Characterizations of scoring methods for preference aggregation

The paper surveys more than forty characterizations of scoring methods for preference aggregation and contains one new result. A general scoring operator is self-consistent if alternative i is assigned a greater score than j whenever i gets no worse (better) results of comparisons and its ‘opponents’ are assigned, respectively, greater (no smaller) scores than those of j. We prove that self-consistency is satisfied if and only if the application of a scoring operator reduces to the solution of a homogeneous system of algebraic equations with a monotone function on the left-hand side.