  The rank of a bimatrix game is the matrix rank of the sum of the two payoff
matrices. This paper comprehensively analyzes games of rank one, and shows the
following: (1) For a game of rank r, the set of its Nash equilibria is the
intersection of a generically one-dimensional set of equilibria of
parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a
rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1
game can be found by following a piecewise linear path. In contrast, such a
path-following method finds only one equilibrium of a bimatrix game. (4) The
number of equilibria of a rank-1 game may be exponential. (5) There is a
homeomorphism between the space of bimatrix games and their equilibrium
correspondence that preserves rank. It is a variation of the homeomorphism used
for the concept of strategic stability of an equilibrium component.
