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

We study the allocation of shared resources over multiple rounds among competing agents, via the dynamic max-min fair (DMMF) mechanism: the good in each round is allocated to the requesting agent with the least number of allocations received to date. We show that in large markets when an agent has i.i.d. values across rounds, under mild distributional assumptions (e.g., bounded PDF function), the DMMF mechanism allows each agent to realize a $1 - o(1)$ fraction of her ideal utility -- her highest achievable utility given her nominal share of resources. This guarantee holds under arbitrary behavior by other agents and is achieved by characterizing the agent's utility under a rich space of strategies, wherein an agent can tune how aggressive to be in requesting the item. Our techniques also allow us to handle settings where an agent's values are correlated across rounds, thereby allowing an adversary to predict and block her future values. By tuning the aggressiveness, an agent can guarantee $\Omega(\gamma)$ fraction of her ideal utility, where $\gamma\in [0, 1]$ is a parameter that quantifies dependence across rounds (with $\gamma = 1$ indicating full independence and lower values indicating more correlation). Finally, we extend our efficiency results to the case of reusable resources, where an agent might need to hold the item over multiple rounds to receive utility. Our results subsume previous guarantees obtained using a more complicated mechanism proving a half ideal utility guarantee under i.i.d. values sampled from worst-case distributions.