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

TL;DR

This paper introduces a diagnostic framework to quantify implementation risk in bilevel decision problems by evaluating the ambiguity premium arising from nonunique or near-optimal follower responses.

Contribution

It operationalizes the ambiguity premium as a diagnostic tool, providing bounds and a workflow to assess policy robustness versus efficiency in hierarchical decision models.

Findings

Established a diameter bound for the ambiguity premium.

Provided an $ ext{O}(\sqrt{ ext{epsilon}})$ estimate under quadratic growth.

Demonstrated the framework with two case studies showing policy sensitivity.

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, \[ \Delta_\epsilon(x):=\psi_\epsilon^p(x)-\psi_\epsilon^o(x), \] 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 $\Delta_\epsilon(x)\le L_F(x)\,\mathrm{diam}(S_\epsilon(x))$ 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.