Fair Division via Quantile Shares

We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely proportionality and maximin share, are not universally feasible, nor are any constant approximations of them. We propose a novel share notion, where an agent assesses the fairness of a bundle by comparing it to her valuation in a random allocation. In this framework, a bundle is considered q-quantile fair, for q∈[0,1], if it is at least as good as a bundle obtained in a uniformly random allocation with probability at least q. Our main question is whether there exists a constant value of q for which the q-quantile share is universally feasible. Our main result establishes a strong connection between the feasibility of quantile shares and the classical ErdHos Matching Conjecture. Specifically, we show that if a version of this conjecture is true, then the frac12e-quantile share is universally feasible. Furthermore, we provide unconditional feasibility results for additive, unit-demand and matroid-rank valuations for constant values of q. Finally, we discuss the implications of our results for other share notions.