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

Abstract

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, \[ Δ_ε(x):=ψ_ε^p(x)-ψ_ε^o(x), \] 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 ${\Delta }_{\epsilon }(x)\le L_F(x)\mathrm{diam}(S_{\epsilon }(x))$ 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.