Softening Bilevel Problems Via Two-scale Gibbs Measures

We introduce a new, and elementary, approximation method for bilevel optimization prob- lems motivated by Stackelberg leader-follower games. Our technique is based on the notion of two-scale Gibbs measures. The first scale corresponds to the cost function of the fol- lower and the second scale to that of the leader. We explain how to choose the weights corresponding to these two scales under very general assumptions and establish rigorous Gamma-convergence results. An advantage of our method is that it is applicable both to optimistic and to pessimistic bilevel problems.