  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.
