  We consider two sided matching markets consisting of agents with
non-transferable utilities; agents from the opposite sides form matching pairs
(e.g., buyers-sellers) and negotiate the terms of their math which may include
a monetary transfer. Competitive equilibria are the elements of the core of
this game.
  We present the first combinatorial characterization of competitive equilibria
that relates the utility of each agent at equilibrium to the equilibrium
utilities of other agents in a strictly smaller market excluding that agent;
thus automatically providing a constructive proof of existence of competitive
equilibria in such markets.
  Our characterization also yields a group strategyproof mechanism for
allocating indivisible goods to unit demand buyers with non-quasilinear
utilities that highly resembles the Vickrey Clarke Groves (VCG) mechanism. As a
direct application of this, we present a group strategyproof welfare maximizing
mechanism for Ad-Auctions without requiring the usual assumption that search
engine and advertisers have consistent estimates of the clickthrough rates.
