Some Tractable Win-Lose Games

Determining a Nash equilibrium in a $2$-player non-zero sum game is known to be PPAD-hard (Chen and Deng (2006), Chen, Deng and Teng (2009)). The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant (2005)). However, there do exist polynomial time tractable classes of win-lose bimatrix games - such as, very sparse games (Codenotti, Leoncini and Resta (2006)) and planar games (Addario-Berry, Olver and Vetta (2007)). We extend the results in the latter work to $K_{3,3}$ minor-free games and a subclass of $K_5$ minor-free games. Both these classes of games strictly contain planar games. Further, we sharpen the upper bound to unambiguous logspace, a small complexity class contained well within polynomial time. Apart from these classes of games, our results also extend to a class of games that contain both $K_{3,3}$ and $K_5$ as minors, thereby covering a large and non-trivial class of win-lose bimatrix games. For this class, we prove an upper bound of nondeterministic logspace, again a small complexity class within polynomial time. Our techniques are primarily graph theoretic and use structural characterizations of the considered minor-closed families.Comment: We have fixed an error in the proof of Lemma 4.5. The proof is in Section 4.1 on "Stitching cycles together", pages 6-7. We have reworded the statement of Lemma 4.5 as well (on page 6

The Complexity of Computational Problems about Nash Equilibria in Symmetric Win-Lose Games

We revisit the complexity of deciding, given a {\it bimatrix game,} whether it has a {\it Nash equilibrium} with certain natural properties; such decision problems were early known to be ${\mathcal{NP}}$-hard~\cite{GZ89}. We show that ${\mathcal{NP}}$-hardness still holds under two significant restrictions in simultaneity: the game is {\it win-lose} (that is, all {\it utilities} are $0$ or $1$) and {\it symmetric}. To address the former restriction, we design win-lose {\it gadgets} and a win-lose reduction; to accomodate the latter restriction, we employ and analyze the classical {\it ${\mathsf{GHR}}$-symmetrization}~\cite{GHR63} in the win-lose setting. Thus, {\it symmetric win-lose bimatrix games} are as complex as general bimatrix games with respect to such decision problems. As a byproduct of our techniques, we derive hardness results for search, counting and parity problems about Nash equilibria in symmetric win-lose bimatrix games