A Diagnostic Framework for Implementation Risk in Bilevel Decision Problems: The Ambiguity Premium and the Robustness--Efficiency Frontier

Hierarchical decision problems are often modeled as bilevel programs in which a leader commits to a policy and a follower responds optimally. When the follower’s optimal response is nonunique, or when only near-optimal follower behavior can be verified, the same leader decision may induce a range of upper-level outcomes. This paper develops a diagnostic framework for quantifying that exposure. For a leader decision $x$, we evaluate the optimistic and pessimistic upper-level values over the $\epsilon$-optimal follower response set $S_{\epsilon }(x)$ and use their difference, $${\mathrm{\Delta }}_{\epsilon }(x):={\psi }_{\epsilon }^p(x)-{\psi }_{\epsilon }^o(x),$$ as an ambiguity premium. The premium itself is classical in the optimistic–pessimistic bilevel distinction; the contribution here is to make it operational as an implementation-risk diagnostic. We establish a diameter bound ${\mathrm{\Delta }}_{\epsilon }(x)\le L_F(x)\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(S_{\epsilon }(x))$ and an $\mathcal{O}(\sqrt{\epsilon })$ estimate under quadratic lower-level growth. We then organize existing bilevel–GNEP reformulations by their computational roles and propose a screening workflow that reports, for each candidate policy, nominal value, ambiguity exposure, and a first-order residual. Two stylized case studies – a parallel-link Stackelberg pricing problem and a convex generation-planning model with diversification constraints – show how the resulting robustness–efficiency frontier can identify policies that are nominally attractive but sensitive to near-optimal follower responses.