Beyond Worst-Case Online Allocation via Dynamic Max-min Fairness

Giannis Fikioris, Siddhartha Banerjee, Éva Tardos

Abstract: We consider the classical Dynamic Max-min fair (DMMF) mechanism for allocating an indivisible resource without money over multiple agents and T rounds. We show that under mild assumption on value distributions, it guarantees every agent close to optimal utility in large markets.Each agent i has a fair share αi (where αi > 0 and Σi αi = 1), representing her nominal share. In every round, each agent has a non-negative value for the item which is a random variable independent of other agents. The DMMF mechanism allocates as follows: every round a subset of agents requests the resource; out of these, the one with the least past allocations gets the item.We use ideal utility to benchmark our results: v*i is agent i's maximum expected average per-round utility if allocated her favorite αi fraction of the rounds. We show that in the DMMF mechanism, any agent i can use a simple strategy to make utility guarantees that hold under arbitrary (potentially adversarial and collusive) behavior by the other agents. Specifically, when her values are i.i.d. across rounds, agent i can guarantee total expected utility by every round t ≤ T:• [EQUATION] under any value distribution.• [EQUATION] when her values are uniformly distributed.• [EQUATION] when her value distribution's pdf max and min value over its support differ by a λ ≥ 1 multiplicative factor.The last two results are approximately optimal in the large market setting (when αi → 0) and greatly outperform the [EQUATION] upper bound that holds for worst-case distributions. With some modifications, similar results are achieved for resources needed for multiple contiguous periods.We also prove that an agent can have utility guarantees under the same mechanism, even when her values are dependent across rounds, generated by a hidden Markov model. We use a parameter γ ∈ (0, 1] to measure the dependence of values across rounds, where γ = 1 indicates that values are independent and lower γ indicates more dependence across rounds. In this case, we prove that agent i can always guarantee Ω(γ)v*i t - O(1) utility for small α. We also offer guarantees that are independent of γ.The full version of our paper can be found in https://arxiv.org/abs/2310.08881.

External IDs:dblp:conf/sigecom/FikiorisBT25