Characterizations of degree one bivariate measures of concordance

We call a measure of concordance κκ of an ordered pair (X,Y)(X,Y) of two continuous random variables a bivariate   measure of concordance. This κκ may be considered to be a function κ(C)κ(C) of the copula CC associated with (X,Y)(X,Y). κκ is considered to be of degree nn if, given any two copulas AA and BB, the value of their convex sum, κ(tA+(1−t)B)κ(tA+(1−t)B), is a polynomial in tt of degree nn. Examples of bivariate measures of concordance are Spearman’s rho, Blomqvist’s beta, Gini’s measure of association, and Kendall’s tau. The first three of these are of degree one, but Kendall’s tau is of degree two. We exhibit three characterizations of bivariate measures of concordance of degree one.