Better together? A performance theory of human-machine skill policies under generalization difficulty

Organizations pairing a human and a machine on a task decide, often by default, which executes each instance and what happens when both do. We account for what this attains under the squared rule, with performance distributed over instances to capture generalization difficulty. A gross stake lies between the skills' mean and best per instance. It splits at the best single skill into a systematic part, known from separate assessments and secured by commitment, and a residual, the expected value of perfect information on which skill each instance favors, known only from paired observation and split further by classes. Skills set the stake; workflows decide the take, via two channels. Routing directs work, weighting or selecting responses; with binary truth it is capped by the best skill per instance. Reading responses as joint evidence is not routing; what they jointly reveal caps it. Synergy lets skills interact before responding; bounded by no skill either way, its lift over independent responses must clear a break-even and can fall below one. As algorithms, arrangements form a ladder stating what each reads, holds and records, and a procedure whose checks keep earlier gains when class means are known and lose little on finite records given a margin. The theory extends to many skills, each worth what it adds where it beats all others held, and to competing skills, where at parity a single failure's allowed cost decides. In a public chest-radiograph study with 227 radiologists, they are near parity with the model and the stake is mostly residual; assisted reading, no better on average originally, returned less than averaging both responses; recalibrating the model alone beats any deployable arrangement of raw responses. Organizations can learn from their records what is at stake, which arrangement collects it and what evidence each step needs; pricing costs and errors is future work.