Intermediate Bilevel Optimization: Modeling Endogenous Follower Tie-Breaking Behavior

In bilevel optimization, optimistic and pessimistic follower behaviors are the most commonly used forms to define how the follower ties-breaks among multiple optimal solutions. In this work, we go beyond these extreme tie-breaking behaviors and investigate the intermediate bilevel optimization program (I-BO), where the follower's selected optimal response is a decision-dependent random event, with a probability measure influenced by the leader's decision. We formally introduce a class of such endogenous measures, including the special case of strong-weak decision-dependent I-BO. We reformulate the I-BO as a Transformed I-BO (T-I-BO) with exogenous uncertainty by defining inverse and Markov-chain transformations, which represent the follower's response as a function of the leader's decision and exogenous randomness. We handle the T-I-BO's uncertainty via sample-average approximation (SAA), and we propose tailored approaches for its SAA program according to the chosen transformation. Computationally, our methods solve reasonable-sized instances efficiently and outperform the deterministic equivalent when available. Furthermore, experiments stress the critical need to accurately model follower tie-breaking behavior, particularly depending on its alignment with the leader's objective, as misspecification leads to suboptimal leader decisions.