  A game theoretic framework is presented to analyze the problem of finding the
optimal number of data streams to transmit in a multi-user MIMO scenario, where
both the transmitters and receivers are equipped with multiple antennas.
Without channel state information (CSI) at any transmitter, and using outage
capacity as the utility function with zero-forcing receiver, each user is shown
to transmit a single data stream at Nash equilibrium in the presence of
sufficient number of users. Transmitting a single data stream is also shown to
be optimal in terms of maximizing the sum of the outage capacities in the
presence of sufficient number of users. With CSI available at each transmitter,
and using the number of successful bits per Joule of energy as the utility
function, at Nash equilibrium, each user is shown to transmit a single data
stream on the best eigen-mode that requires the least transmit power to achieve
a fixed signal-to-interference ratio. Using the concept of locally gross
direction preserving maps, existence of Nash equilibrium is shown when the
number of successful bits per Joule of energy is used as the utility function.
