We are concerned with a class of Nash games in which the players' strategy sets are coupled by a shared constraint. A widely employed solution concept for such games, referred to as generalized Nash games, is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) [6] 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 a theory that provides sufficient conditions for ensuring that the existence of a GNE implies the existence of a VE; in such an instance, the VE is said to be a refinement of the GNE. For certain games our conditions are shown to be necessary. This theory rests on a result showing that, in both the primal and the primal-dual space, the GNE and the VE are equivalent upto the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. The refinement of the GNE is of relevance to pure, applied and computational game theory. Our results unify some previously known facts pertaining to such equilibria and are utilized in showing that shared-constraint Nash-Cournot games arising in power markets do indeed admit a refinement.

Explicit nonasymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for -ary alphabet and string length is shown to be of size at most . An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The nonasymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known nonasymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.

Neuro-dynamic programming is a class of powerful techniques for approximating the solution to dynamic programming equations. In their most computationally attractive formulations, these techniques provide the approximate solution only within a prescribed finite-dimensional function class. Thus, the question that always arises is how should the function class be chosen? The goal of this paper is to propose an approach using the solutions to associated fluid and diffusion approximations. In order to illustrate this approach, the paper focuses on an application to dynamic speed scaling for power management in computer processors.

Abstract-In Stackelberg v/s Stackelberg games a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problems are plagued by the nonuniqueness of follower equilibria and nonconvexity of leader problems whereby the problem of providing sufficient conditions for existence of global or even local equilibria remains largely open. Indeed available existence statements are restrictive and model specific. In this paper, we present what is possibly the first general existence result for equilibria for this class of games. Importantly, we impose no single-valuedness assumption on the equilibrium of the follower-level game. Specifically, under the assumption that the objectives of the leaders admit a quasipotential function, a concept we introduce in this paper, the global and local minimizers of a suitably defined optimization problem are shown to be the global and local equilibria of the game. In effect existence of equilibria can be guaranteed by the solvability of an optimization problem, which holds under mild and verifiable conditions. We motivate quasi-potential games through an application in communication networks.