A measure of distance between judgment sets

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 Financial support from the Irish Research Council for the Humanities and Social Sciences, the Spanish Ministry of Science and Innovation through MICINN/FEDER grant ECO2010-21624 and the NUI Galway Millennium Fund is gratefully acknowledged. We are grateful for the comments of Maurice Salles, Nick Tosh, an Associate Editor and two referees. Government of Ireland Scholar, J.E. Cairnes School of Business and Economics, National University of Ireland Galway, University Road, Galway, Ireland. Email: [email protected] J.E. Cairnes School of Business and Economics, National University of Ireland Galway, University Road, Galway, Ireland. Email: [email protected]