  We study a game with \emph{strategic} vendors who own multiple items and a
single buyer with a submodular valuation function. The goal of the vendors is
to maximize their revenue via pricing of the items, given that the buyer will
buy the set of items that maximizes his net payoff.
  We show this game may not always have a pure Nash equilibrium, in contrast to
previous results for the special case where each vendor owns a single item. We
do so by relating our game to an intermediate, discrete game in which the
vendors only choose the available items, and their prices are set exogenously
afterwards.
  We further make use of the intermediate game to provide tight bounds on the
price of anarchy for the subset games that have pure Nash equilibria; we find
that the optimal PoA reached in the previous special cases does not hold, but
only a logarithmic one.
  Finally, we show that for a special case of submodular functions, efficient
pure Nash equilibria always exist.
