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A Ratio-Based Shapley Value for Collaborative Machine Learning - Extended Version

Björn Filter, Ralf Möller, Özgür Lütfü Özçep

TL;DR

This work addresses incentive-compatible, fair allocation of replicable model rewards in collaborative machine learning by introducing a ratio-based Shapley value. It defines relative marginal contributions Delta^{rel}_{i,C}, derives phi^{rel}_i, and computes rewards via r_i = (phi^{rel}_i/phi^{rel}_*)^{rho} * v_C within the existing rho-scaled framework. The authors prove that the ratio-based value satisfies the core rationality and fairness axioms (R1–R3, F1–F4) under monotone value functions and give conditions for individual rationality and stability (R4, grand-coalition stability) through bounds on rho. A comparison with additive Shapley rewards shows how proportional improvements can yield different allocations, particularly in heterogeneous or redundant data settings. The paper broadens the toolkit for fair, incentive-aware collaboration by demonstrating a mathematically grounded yet distinct alternative to additive valuation, and it identifies open questions about the full space of mechanisms that satisfy the proposed axioms and how to tailor them to specific application domains.

Abstract

Collaborative machine learning enables multiple data owners to jointly train models for improved predictive performance. However, ensuring incentive compatibility and fair contribution-based rewards remains a critical challenge. Prior work by Sim and colleagues (Rachel Hwee Ling Sim et al: Collaborative machine learning with incentive-aware model rewards. In: International conference on machine learning. PMLR. 2020, pp. 8927-8963) addressed this by allocating model rewards, which are non-monetary and freely replicable, based on the Shapley value of each party's data contribution, measured via information gain. In this paper, we introduce a ratio-based Shapley value that replaces the standard additive formulation with a relative contribution measure. While our overall reward framework, including the incentive definitions and model-reward setting, remains aligned with that of Sim and colleagues, the underlying value function is fundamentally different. Our alternative valuation induces a different distribution of model rewards and offers a new lens through which to analyze incentive properties. We formally define the ratio-based value and prove that it satisfies the same set of incentive conditions as the additive formulation, including adapted versions of fairness, individual rationality, and stability. Like the original approach, our method faces the same fundamental trade-offs between these incentives. Our contribution is a mathematically grounded alternative to the additive Shapley framework, potentially better suited to contexts where proportionality among contributors is more meaningful than additive differences.

A Ratio-Based Shapley Value for Collaborative Machine Learning - Extended Version

TL;DR

This work addresses incentive-compatible, fair allocation of replicable model rewards in collaborative machine learning by introducing a ratio-based Shapley value. It defines relative marginal contributions Delta^{rel}_{i,C}, derives phi^{rel}_i, and computes rewards via r_i = (phi^{rel}_i/phi^{rel}_*)^{rho} * v_C within the existing rho-scaled framework. The authors prove that the ratio-based value satisfies the core rationality and fairness axioms (R1–R3, F1–F4) under monotone value functions and give conditions for individual rationality and stability (R4, grand-coalition stability) through bounds on rho. A comparison with additive Shapley rewards shows how proportional improvements can yield different allocations, particularly in heterogeneous or redundant data settings. The paper broadens the toolkit for fair, incentive-aware collaboration by demonstrating a mathematically grounded yet distinct alternative to additive valuation, and it identifies open questions about the full space of mechanisms that satisfy the proposed axioms and how to tailor them to specific application domains.

Abstract

Collaborative machine learning enables multiple data owners to jointly train models for improved predictive performance. However, ensuring incentive compatibility and fair contribution-based rewards remains a critical challenge. Prior work by Sim and colleagues (Rachel Hwee Ling Sim et al: Collaborative machine learning with incentive-aware model rewards. In: International conference on machine learning. PMLR. 2020, pp. 8927-8963) addressed this by allocating model rewards, which are non-monetary and freely replicable, based on the Shapley value of each party's data contribution, measured via information gain. In this paper, we introduce a ratio-based Shapley value that replaces the standard additive formulation with a relative contribution measure. While our overall reward framework, including the incentive definitions and model-reward setting, remains aligned with that of Sim and colleagues, the underlying value function is fundamentally different. Our alternative valuation induces a different distribution of model rewards and offers a new lens through which to analyze incentive properties. We formally define the ratio-based value and prove that it satisfies the same set of incentive conditions as the additive formulation, including adapted versions of fairness, individual rationality, and stability. Like the original approach, our method faces the same fundamental trade-offs between these incentives. Our contribution is a mathematically grounded alternative to the additive Shapley framework, potentially better suited to contexts where proportionality among contributors is more meaningful than additive differences.
Paper Structure (10 sections, 3 theorems, 25 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Axioms for Incentivization and Fairness
  4. Incentive Axioms
  5. Fairness Axioms
  6. A Ratio-Based Shapley Value
  7. Ratio-based Scaled Shapley Value Fulfills Incentive and Fairness Axioms
  8. Proof of Individual Rationality (R4) and Stability
  9. Comparison to Additive Shapley Rewards
  10. Conclusion

Key Result

theorem thmcountertheorem

The ratio-based Shapley value described in equation ratioShapleyReward fulfills the Axioms R1 to R3 as well as F1 to F4.

Figures (1)

  • Figure 1: Rewards $R_i$ (solid lines) and $A_i$ (dashed lines) achieved by each agent $i$ within the grand coalition $N$ for different values of $\rho$.

Theorems & Definitions (4)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem