Kamm’s sore throat case gives us a choice: save one life, or save a distinct life and cure a sore throat. We defend the fairness explanation of the judgement that one should flip a coin to decide whom to save: it is disrespectful to let a sore throat act as a tie-breaker, because an individual would be forced to forgo a 50% fair chance of living (given to them by a coin flip), which cannot be outweighed by any number of (...) sore throats. We show that this explanation of when and why claims can permissibly break ties generates new problem cases for theories of aggregation, including theories that have been thought to accommodate the judgement that one should flip a coin in Kamm’s sore throat case. We then generalise the fairness explanation to cases involving multiple groups. (shrink)

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. (...) Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability. (shrink)

Dwork, Hardt, Pitassi, Reingold, & Zemel [6] introduced two notions of fairness, each of which is meant to formalize the notion of similar treatment for similarly qualified individuals. The first of these notions, which we call additive metric fairness, has received much attention in subsequent work studying the fairness of a system composed of classifiers which are fair when considered in isolation [3, 4, 7, 8, 12] and in work studying the relationship between fair treatment of individuals and fair treatment (...) of groups [6, 7, 13]. Here, we extend these lines of research to the second, less-studied notion, which we call multiplicative metric fairness. In particular, we exactly characterize the fairness of conjunctions and disjunctions of multiplicative metric fair classifiers, and the extent to which a classifier which satisfies multiplicative metric fairness also treats groups fairly. This characterization reveals that whereas additive metric fairness becomes easier to satisfy when probabilities of acceptance are small, leading to unfairness under functional and group compositions, multiplicative metric fairness is better-behaved, due to its scale-invariance. (shrink)