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We consider the problem of dividing limited resources to individuals arrivingover $T$ rounds. Each round has a random number of individuals arrive, andindividuals can be characterized by their type (i.e. preferences over thedifferent resources). A standard notion of 'fairness' in this setting is thatan allocation simultaneously satisfy envy-freeness and efficiency. The formeris an individual guarantee, requiring that each agent prefers their ownallocation over the allocation of any other; in contrast, efficiency is aglobal property, requiring that the allocations clear the available resources.For divisible resources, when the number of individuals of each type are knownupfront, the above desiderata are simultaneously achievable for a large classof utility functions. However, in an online setting when the number ofindividuals of each type are only revealed round by round, no policy canguarantee these desiderata simultaneously, and hence the best one can do is totry and allocate so as to approximately satisfy the two properties.  We show that in the online setting, the two desired properties (envy-freenessand efficiency) are in direct contention, in that any algorithm achievingadditive counterfactual envy-freeness up to a factor of $L_T$ necessarilysuffers a efficiency loss of at least $1 / L_T$. We complement this uncertaintyprinciple with a simple algorithm, HopeGuardrail, which allocates resourcesbased on an adaptive threshold policy and is able to achieve anyfairness-efficiency point on this frontier. In simulation results, ouralgorithm provides allocations close to the optimal fair solution in hindsight,motivating its use in practical applications as the algorithm is able to adaptto any desired fairness efficiency trade-off.

Sequential Fair Allocation: Achieving the Optimal Envy-Efficiency Tradeoff Curve