Meritocratic Fairness in Budgeted Combinatorial Multi-armed Bandits via Shapley Values

We study multi-objective multi-agent multi-armed bandits (MO-MA-MAB) under stochastic rewards, where agents observe heterogeneous reward vectors and communicate over time-varying graphs. We formulate this emerging problem setting to address \emph{efficient learning}, measured by Pareto regret, and incorporate \emph{fair learning} as an additional goal, captured via social welfare. To measure efficiency, we formulate Pareto regret and develop \textsc{Pareto UCB1 Gossip}, whose novel exploration radius explicitly separates statistical uncertainty in Pareto-based inference from consensus error. To express the fairness constraint, we formulate a Nash Social Welfare objective over preference-scalarized rewards and propose \textsc{Simulated NSW UCB Gossip}, which integrates preference-based reward simulation, gossip-based utility estimation, and UCB-style exploration. We prove that \textsc{Pareto UCB1 Gossip} achieves $\mathcal{O}(\log T)$ regret and an instance-independent rate of $\mathcal{O}(\sqrt{T})$, while \textsc{Simulated NSW UCB Gossip} achieves an instance-independent regret bound of $\mathcal{O}(T^{3/4})$. This separation reveals the cost of imposing the fairness constraint to our efficiency objective: fairness limits information aggregation and slows convergence. Experiments show that our methods consistently outperform baselines, improving performance by approximately $100\%$ and $50\%$ in the efficiency and fairness settings, respectively.

We study two-player zero-sum games (TPZSGs) with bandit feedback under fairness constraints requiring every action to be played with probability at least $α/m$. Existing instance-dependent results target $\textit{pure}$ Nash equilibria, while fairness generically produces $\textit{mixed}$ equilibria, a harder learning target. Our key technical tool is a reparametrization: every fair strategy decomposes as $p = (α/m)\mathbf{1} + (1-α)\widetilde{p}$ with $\widetilde{p} \in Δ_m$, and substituting into the payoff form yields $p^{\top}Aq = \widetilde{p}^{\top}\widetilde{A} q$ for a fair payoff matrix $\widetilde{A} := (1-α)A + α\mathbf{1} c^{\top}$, where $c_j = \tfrac{1}{m}\sum_i A(i,j)$ is the column-mean vector. The fair game on $A$ is then equivalent to a standard zero-sum game on $\widetilde{A}$, so equilibrium existence, KKT structure, and LP basis stability reduce to classical results applied to $\widetilde{A}$. We derive the fair minimax value, fair Nash equilibrium, fair regret, and a clean dual representation showing the price of fairness is at most $α(1-1/m)$ and vanishes whenever the unconstrained equilibrium already has full support. Our main result is an $\widetilde{O}(T^{2/3})$ regret bound for an Explore-Then-Commit algorithm, $\texttt{Fair-ETC-TPZSG}$, applicable to general mixed fair equilibria, together with a discussion of why naive action elimination does not readily improve it. When the fair equilibrium has a single dominant action, equivalently when $\widetilde{p}^{\star}$ is a vertex of $Δ_m$, the bound sharpens to instance-dependent $\widetilde{O}(1/\widetildeΔ(α)^{2})$, where $\widetildeΔ(α)$ is the LP-margin gap.

Nash regret has recently emerged as a principled fairness-aware performance metric for stochastic multi-armed bandits, motivated by the Nash Social Welfare objective. Although this notion has been extended to linear bandits, existing results suffer from suboptimality in ambient dimension $d$, stemming from proof techniques that rely on restrictive concentration inequalities. In this work, we resolve this open problem by introducing new analytical tools that yield an order-optimal Nash regret bound in linear bandits. Beyond Nash regret, we initiate the study of $p$-means regret in linear bandits, a unifying framework that interpolates between fairness and utility objectives and strictly generalizes Nash regret. We propose a generic algorithmic framework, FairLinBandit, that works as a meta-algorithm on top of any linear bandit strategy. We instantiate this framework using two bandit algorithms: Phased Elimination and Upper Confidence Bound, and prove that both achieve sublinear $p$-means regret for the entire range of $p$. Extensive experiments on linear bandit instances generated from real-world datasets demonstrate that our methods consistently outperform the existing state-of-the-art baseline.