Membership design involves allocating an economic good whose value to any individual depends on who else receives it. We introduce a framework for optimal membership design by combining an otherwise standard mechanism-design model with allocative externalities that depend flexibly on agents' observable and unobservable characteristics. Our main technical result demonstrates that the optimal mechanism offers distinct membership tiers—differing in prices and level of access—and that the number of membership tiers is increasing in the complexity of the externalities. This insight may help explain a number of mechanisms used in practice to sell membership goods, including musical artists charging below-market-clearing prices for concert tickets, certain admission procedures used by colleges concerned about the diversity of the student body, heterogeneous pricing tiers for access to digital communities, and the use of vesting and free allocation in the distribution of network tokens.

We show how to achieve the notion of "multicalibration" from Hebert-Johnson et al. (2018) not just for means, but also for variances and other higher moments. Informally, this means that we can find regression functions which, given a data point, can make point predictions not just for the expectation of its label, but for higher moments of its label distribution as well—and those predictions match the true distribution quantities when averaged not just over the population as a whole, but also when averaged over an enormous number of finely defined subgroups. It yields a principled way to estimate the uncertainty of predictions on many different subgroups—and to diagnose potential sources of unfairness in the predictive power of features across subgroups. As an application, we show that our moment estimates can be used to derive marginal prediction intervals that are simultaneously valid as averaged over all of the (sufficiently large) subgroups for which moment multicalibration has been obtained.

There is great interest in whether machine learning algorithms deployed in consequential domains (e.g. in criminal justice) treat different demographic groups "fairly." However, there are several proposed notions of fairness, typically mutually incompatible. Using criminal justice as an example, we study a model in which society chooses an incarceration rule. Agents of different demographic groups differ in their outside options (e.g. opportunity for legal employment) and decide whether to commit crimes. We show that equalizing type I and type II errors across groups is consistent with the goal of minimizing the overall crime rate; other popular notions of fairness are not.