The Four Equivalences

Optimizing a linear functional over probability measures is the core problem of information design, mechanism design, and decision theory under uncertainty. When does the solution behave well — admit clean characterizations, vary smoothly with parameters, extend to comparisons between experiments?

Four conditions are equivalent (arXiv:2603.11448): (i) the cone of test functions is closed under pointwise minimum, (ii) the value function is affine in the parameter, (iii) the solution correspondence has a convex graph, and (iv) any two measures in the constraint set can be coupled in an order-preserving way. Each condition lives in a different mathematical universe — function spaces, optimization, correspondence theory, coupling — yet they’re the same condition.

The structural insight: the equivalence connects what a function class looks like (closed under min), how the optimum behaves (affine), what the solution set looks like (convex graph), and how measures relate to each other (order-preserving coupling). The equivalence means you can prove theorems about couplings by checking closure properties of function cones, or vice versa. Blackwell’s theorem on experiment comparison — a foundational result in statistical decision theory — follows as a special case. The deep structure is that all four conditions express the same underlying lattice-theoretic property: the order structure of the probability measures is compatible with the optimization structure of the functionals.