Abstract
Finding a Nash equilibrium in a bimatrix game is PPAD-hard (Chen and Deng, 2006 [3], Chen, Deng and Teng, 2009 [6]). The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant, 2005 [1]). However, there do exist polynomial time tractable classes of win-lose bimatrix games - such as, very sparse games (Codenotti, Leoncini and Resta, 2006 [8]) and planar games (Addario-Berry, Olver and Vetta, 2007 [2]).
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 strictly contain planar games. Further, we sharpen the upper bound to unambiguous logspace UL, a small complexity class contained well within polynomial time P. 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 NL, again a small complexity class in P. Our techniques are primarily graph theoretic and use structural characterizations of the considered minor-closed families.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abbott, T.G., Kane, D.M., Valiant, P.: On the complexity of two-player win-lose games. In: Proc. Symp. Found. of Comp. Sc (FOCS), pp. 113–122. IEEE Computer Society, Los Alamitos (2005)
Addario-Berry, L., Olver, N., Vetta, A.: A polynomial time algorithm for finding Nash equilibria in planar win-lose games. J. Graph Algo. Appl. 11(1), 309–319 (2007)
Asano, T.: An approach to the subgraph homeomorphism problem. Theoretical Comp. Sc. 38, 249–267 (1985)
Chen, X., Deng, X.: 3-NASH is PPAD-complete. Electronic Colloquium on Computational Complexity (ECCC)Â (134) (2005)
Chen, X., Deng, X.: Settling the complexity of two-player Nash-equilibrium. In: Proc. Symp. Found. of Comp. Sc. (FOCS), pp. 261–272 (2006)
Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACMÂ 56(3) (2009)
Chen, X., Teng, S.-H., Valiant, P.: The approximation complexity of win-lose games. In: Proc. Symp. Discrete Algo (SODA), pp. 159–168. SIAM, Philadelphia (2007)
Codenotti, B., Leoncini, M., Resta, G.: Efficient computation of Nash equilibria for very sparse win-lose bimatrix games. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 232–243. Springer, Heidelberg (2006)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)
Datta, S., Limaye, N., Nimbhorkar, P., Thierauf, T., Wagner, F.: Planar graph isomorphism is in log-space. In: Proc. Conf. Comput. Compl (CCC), pp. 203–214 (2009)
Datta, S., Nimbhorkar, P., Thierauf, T., Wagner, F.: Graph isomorphism for K 3, 3-free and K 5-free graphs is in log-space. In: Kannan, R., Kumar, N. (eds.) IARCS Annual Conf. on Found. of Software Tech. and Theor. Comp. Sc (FSTTCS 2009). LIPIcs, vol. 4, pp. 145–156. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2009)
Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 2(3), 135–158 (1973)
Khuller, S.: Parallel algorithms for K 5-minor free graphs. Technical Report TR88-909, Cornell Univ., Comp. Sc. Dept. (1988)
Kuratowski, K.: Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)
Thierauf, T., Wagner, F.: Reachability in K 3,3-free graphs and K 5-free graphs is in unambiguous log-space. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds.) FCT 2009. LNCS, vol. 5699, pp. 323–334. Springer, Heidelberg (2009)
Vazirani, V.V.: NC algorithms for computing the number of perfect matchings in K 3,3-free graphs and related problems. Info. and Comput. 80(2), 152–164 (1989)
Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114, 570–590 (1937)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Datta, S., Krishnamurthy, N. (2011). Some Tractable Win-Lose Games. In: Ogihara, M., Tarui, J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20877-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-642-20877-5_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20876-8
Online ISBN: 978-3-642-20877-5
eBook Packages: Computer ScienceComputer Science (R0)