Cost of transformation: a measure on matchings

This paper constructs a normative framework to quantify the difference between two matchings in roommate markets. We investigate the “cost of transformation” from one mechanism to another, based on the differences in the outputs of these mechanisms. Several conditions are introduced to ensure this cost reflects the welfare effect of the transformation on individuals. We introduce a measure called the Borda measure, which is fully characterized by these conditions. Several possible applications of this measure under different contexts interpretations are also discussed, such as measuring how unstable, how unfair, or how inefficient a matching is.