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J Optim Theory Appl (2012) 154:175186
DOI 10.1007/s10957-011-9981-5
Received: 25 March 2011 / Accepted: 17 December 2011 / Published online: 6 January 2012 Springer Science+Business Media, LLC 2012
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):8194, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately dened 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):159164, 2007) suggested that the hypotheses of this result are difcult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harkers result for these games with the aim of formally establishing its reach. Specically, we show that if Harkers 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 Harkers result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harkers result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159164, 2007), in which it is shown that a suitably dened VI provides a solution to the QVI of a shared-constraint game.
This work was done while Kulkarni was at the latter department.
A.A. Kulkarni
Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, USA e-mail: [email protected]
U.V. Shanbhag ( )
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USAe-mail: mailto:[email protected]
Web End [email protected]
Revisiting Generalized Nash Games and Variational Inequalities
Ankur A. Kulkarni Uday V. Shanbhag
176 J Optim Theory Appl (2012) 154:175186
Keywords Variational inequalities Quasi-variational inequalities Generalized
Nash games Shared constraints Game theory
1 Introduction
Variational inequalities (VIs) provide an avenue for compactly articulating equilibrium conditions for continuous-strategy convex Nash games. When these games are generalized to...