Chicago Law & Economics, Olin Working Paper, 164.

  67. Tannenhaus, S. (2003). Deputy secretary Wolfowitz interview with Sam Tannenhaus. Vanity Fair.

  68. Vacca, R. (1921). Opinioni individuali e deliberazioni collettive. Rivista Internazionale di Filosofia del Diritto, 52, 52–59.

    Google Scholar 

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    Gabriella Pigozzi

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Pigozzi, G. The Logic of Group Decisions: Judgment Aggregation. J Philos Logic 44, 755–769 (2015). https://doi.org/10.1007/s10992-015-9357-7

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