  We introduce and study the class of semidefinite games, which generalizes
bimatrix games and finite $N$-person games, by replacing the simplex of the
mixed strategies for each player by a slice of the positive semidefinite cone
in the space of real symmetric matrices.
  For semidefinite two-player zero-sum games, we show that the optimal
strategies can be computed by semidefinite programming. Furthermore, we show
that two-player semidefinite zero-sum games are almost equivalent to
semidefinite programming, generalizing Dantzig's result on the almost
equivalence of bimatrix games and linear programming.
  For general two-player semidefinite games, we prove a spectrahedral
characterization of the Nash equilibria. Moreover, we give constructions of
semidefinite games with many Nash equilibria. In particular, we give a
construction of semidefinite games whose number of connected components of Nash
equilibria exceeds the long standing best known construction for many Nash
equilibria in bimatrix games, which was presented by von Stengel in 1999.
