Price of Fairness in Bandits: A Tight Minimax Characterization

In bandit problems, standard regret-minimizing algorithms treat exploration as an amortized cost, which can expose early participants to unfair ex-ante losses in settings such as clinical trials. Recent work addresses this by evaluating the sequence of per-round expected rewards through the generalized p-mean, interpolating between utilitarian welfare (p=1), Nash welfare (p to0), and Rawlsian fairness (p to-infinity). Although tight guarantees are known for p 0, the strictly fair regime q=-p>0 remains unresolved because negative-power means are dominated by the smallest per-round rewards. For sigma-sub-Gaussian rewards with nonnegative means, the best prior algorithm relied on uniform early exploration and achieved regret O(k(q+1)/2/sqrt T), while the only general lower bound was the classical (sigma sqrt k/T). Thus it was unclear whether the extra dependence on k was intrinsic to strict fairness or an artifact of uniform exploration. We close this gap by identifying the exact polynomial price of strict fairness. Using a needle-in-haystack construction, we prove an algorithm-independent lower bound (sigma sqrt k(1,q)/T); for q>1, this shows that the penalty k q/2 is information-theoretically unavoidable. We then introduce (Harmonic Anchored Rank Exploration), which replaces uniform exploration with an inverse-weighted harmonic rank schedule protected by a certified positive-mean anchor. Its regret is (sigma sqrt k(1,q)/T), matching the lower bound up to logarithmic factors. Experiments on synthetic instances confirm that improves over uniform-exploration baselines, with gains increasing as q grows.