A Bridge Between Bilevel Programs and Nash Games

Abstract

We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our "uneven" horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our "uneven" horizontal model, in some sense, lies between the vertical bilevel model and a "pure" horizontal game.

The Standard Pessimistic Bilevel Problem

“…Note that, under the initial assumptions, each player's problem is convex. As for the relation between equilibria of (10) and global optimal points of (SPBP ε ) (and, thus, global optimal points of (PBP ε ), see Proposition 4.1 (ii)), the following result, which is reminiscent of [16,Theorem 3.1], holds.…”

“…• the bilevel problem (5), for every fixed x, is pure hierarchical [16] (or simple [5]), i.e., S ε (x) is a fixed set.…”

“…Considering the classical counterexample (see, e.g., [16,Example 3.6]), one can easily show that a local optimal point of the resulting (MFG ε ) might not be local optimal for the (SPBP ε ). We observe that a sufficient condition, which is not necessary, for the assumption in Proposition 5.2 (ii) to hold is the following: ( x, y, z) is a local optimal point for (MFG ε ), being S inner semicontinuous relative to X at x.…”

“…Leveraging some results in [16], but differently from what done in subsection 5.1, here we introduce a (single-level) GNEP with three players which turns out to have some interesting connections with the (SPBP ε ). Preliminarily, we consider the following Nash game with a hierarchical structure:…”

“…Clearly, when S ε is fixed for every x ∈ X, (ii) is verified; moreover, if F is separable, then (SPBP ε ) and GNEP (10) turn out to be equivalent in a global sense. The assertion (ii) in Proposition C.1 can be stated in a local sense (see also [16,Theorem 3.3]).…”