A measure of distance between judgment sets

Abstract

In the literature on judgment aggregation, an important open question is how to measure the distance between any two judgment sets. This is relevant for issues of social choice: if two individuals hold different beliefs then we might want to find a compromise that lies somewhere between them. We propose a set of axioms that determine a measure of distance uniquely. This measure differs from the widely used Hamming metric. The difference between Hamming's metric and ours boils down to one axiom. Given judgment sets A and B, this axiom says that if the propositions in A ∩ B jointly imply that the propositions in A−B share the same truth value, then the disagreement between A and B over those propositions in A−B should be counted as a single disagreement. We consider the application of our metric to judgment aggregation, and also use the metric to measure the distance between preference rankings.

Efficient minimal preference change

“…There is a large literature on alternative distance measures between preferences, for example [5,13,21,22,24,31,36]. It has been argued in for example [34,12], that the Hamming distance as a measure of dissimilarity may not be appropriate when the elements in the preference set are logically related. For example, A ≡ B and B ≡ A are logically related (logically equivalent in fact).…”

Scoring rules for judgment aggregation

“…The recent conference 'Judgment aggregation and voting' in Freudenstadt in 2011 however marks a visible shift of attention towards constructing concrete aggregation rules and finding 'second best' solutions in the face of impossibility results. The new proposals range from a first Borda-type aggregation rule (Zwicker 2011) to, among others, new distance-based rules (Duddy and Piggins 2012) and rules which approximate the majority judgments when these are inconsistent (Nehring et al 2011). The more traditional proposals include premise-and conclusion-based rules (e.g., Kornhauser and Sager 1986;Pettit 2001;List and Pettit 2002;Dietrich 2006a;Dietrich and Mongin 2010), sequential rules (e.g., List 2004;Dietrich and List 2007b), distance-based rules (e.g., Konieczny and Pino-Perez 2002;Pigozzi 2006;Miller and Osherson 2008;Eckert and Klamler 2009;Hartmann et al 2010;Lang et al 2011), and quota rules with well-calibrated acceptance thresholds and various degrees of collective rationality (e.g., Dietrich and List 2007b; see also Nehring and Puppe 2010a).…”

Aggregation of Value Judgments Differs from Aggregation of Preferences

“…Thus, the Hamming distance is the same between, say, a judgment that the value of a certain parameter is 1 and the judgment that this value is 0 as between the judgments that this value is 1 and .9, respectively. Another kind of criticism has been levelled by Duddy & Piggins (2012), who argue that the Hamming metric will sometimes involve double-counting if the propositions on the agenda are allowed to be logically interconnected. To use their example, if two individuals both accept a proposition p, then they disagree on the conjunction p  q iff they disagree on q.…”

“…In view of the relationship between the Hamming metric and the KS-measure, this objection might have implications for the use of the latter measure as well. And indeed, Duddy and Piggins (2012) argue that it does. For their own proposal of a measure of distance between rankings, see Appendix.…”

“…One such proposal is due to Duddy and Piggins (2012). Essentially, their idea is to look at the distance between two rankings as the smallest number of steps needed to transform one ranking into the other.…”