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Deterministic algorithms for inhomogeneous Bernoulli trials: Shapley value of network devices

Jesse D Wei, Guo Wei

TL;DR

The paper tackles the exponential complexity of computing Shapley values for networks formed by inhomogeneous Bernoulli trials. It develops deterministic, scalable algorithms that approximate the Shapley value in linear or quadratic time by leveraging a random-set (RACS) representation and independence assumptions, with rigorous error analyses. A second algorithm provides an alternative approximation under a block-wise, sorted-probability scheme, accompanied by exact-form expressions and error bounds. Additionally, the authors supplement Shapley’s original proof with a rigorous integral-combinatorial derivation that highlights the roles of the Binomial Theorem and the Beta function. The work yields practical, real-time risk contributions for network devices and offers guidance on when to use exact versus approximate Shapley values in large-scale systems.

Abstract

Suppose that $n$ computer devices are to be connected to a network via inhomogeneous Bernoulli trials. The Shapley value of a device quantifies how much the network's value increases due to the participation of that device. Characteristic functions of such games are naturally taken as the belief function (containment function) and Choquet capacity (hitting probability) of a random set (random network of devices). Traditionally, the Shapley value is either calculated as the expected marginal contribution over all possible coalitions (subnetworks), which results in exponential computational complexity, or approximated by the Monte Carlo sampling technique, where the performance is highly dependent on the stochastic sampling process. The purpose of this study is to design deterministic algorithms for games formulated via inhomogeneous Bernoulli trials that approximate the Shapley value in linear or quadratic time, with rigorous error analysis (Sections 3 and 4). Additionally, we provide a review of relevant literature on existing calculation methods in Remark 3.1 and Appendix I. A further goal is to supplement Shapley's original proof by deriving the Shapley value formula using a rigorous approach based on definite integrals and combinatorial analysis. This method explicitly highlights the roles of the Binomial Theorem and the Beta function in the proof, addressing a gap in Shapley's work (Appendix II).

Deterministic algorithms for inhomogeneous Bernoulli trials: Shapley value of network devices

TL;DR

The paper tackles the exponential complexity of computing Shapley values for networks formed by inhomogeneous Bernoulli trials. It develops deterministic, scalable algorithms that approximate the Shapley value in linear or quadratic time by leveraging a random-set (RACS) representation and independence assumptions, with rigorous error analyses. A second algorithm provides an alternative approximation under a block-wise, sorted-probability scheme, accompanied by exact-form expressions and error bounds. Additionally, the authors supplement Shapley’s original proof with a rigorous integral-combinatorial derivation that highlights the roles of the Binomial Theorem and the Beta function. The work yields practical, real-time risk contributions for network devices and offers guidance on when to use exact versus approximate Shapley values in large-scale systems.

Abstract

Suppose that computer devices are to be connected to a network via inhomogeneous Bernoulli trials. The Shapley value of a device quantifies how much the network's value increases due to the participation of that device. Characteristic functions of such games are naturally taken as the belief function (containment function) and Choquet capacity (hitting probability) of a random set (random network of devices). Traditionally, the Shapley value is either calculated as the expected marginal contribution over all possible coalitions (subnetworks), which results in exponential computational complexity, or approximated by the Monte Carlo sampling technique, where the performance is highly dependent on the stochastic sampling process. The purpose of this study is to design deterministic algorithms for games formulated via inhomogeneous Bernoulli trials that approximate the Shapley value in linear or quadratic time, with rigorous error analysis (Sections 3 and 4). Additionally, we provide a review of relevant literature on existing calculation methods in Remark 3.1 and Appendix I. A further goal is to supplement Shapley's original proof by deriving the Shapley value formula using a rigorous approach based on definite integrals and combinatorial analysis. This method explicitly highlights the roles of the Binomial Theorem and the Beta function in the proof, addressing a gap in Shapley's work (Appendix II).

Paper Structure

This paper contains 35 sections, 3 theorems, 99 equations, 7 tables.

Table of Contents

  1. Known approximation methods to the Shapley value and the existing research gap
  2. Random set representation of characteristic functions
  3. Lowering the computational complexity of Shapley value
  4. Key assumption: Independence
  5. Methods to reduce complexity
  6. 1. Assume Independence
  7. Complexity Analysis
  8. Accuracy analysis of our binomial approximation method via an example
  9. Rigorous error analysis of the approximation
  10. Error decomposition
  11. Main error bound
  12. Examples and application
  13. Examples
  14. Application
  15. Computational complexity comparison: $\phi_i(\mu)$ vs $\phi_i(T)$
  16. ...and 20 more sections

Key Result

Theorem 2.3

If $v$ satisfies $v(E) = 1$ and $m_v(S) \geq 0$ for all $S \subseteq E$, then there exists a (unique) random set $X$ such that

Theorems & Definitions (13)

  • Definition 2.1: Möbius Coefficients
  • Definition 2.2: Random Set
  • Theorem 2.3: See Shafer, 1976
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 3 more