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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].

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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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