We consider a variation on the classical finance problem of optimal portfolio design. In our setting, a large population of consumers is drawn from some distribution over risk tolerances, and each consumer must be assigned to a portfolio of lower risk than her tolerance. The consumers may also belong to underlying groups (for instance, of demographic properties or wealth), and the goal is to design a small number of portfolios that are fair across groups in a particular and natural technical sense. Our main results are algorithms for optimal and near-optimal portfolio design for both social welfare and fairness objectives, both with and without assumptions on the underlying group structure. We describe an efficient algorithm based on an internal two-player zero-sum game that learns near-optimal fair portfolios ex ante and show experimentally that it can be used to obtain a small set of fair portfolios ex post as well. For the special but natural case in which group structure coincides with risk tolerances (which models the reality that wealthy consumers generally tolerate greater risk), we give an efficient and optimal fair algorithm. We also provide generalization guarantees for the underlying risk distribution that has no dependence on the number of portfolios and illustrate the theory with simulation results.

中文翻译：

---

公平投资组合设计的算法和学习

我们考虑优化投资组合设计的经典金融问题的变体。在我们的设置中，大量消费者来自风险容忍度的某种分布，并且每个消费者必须被分配到风险低于其容忍度的投资组合。消费者也可能属于潜在群体（例如，人口属性或财富），其目标是设计少数在特定和自然技术意义上跨群体公平的投资组合。我们的主要结果是针对社会福利和公平目标的最优和接近最优的投资组合设计算法，无论是否对基础组结构进行假设。我们描述了一种基于内部两人零和游戏的有效算法，该算法事前学习接近最优的公平投资组合，并通过实验表明它也可用于事后获得一小组公平投资组合。对于群体结构与风险容忍度相吻合的特殊但自然的情况（它模拟了富裕消费者通常能够承受更大风险的现实），我们给出了一个有效且最优的公平算法。我们还为不依赖于投资组合数量的基础风险分布提供泛化保证，并用模拟结果说明理论。对于群体结构与风险容忍度相吻合的特殊但自然的情况（它模拟了富裕消费者通常能够承受更大风险的现实），我们给出了一个有效且最优的公平算法。我们还为不依赖于投资组合数量的基础风险分布提供泛化保证，并用模拟结果说明理论。对于群体结构与风险容忍度相吻合的特殊但自然的情况（它模拟了富裕消费者通常能够承受更大风险的现实），我们给出了一个有效且最优的公平算法。我们还为不依赖于投资组合数量的基础风险分布提供泛化保证，并用模拟结果说明理论。