Axiomatic Attribution for Multilinear Functions

TL;DR

This paper introduces a unique attribution method for multilinear functions, formalizes the attribution problem, and demonstrates its practical application in advertising and finance.

Contribution

It formalizes the attribution problem, proves the uniqueness of the Aumann-Shapley-Shubik method for multilinear functions, and provides an efficient implementation with practical applications.

Findings

Unique attribution method for multilinear functions identified

Efficient computational implementation developed

Applications demonstrated in advertising and portfolio analysis

Abstract

We study the attribution problem, that is, the problem of attributing a change in the value of a characteristic function to its independent variables. We make three contributions. First, we propose a formalization of the problem based on a standard cost sharing model. Second, we show that there is a unique attribution method that satisfies Dummy, Additivity, Conditional Nonnegativity, Affine Scale Invariance, and Anonymity for all characteristic functions that are the sum of a multilinear function and an additive function. We term this the Aumann-Shapley-Shubik method. Conversely, we show that such a uniqueness result does not hold for characteristic functions outside this class. Third, we study multilinear characteristic functions in detail; we describe a computationally efficient implementation of the Aumann-Shapley-Shubik method and discuss practical applications to pay-per-click advertising and portfolio analysis.