Characterize Lin-Bellman systems that satisfy the P-Lin-Bellman promise

Characterize all triples (L, R, S) for which the Lin-Bellman system (L, R, q, S) has a unique solution for every q ∈ R^n. Develop necessary and sufficient conditions (beyond the current invertibility of R − I) that precisely capture when a Lin-Bellman instance meets the P-Lin-Bellman promise.

Background

The paper introduces P-Lin-Bellman, a promise problem where each coordinate x_i equals a min or max of two affine forms in x, and the promise guarantees a unique solution for every right-hand side q. This generalizes Bellman equations used in discounted stochastic games.

While a necessary condition (R − I is invertible) is identified, a complete characterization analogous to the P-matrix condition for LCPs is lacking. Establishing such a characterization would deepen understanding of when these systems are well-posed and uniquely solvable for all q.