Games of fixed rank: A hierarchy of bimatrix games
<|reference_start|>Games of fixed rank: A hierarchy of bimatrix games: 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.<|reference_end|>
@article{kannan2005games,
    title={Games of fixed rank: A hierarchy of bimatrix games},
    author={Ravi Kannan and Thorsten Theobald},
    journal={arXiv preprint arXiv:cs/0511021},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0511021},
    primaryClass={cs.GT math.CO}
}