更好地在一起？规模的外部性如何使团结和精算公平的概念复杂化
Better Together? How Externalities of Size Complicate Notions of Solidarity and Actuarial Fairness
摘要：
考虑一个成本分担博弈，参与者对总成本的贡献不同：一个例子可能是保险公司为一群混合风险的个人计算保费。两个自然的和相互竞争的公平概念可能是a）向每个人收取相同的价格或b）根据他们给游泳池带来的成本向每个人收费。在保险文献中，这些一般方法被称为“团结”和“精算公平”，通常被视为对立的。然而，在保险业（以及许多其他自然环境）中，成本分摊博弈也表现出“规模外部性”：在其他条件相同的情况下，规模较大的群体平均成本较低。在保险的情况下，我们分析了一个模型的外部性的大小，由于减少了损失的变化。我们探讨这是如何复杂化的传统理解的公平，借鉴文献合作博弈论。首先，我们探讨团结：我们表明，这是可能的两个群体（高风险和低风险），严格受益于加入保险池的费用是平均分摊，而不是在单独的风险池。在此基础上，我们制定了一个定价计划，最大限度地补贴高风险群体，同时保持激励低风险人群留在保险池中。接下来，我们证明了在这个新模型中，向每个人收取的价格必须取决于其他参与者的风险，这使得天真的精算公平性效率低下。此外，我们证明了稳定的定价方案必须是那些球员有反社会的激励，渴望风险更大的合作伙伴，矛盾的动机，使用精算公平。最后，我们描述了这些结果如何与机器学习的公平性和未来研究的潜在途径有关。
英文摘要：
Consider a cost-sharing game with players of different contribution to the total cost: an example might be an insurance company calculating premiums for a population of mixed-risk individuals. Two natural and competing notions of fairness might be to a) charge each individual the same price or b) charge each individual according to the cost that they bring to the pool. In the insurance literature, these general approaches are referred to as "solidarity" and "actuarial fairness" and are commonly viewed as opposites. However, in insurance (and many other natural settings), the cost-sharing game also exhibits "externalities of size": all else being equal, larger groups have lower average cost. In the insurance case, we analyze a model with externalities of size due to a reduction in the variability of losses. We explore how this complicates traditional understandings of fairness, drawing on literature in cooperative game theory. First, we explore solidarity: we show that it is possible for both groups (high and low risk) to strictly benefit by joining an insurance pool where costs are evenly split, as opposed to being in separate risk pools. We build on this by producing a pricing scheme that maximally subsidizes the high risk group, while maintaining an incentive for lower risk people to stay in the insurance pool. Next, we demonstrate that with this new model, the price charged to each individual has to depend on the risk of other participants, making naive actuarial fairness inefficient. Furthermore, we prove that stable pricing schemes must be ones where players have the anti-social incentive of desiring riskier partners, contradicting motivations for using actuarial fairness. Finally, we describe how these results relate to debates about fairness in machine learning and potential avenues for future research.
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