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T HE C OMPLEXITY OF P LAYING D URAK É DOUARD B ONNET I NTRODUCTION The complexity of board games has been quite extensively studied since the dawn of complexity theory. However, despite some works on Whist and Bridge [5, 2], Uno [3], and algorithmic questions related to Set [4] and Hanabi [1], there is few known about the computational complexity of card games. It is still a vastly unexplored line of re- search with connections to combinatorial game theory and parameterized complex- ity. We tackle the russian game Durak whose gameplay is significantly different from the card games mentioned above. R ULES OF D URAK Simplified rules with 2 players, no trump suit, and an empty drawing pile: • goal: getting rid of all one’s cards. • round: sequence of moves between the attacker and her opponent the defender. • attacker’s moves: play a card matching the rank of a played card. • defender’s moves: defend with a higher card in the same suit. • if the defender ceases to defend, he takes all the cards in his hand; he re- mains the defender for the next round. • otherwise, all the cards are discarded; defender/attacker switch roles. O N DEFENDING AN ATTACK 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s x 1 s x 2 s x 3 s x 4 s x 5 s x 6 s C 1 s C 2 s C 3 s C 4 s C 5 s C 6 s C 7 s C 8 W EAKNESS AND STRONG SUIT • weakness: rank where one player has only cards dominated by cards of her/his opponent. • well-covered weakness: weakness such that all the dominating cards are of rank not owned by the player with the weakness. • strong suit: only owned by one player. O N FINDING OPTIMAL PLAY ∀x 0 ∃x 1 ∀x 2 ∃x 3 ∀x 4 ... ... ... ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 s 1 0 s 2 0 s 3 0 s 1 2 s 2 2 s 3 2 s 1 4 s 2 4 s 3 4 s 1 1 s 2 1 s 3 1 s 4 1 s 1 3 s 2 3 s 3 3 s 4 3 s C 0 1 s C 0 2 s C 0 3 s C 0 4 s C 00 1 s C 00 2 s C 00 3 s C 00 4 s 1 w ... s 2m w s O s P s 1,1 d ...... s 2 d s 3 d s 4 d O PEN QUESTIONS • Is single-suit Durak tractable? • What about Durak with a bounded number of suits? • In Durak with more than two players, games are not polynomially bounded. Is deciding if one player has a winning strategy still in PSPACE? • Is deciding if one player can defend an attack in NP? • What is the complexity of Durak with a bounded number of ranks? • Removing the threshold feature in the PSPACE-hardness construction. R EFERENCES [1] J. Baffier, M. Chiu, Y. Diez, M. Korman, V. Mitsou, A. van Renssen, M. Roeloffzen, and Y. Uno. Hanabi is np-complete, even for cheaters who look at their cards. In 8th International Conference on Fun with Algorithms, FUN 2016, June 8-10, 2016, La Maddalena, Italy, pages 4:1–4:17, 2016. [2] E. Bonnet, F. Jamain, and A. Saffidine. On the complexity of trick-taking card games. In IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9, 2013, 2013. [3] E. D. Demaine, M. L. Demaine, N. J. A. Harvey, R. Uehara, T. Uno, and Y. Uno. UNO is hard, even for a single player. Theor. Comput. Sci., 521:51–61, 2014. [4] M. Lampis and V. Mitsou. The computational complexity of the game of set and its theoretical applications. In LATIN 2014: Theoretical Informatics - 11th Latin American Symposium, Montevideo, Uruguay, March 31 - April 4, 2014. Proceedings, pages 24–34, 2014. [5] J. Wästlund. A solution of two-person single-suit whist. The Electronic Journal of Combinatorics, 12(1):R43, 2005. R ESULTS • Deciding if one player can defend until the end is NP-hard. • Deciding if one player has a winning strategy is PSPACE-complete. Question for you: why is the game length polynomially bounded?
