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Priority-Aware Shapley Value

Kiljae Lee, Ziqi Liu, Weijing Tang, Yuan Zhang

TL;DR

PASV introduces a unified Shapley-type framework that jointly enforces hard precedence constraints and soft, contributor-specific priorities. By formulating the sampling distribution p^{(\\preceq,\\lambda)} over precedence-respecting orders, PASV recovers PSV and WSV as special cases and is computable via an adjacent-swap MCMC scheme. The paper provides axioms (PROV, SCF) that characterize PASV and analyzes limiting regimes under extreme weights, with a practical priority sweeping diagnostic to assess robustness. Empirical studies on data valuation (MNIST/CIFAR10) and feature attribution (Census Income) show PASV yields more structure-faithful allocations and useful sensitivity analyses for trust/risk considerations. Overall, PASV offers a scalable, principled approach to priority-aware attribution with broad applicability in data-centric AI.

Abstract

Shapley values are widely used for model-agnostic data valuation and feature attribution, yet they implicitly assume contributors are interchangeable. This can be problematic when contributors are dependent (e.g., reused/augmented data or causal feature orderings) or when contributions should be adjusted by factors such as trust or risk. We propose Priority-Aware Shapley Value (PASV), which incorporates both hard precedence constraints and soft, contributor-specific priority weights. PASV is applicable to general precedence structures, recovers precedence-only and weight-only Shapley variants as special cases, and is uniquely characterized by natural axioms. We develop an efficient adjacent-swap Metropolis-Hastings sampler for scalable Monte Carlo estimation and analyze limiting regimes induced by extreme priority weights. Experiments on data valuation (MNIST/CIFAR10) and feature attribution (Census Income) demonstrate more structure-faithful allocations and a practical sensitivity analysis via our proposed "priority sweeping".

Priority-Aware Shapley Value

TL;DR

PASV introduces a unified Shapley-type framework that jointly enforces hard precedence constraints and soft, contributor-specific priorities. By formulating the sampling distribution p^{(\\preceq,\\lambda)} over precedence-respecting orders, PASV recovers PSV and WSV as special cases and is computable via an adjacent-swap MCMC scheme. The paper provides axioms (PROV, SCF) that characterize PASV and analyzes limiting regimes under extreme weights, with a practical priority sweeping diagnostic to assess robustness. Empirical studies on data valuation (MNIST/CIFAR10) and feature attribution (Census Income) show PASV yields more structure-faithful allocations and useful sensitivity analyses for trust/risk considerations. Overall, PASV offers a scalable, principled approach to priority-aware attribution with broad applicability in data-centric AI.

Abstract

Shapley values are widely used for model-agnostic data valuation and feature attribution, yet they implicitly assume contributors are interchangeable. This can be problematic when contributors are dependent (e.g., reused/augmented data or causal feature orderings) or when contributions should be adjusted by factors such as trust or risk. We propose Priority-Aware Shapley Value (PASV), which incorporates both hard precedence constraints and soft, contributor-specific priority weights. PASV is applicable to general precedence structures, recovers precedence-only and weight-only Shapley variants as special cases, and is uniquely characterized by natural axioms. We develop an efficient adjacent-swap Metropolis-Hastings sampler for scalable Monte Carlo estimation and analyze limiting regimes induced by extreme priority weights. Experiments on data valuation (MNIST/CIFAR10) and feature attribution (Census Income) demonstrate more structure-faithful allocations and a practical sensitivity analysis via our proposed "priority sweeping".
Paper Structure (45 sections, 7 theorems, 21 equations, 11 figures, 2 tables, 2 algorithms)

This paper contains 45 sections, 7 theorems, 21 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From Shapley Value to Random Order Values
  4. Hard Priority: Partial Order, Directed Acyclic Graph and Linear Extension
  5. Precedence Shapley Value (PSV) faigle1992shapley.
  6. Soft Priority: Individual (Soft) Priority Weights
  7. Ordered Partition.
  8. Weighted Shapley Value (WSV)
  9. Our Method
  10. Desired Properties
  11. Priority-Aware Shapley Value (PASV)
  12. Axiomatization of PASV
  13. Precedence Random Order Value (PROV)
  14. State-Choice Factorization (SCF)
  15. Boundary case axioms
  16. ...and 30 more sections

Key Result

Proposition 3.1

If $\lambda_1=\cdots=\lambda_n$, then $p^{(\preceq,\lambda)}(\pi)=p_{\mathrm{PSV}}(\pi), \forall\pi\in\Pi^{\preceq}.$

Figures (11)

  • Figure 1: Illustration of priority structures. Dashed arrows in (c): existing works on (c) require special precedence structures.
  • Figure 2: Extreme $\lambda$ may translate into DAG edges: Top: sending $\lambda_4\to\infty$ adds an edge $2\to4$ (on the right we omitted the edge $3\to 4$ as it is implied by the path $3\to 2\to 4$). Bottom: sending $\lambda_3\to\infty$ adds an edge $4\to 3$, forming a refined layer structure.
  • Figure 3: Extreme $\lambda$ may not always translate into a new DAG edge: if $\pi_4=2$, then $1$ and $4$ are both eligible for $\pi_3$; the (only possible) edge to add $1\to 3$ would incorrectly deny $1$'s candidacy.
  • Figure 4: The DAG used for MNIST/CIFAR10. Only one Booster is shown. Here, (for example) a blue edge from Owner to half of Copier represents $100\times50$ edges between their members.
  • Figure 5: MNIST results. Top: Provider-level values (summed value for each provider lee2025faithful; $10$ repetitions, report mean & $\pm 1$ std. bars; Middle: Marginal contributions: $\mathbb{E}[U(\pi^i\cup\{i\})-U(\pi^i)||\pi^i|=s]$ vs $s$, c.f. \ref{['eqn:rov']}, $p:=$ Uniform$(\Pi^\preceq)$; Bottom: Priority sweeping: $c\in\{(0,1,0,0,0),\ldots,(0,0,0,0,1)\}$, $b\in\{2^k:k\in[11]\}$, Owner not swept (alone in maximal set, weight has no effect).
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 2.1: Partial order set (poset)
  • Definition 2.2: Linear extension (LE)
  • Remark 2.3
  • Definition 2.4: Feasible set
  • Definition 2.5: Maximal elements
  • Proposition 3.1: Reduction to PSV
  • Proposition 3.2: Reduction to WSV
  • Remark 3.3
  • Definition 3.4: Precedence random order value (PROV)
  • Definition 3.6: Elementary game
  • ...and 6 more