We are concerned with generalized Nash games in which the players' strategy sets are coupled by a shared constraint. A widely employed solution concept for these games is the generalized Nash equilibrium (GNE), which is the natural generalization of the Nash equilibrium. The variational equilibrium (VE) is a specific kind of GNE given by a solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives. Our contribution is in providing conditions under which the existence of a GNE is necessary and sufficient for the existence of a VE in such an instance, the VE is said to be a ``refinement of the GNE. Establishment of this result is seen to be of relevance to pure, applied and computational game theorists. We present a theory that gives sufficient conditions for the VE to be a refinement of the GNE. For certain games these conditions are shown to be necessary. This theory rests on a result showing that the GNE and the VE are equivalent upto the Brouwer degree of certains functions whose zeros are the GNE and VE respectively. In the primal space we show equality between the Brouwer degrees of the natural maps of the quasi-variational inequality and the variational inequality, whose solutions are the GNE and VE respectively. Using a novel equation reformulation of the VE, this result is extended to the primal-dual space. These degree theoretic relationships pave the way for the aforesaid sufficient conditions. Our results unify some known results about shared constraint games and provide mathematical justification for ideas that were known to be appealing to economic intuition.
Bio: Ankur is a doctoral student at the University of Illinois at Urbana-Champaign (UIUC), USA. He received his B.Tech. in Aerospace Engineering from Indian Institute of Technology, Bombay in 2006 and M.S. in General Engineering from UIUC in 2008. He was a visiting scholar at the Tata Institute of Fundamental Research, Mumbai in 2008. His interests are in the theory of games, economics, mathematical programming and applied probability. His work has been based on providing topological insights into equilibria of generalized Nash games and multi-leader-multi-follower games.