  Much work has been done on the computation of market equilibria. However due
to strategic play by buyers, it is not clear whether these are actually
observed in the market. Motivated by the observation that a buyer may derive a
better payoff by feigning a different utility function and thereby manipulating
the Fisher market equilibrium, we formulate the {\em Fisher market game} in
which buyers strategize by posing different utility functions. We show that
existence of a {\em conflict-free allocation} is a necessary condition for the
Nash equilibria (NE) and also sufficient for the symmetric NE in this game.
There are many NE with very different payoffs, and the Fisher equilibrium
payoff is captured at a symmetric NE. We provide a complete polyhedral
characterization of all the NE for the two-buyer market game. Surprisingly, all
the NE of this game turn out to be symmetric and the corresponding payoffs
constitute a piecewise linear concave curve. We also study the correlated
equilibria of this game and show that third-party mediation does not help to
achieve a better payoff than NE payoffs.
