A Bridge Between Bilevel Programs and Nash Games

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.