On Cofactuals
Research question
When can policy-regime marginals determine how outcomes are paired across worlds, and therefore the distribution of policy gains?
Abstract
The distribution of policy gains depends on how outcomes are paired across regimes. I study a shared-component model in which policy labels fix the loadings of independent interaction terms. Under local exponential integrability, the marginal laws admit this model if and only if their Möbius-transformed cumulant generating functions are admissible; the same transforms construct the joint law within the class. A declared collection of k regimes requires only its intersection closure, containing at most 2^k-1 worlds. For two regimes, their two marginals and intersection marginal determine the full gain distribution. A row-space condition shows which marginal cumulants identify a declared contrast, and a Poisson construction gives necessity at every order. For several regimes, the endpoints and distinct pairwise intersections form the unique inclusion-minimal bundle for all pairwise odd and even gain cumulants, including under heterogeneous positive costs. Transform admissibility is equivalent to a hierarchy of positive-semidefinite matrices, yielding a consistent sieve and finite-sample size control under bounded or exponentially integrable unbounded outcomes. With paired observations, a joint-CGF defect records the complete departure of the gain law from its shared-component reference and bounds every cumulant error. A global positive-definiteness criterion gives the attained set of gain laws within a declared sensitivity band. In a discounted decision model, an action-gap condition preserves the optimal policy under reward and transition perturbations; a common-resolvent bound and a margin condition quantify departures. Paired and state-adjusted moment procedures assess selected restrictions implied by the model.
In one sentence
Marginal outcome distributions determine policy-gain distributions only after the cross-world coupling is disciplined; cofactual components provide one such structure and observable restrictions.