Revisiting Generalized Nash Games and Variational Inequalities

Abstract

Generalized Nash games with shared constraints represent an extension of Nash games in which strategy sets are coupled across players through a shared or common constraint. The equilibrium conditions of such a game can be compactly stated as a quasi-variational inequality (QVI), an extension of the variational inequality (VI). In (Eur. J. Oper. Res. 54(1):81–94, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately defined VI solves the QVI. This is a particularly important result, given that VIs are generally far more tractable than QVIs. However Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007) suggested that the hypotheses of this result are difficult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harker’s result for these games with the aim of formally establishing its reach. Specifically, we show that if Harker’s result is applied in a natural manner, its hypotheses are impossible to satisfy in most settings, thereby supporting the observations of Facchinei et al. But we also show that an indirect application of the result extends the realm of applicability of Harker’s result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harker’s result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007), in which it is shown that a suitably defined VI provides a solution to the QVI of a shared-constraint game.