Desirability and social ranking

Abstract We present an axiomatic study of various solutions to the social ranking problem, where a solution links any ranking of coalitions of players to a binary relation between individual players. We focus on solutions that align with the desirability relation, asserting that player i is more desirable than player j if any coalition including i but not j ranks higher than the corresponding coalition formed by replacing i with j . Unlike previous characterizations, our study highlights the central role of the desirability property as a foundational axiom in the characterization of five solutions from the related literature: Ceteris Paribus majority, lexicographic excellence and its dual, $$L^{(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> solution and its dual. Our main results reveal additional similarities among these five solutions and emphasize the essential features that should be considered when selecting the most appropriate solution for a given scenario. A practical application involving a bicameral legislature is also presented.