Revisiting Generalized Nash Games and Variational Inequalities

Generalized Nash games represent an extension of Nash games in which strategy sets are coupled across players. The equilibrium conditions of such a game can be compactly stated as a quasivariational inequality (QVI), an extension of the variational inequality (VI). Harker [9] showed that under certain conditions on the maps defining the QVI, a solution to a related VI solves the QVI. This is a particularly important result, given that variational inequalities are generally far more tractable than quasi-variational inequalities. This paper investigates the applicability of Harker’s result to the class of generalized Nash games where the strategy sets are linked through a shared or common constraint. The application of Harker’s result to the QVI associated with such games proves difficult because the hypotheses, that require that a set with certain properties exist, can fail to hold even for simple shared-constraint games. We show these hypotheses are in fact impossible to satisfy in most settings. But we show that for a modified QVI, whose solution set equals that of the original QVI, the hypothesis of Harker’s result always hold. This paves the way for applying this result to shared-constraint games, albeit in an indirect manner. This avenue allows us to recover as a special case, a result proved by Facchinei et al. [4], in which it is shown that a suitably defined variational inequality provides a solution to the QVI of a shared-constraint Nash game.