We show that if y is an odd integer between 1 and 2 n - 1, there is an n × n bimatrix game with exactly y Nash equilibria (NE). We conjecture that this 2 n - 1 is a tight upper for n < 3, and provide bounds on the number of NEs in m × n nondegenerate games when min( m,n ) < 4.
Quint, Thomas and Shubik, Martin, "On the Number of Nash Equilibria in a Bimatrix Game" (1994). Cowles Foundation Discussion Papers. 1332.