Sketching Stochastic Valuation Functions

We consider the problem of sketching set valuation functions, defined as the expectation of a valuation function applied to independent random item values. For valuation functions that are monotone and either subadditive or submodular and that satisfy a weak homogeneity condition (or other structural conditions), we show that there exist discretized versions of the item value distributions—each with support size $O(k\mathrm{\hspace{0.17em}log\hspace{0.17em}}k)$—that yield a sketch valuation function providing a constant-factor approximation to the true valuation for any subset of items of size at most k. These discretized distributions can be computed efficiently for each item independently, making the approach highly scalable. Our results apply broadly to valuation functions commonly encountered in practice, including team performance based on the best member (e.g., maximum functions), constant elasticity of substitution (CES) production functions with diminishing returns in economics, and others. Sketch valuation functions are especially useful in optimization problems such as best set selection and welfare maximization, where exact value computations are costly or intricate. They enable efficient approximate evaluation of value oracle queries while preserving provable approximation guarantees for the original stochastic optimization problem.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2024.0039.