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Meritocratic Fairness via $K$-Shapley Values in Budgeted Combinatorial Bandits with Full-Bandit Feedback

We study meritocratic fairness in budgeted combinatorial multi-armed bandits with full-bandit feedback, where a learner selects at most $K$ arms per time step and observes only the noisy aggregate reward of the selected set. To define merit under budgeted coalition constraints, we introduce the $K$-Shapley value, an adaptation of the classical Shapley value that measures marginal contributions using only coalitions of size at most $K$. We show that the $K$-Shapley value is the unique solution concept satisfying symmetry, linearity, null player, and $K$-efficiency axioms. We then establish an $Ω(T^{2/3})$ lower bound on fairness regret for monotone submodular valuation functions. We show that an explore-then-commit algorithm MURaS (Meritocratic Uniform Random Sampling) achieves $\tilde O(T^{2/3})$ fairness regret by exploring all arms uniformly in exploration phase. To improve empirical regret, we propose IW-KSVFair, a meritocratic full-bandit algorithm that learns a selection policy whose arm marginals are proportional to the unknown $K$-Shapley values. To correct the bias induced by adaptive sampling, IW-KSVFair uses importance-weighted estimation and mixes the adaptive set distribution with a uniform distribution to keep importance weights bounded. We prove that IW-KSVFair achieves $\tilde O(T^{2/3})$ fairness regret, matching the lower bound up to logarithmic factors. Experiments on synthetic and real-world datasets show that IW-KSVFair achieves low cumulative fairness regret and closely aligns empirical selection frequencies with $K$-Shapley value-based merit.

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