Today I read a paper titled “Games of fixed rank: A hierarchy of bimatrix games”
The abstract is:
We propose a new hierarchical approach to understand the complexity of the open problem of computing a Nash equilibrium in a bimatrix game.
Specifically, we investigate a hierarchy of bimatrix games $(A,B)$ which results from restricting the rank of the matrix $A+B$ to be of fixed rank at most $k$.
For every fixed $k$, this class strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games.
We show that even for $k=1$ the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components.
While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we can provide polynomial time algorithms for finding an $\epsilon$-approximation.