For the fundamental problem of allocating a set of resources among individuals with varied preferences, the quality of an allocation relates to the degree of fairness and the collective welfare achieved. Unfortunately, in many resource-allocation settings, it is computationally hard to maximize welfare while achieving fairness goals. In this work, we consider ex-ante notions of fairness; popular examples include the \emph{randomized round-robin algorithm} and \emph{sortition mechanism}. We propose a general framework to systematically study the \emph{interpolation} between fairness and welfare goals in a multi-criteria setting. We develop two efficient algorithms ($\varepsilon-Mix$ and $Simple-Mix$) that achieve different trade-off guarantees with respect to fairness and welfare. $\varepsilon-Mix$ achieves an optimal multi-criteria approximation with respect to fairness and welfare, while $Simple-Mix$ achieves optimality up to a constant factor with zero computational overhead beyond the underlying \emph{welfare-maximizing mechanism} and the \emph{ex-ante fair mechanism}. Our framework makes no assumptions on either of the two underlying mechanisms, other than that the fair mechanism produces a distribution over the set of all allocations. Indeed, if these mechanisms are themselves approximation algorithms, our framework will retain the approximation factor, guaranteeing sensitivity to the quality of the underlying mechanisms, while being \emph{oblivious} to them. We also give an extensive experimental analysis for the aforementioned ex-ante fair mechanisms on real data sets, confirming our theoretical analysis.

翻译：针对在具有不同偏好的个体间分配一组资源这一基本问题，分配质量与公平程度及所实现的集体福利相关。然而，在许多资源分配场景中，在实现公平目标的同时最大化福利在计算上是困难的。本文考虑事前公平概念；常见例子包括随机轮询算法和抽签机制。我们提出一个通用框架，系统研究多准则设置中公平与福利目标之间的插值。我们开发了两种高效算法（ε-Mix和Simple-Mix），它们实现了公平与福利方面不同的权衡保证。ε-Mix在公平与福利方面实现了最优多准则近似，而Simple-Mix在底层福利最大化机制和事前公平机制之外，以零计算开销实现了至多常数因子内的最优性。我们的框架对这两种底层机制不作任何假设，仅要求公平机制在全体分配的集合上产生一个分布。实际上，若这些机制本身是近似算法，我们的框架将保留近似因子，确保对底层机制质量的敏感性，同时对其保持无关性。我们还针对上述事前公平机制在真实数据集上进行了广泛的实验分析，验证了我们的理论分析。