We consider the decentralized power allocation and spectrum sharing problem
in multi-user, multi-channel systems with strategic users. We present a
mechanism/game form that has the following desirable features. (1) It is
individually rational. (2) It is budget balanced at every Nash equilibrium of
the game induced by the game form as well as off equilibrium. (3) The
allocation corresponding to every Nash equilibrium (NE) of the game induced by
the mechanism is a Lindahl allocation, that is a weakly Pareto optimal
allocation.

In this paper we present an algorithm to find an integral vector in the
polyhedral cone $\Gamma=\{X | \textbf{A}X \leq 0\}$, without assuming the
explicit knowledge of $\textbf{A}$, only the knowledge of some solution to
$\Gamma$, say $Y$. Furthermore, we show that under assumption that the entries
of $\textbf{A}$, are in $\{-d,-d+1,...,0,...,d-1,d\}$, the maximum component of
the derived integral vector will be at most $(2d)^{2^{n-1}-n}$.

Within the context of games on networks S. Goyal [1, pg. 39] posed the
following problem. Under any arbitrary but fixed topology, does there exist at
least one pure Nash equilibrium that exhibits a positive relation between the
cardinality of a player's set of neighbors and its utility payoff? In this
paper we present a class of topologies in which pure Nash equilibria with the
above property do not exist.