Axiomatic attribution for multilinear functions

We study the attribution problem. That is, given a real-valued characteristic function f of n variables and initial and final values r and s for its independent variables, our objective is to divide the responsibility for the change f(s) - f(r) in the characteristic function among each of its independent variables. We call these assigned responsibilities attributions, and we would like the attributions to form a complete partition of the total change. When r=0, the attribution problem coincides with a standard cost sharing model from the social choice literature (cf. Moulin 2), where the characteristic function is the cost function, the independent variables are the demands of the agents, and the attributions are cost shares for the agents. We follow the cost sharing literature in identifying good attribution methods axiomatically (for a classical example, see Friedman and Moulin 1). We consider: Additivity -- attributions are additive in the characteristic function, Dummy -- if the characteristic function does not depend on a variable, then its attribution is zero, and Affine Scale Invariance -- attributions are invariant under simultaneous affine transformation of the characteristic function and the variables.