Axiomatic characterization of pointwise Shapley decompositions

Paper: arXiv 2303.07773

Authors: Marcus C Christiansen

Abstract

A common problem in various applications is the additive decomposition of the output of a function with respect to its input variables. Functions with binary arguments can be axiomatically decomposed by the famous Shapley value. For the decomposition of functions with real arguments, a popular method is the pointwise application of the Shapley value on the domain. However, this pointwise application largely ignores the overall structure of functions. In this paper, axioms are developed which fully preserve functional structures and lead to unique decompositions for all Borel measurable functions.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework and axiomatic characterization to extend the Shapley value to continuous domain functions (Borel measurable functions). While it is highly novel and theoretically dense, it is purely mathematical without empirical validation or backtesting on real-world datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Axiomatize additive decomposition for real-argument functions] --> B[Problem: Standard pointwise Shapley ignores overall functional structure]
    B --> C[Input: Real-valued domain with Borel measurable functions]
    C --> D[Methodology: Develop new axioms fully preserving functional structure]
    D --> E[Outcome: Unique, structure-preserving Shapley decomposition for all Borel functions]