Randomized Online Fair Division: High-Probability and Expected Realized Fairness

Abstract

We study randomized algorithms for the fully online allocation of indivisible goods among $n\ge2$ agents with nonnegative additive valuations. Goods arrive one by one and must be allocated immediately and irrevocably. Nothing is known in advance except the number of agents. Since exact ex-ante envy freeness and proportionality are readily achievable, while no positive ex-post approximation is possible for the fairness notions considered here, we study the intermediate notions of high-probability fairness and expected realized fairness. Against a non-adaptive adversary, we give a randomized algorithm that achieves an $Ω(\sqrt{\log n}/\log(n/δ))$-approximation to proportionality up to one good (PROP1) with probability at least $1-δ$ for every $δ\in(0,1)$, and an expected realized PROP1 factor of $Ω(1/\sqrt{\log n})$. Compared with independent uniform allocation (Rand), these guarantees yield an $Ω(\sqrt{\log n})$ improvement under both the high-probability and expected realized criteria. For envy freeness up to any good (EFX), the success probability of any positive factor can be made arbitrarily small, even for identical valuations. Consequently, the expected realized factor of every randomized fully online algorithm is zero. For envy freeness up to one good (EF1), no randomized fully online algorithm can guarantee a positive factor with confidence exceeding $\frac{n+1}{2n}$. The expected realized EF1 guarantee is also at most $\frac{n+1}{2n}$.