Abstract
Monte-Carlo Tree Search (MCTS) has been found to play suboptimally in some tactical domains due to its highly selective search, focusing only on the most promising moves. In order to combine the strategic strength of MCTS and the tactical strength of minimax, MCTS-minimax hybrids have been introduced, embedding shallow minimax searches into the MCTS framework. Their results have been promising even without making use of domain knowledge such as heuristic evaluation functions. This paper continues this line of research for the case where evaluation functions are available. Three different approaches are considered, employing minimax with an evaluation function in the rollout phase of MCTS, as a replacement for the rollout phase, and as a node prior to bias move selection. The latter two approaches are newly proposed. The MCTS-minimax hybrids are tested and compared to their counterparts using evaluation functions without minimax in the domains of Othello, Breakthrough, and Catch the Lion. Results showed that introducing minimax search is effective for heuristic node priors in Othello and Catch the Lion. The MCTS-minimax hybrids are also found to work well in combination with each other. For their basic implementation in this investigative study, the effective branching factor of a domain is identified as a limiting factor of the hybrid’s performance.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Auer, P., Cesa-Bianchi, N., Fischer, P.: Finite-Time Analysis of the Multiarmed Bandit Problem. Machine Learning 47(2-3), 235–256 (2002)
Baier, H., Winands, M.H.M.: Monte-Carlo Tree Search and Minimax Hybrids. In: 2013 IEEE Conference on Computational Intelligence and Games, CIG 2013, pp. 129–136 (2013)
Bouzy, B.: Associating Domain-Dependent Knowledge and Monte Carlo Approaches within a Go Program. Information Sciences 175(4), 247–257 (2005)
Browne, C., Powley, E.J., Whitehouse, D., Lucas, S.M., Cowling, P.I., Rohlfshagen, P., Tavener, S., Perez, D., Samothrakis, S., Colton, S.: A Survey of Monte Carlo Tree Search Methods. IEEE Transactions on Computational Intelligence and AI in Games 4(1), 1–43 (2012)
Chaslot, G.M.J.B., Winands, M.H.M., van den Herik, H.J., Uiterwijk, J.W.H.M., Bouzy, B.: Progressive Strategies for Monte-Carlo Tree Search. New Mathematics and Natural Computation 4(3), 343–357 (2008)
Clune, J.E.: Heuristic Evaluation Functions for General Game Playing. Ph.D. thesis, University of California, Los Angeles, USA (2008)
Coulom, R.: Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M(J.) (eds.) CG 2006. LNCS, vol. 4630, pp. 72–83. Springer, Heidelberg (2007)
Finnsson, H., Björnsson, Y.: Game-Tree Properties and MCTS Performance. In: IJCAI 2011 Workshop on General Intelligence in Game Playing Agents (GIGA 2011), pp. 23–30 (2011)
Gelly, S., Silver, D.: Combining Online and Offline Knowledge in UCT. In: Ghahramani, Z. (ed.) 24th International Conference on Machine Learning, ICML 2007. ACM International Conference Proceeding Series, vol. 227, pp. 273–280 (2007)
Gelly, S., Wang, Y., Munos, R., Teytaud, O.: Modification of UCT with Patterns in Monte-Carlo Go. Tech. rep., HAL - CCSd - CNRS, France (2006)
van den Herik, H.J., Xu, X., Ma, Z., Winands, M.H.M. (eds.): CG 2008. LNCS, vol. 5131. Springer, Heidelberg (2008)
Knuth, D.E., Moore, R.W.: An Analysis of Alpha-Beta Pruning. Artificial Intelligence 6(4), 293–326 (1975)
Kocsis, L., Szepesvári, C.: Bandit Based Monte-Carlo Planning. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) ECML 2006. LNCS (LNAI), vol. 4212, pp. 282–293. Springer, Heidelberg (2006)
Lanctot, M., Winands, M.H.M., Pepels, T., Sturtevant, N.R.: Monte Carlo Tree Search with Heuristic Evaluations using Implicit Minimax Backups. In: 2014 IEEE Conference on Computational Intelligence and Games, CIG 2014, pp. 341–348 (2014)
Lorentz, R.J.: Amazons Discover Monte-Carlo. In: van den Herik (ed.) [11], pp. 13–24
Lorentz, R., Horey, T.: Programming breakthrough. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2013. LNCS, vol. 8427, pp. 49–59. Springer, Heidelberg (2014)
Lorentz, R.J.: Experiments with Monte-Carlo Tree Search in the Game of Havannah. ICGA Journal 34(3), 140–149 (2011)
Nijssen, J(P.) A.M., Winands, M.H.M.: Playout Search for Monte-Carlo Tree Search in Multi-player Games. In: van den Herik, H.J., Plaat, A. (eds.) ACG 2011. LNCS, vol. 7168, pp. 72–83. Springer, Heidelberg (2012)
Ramanujan, R., Sabharwal, A., Selman, B.: On Adversarial Search Spaces and Sampling-Based Planning. In: Brafman, R.I., Geffner, H., Hoffmann, J., Kautz, H.A. (eds.) 20th International Conference on Automated Planning and Scheduling, ICAPS 2010, pp. 242–245. AAAI (2010)
Ramanujan, R., Sabharwal, A., Selman, B.: Understanding Sampling Style Adversarial Search Methods. In: Grünwald, P., Spirtes, P. (eds.) 26th Conference on Uncertainty in Artificial Intelligence, UAI 2010, pp. 474–483 (2010)
Ramanujan, R., Sabharwal, A., Selman, B.: On the Behavior of UCT in Synthetic Search Spaces. In: ICAPS 2011 Workshop on Monte-Carlo Tree Search: Theory and Applications (2011)
Ramanujan, R., Selman, B.: Trade-Offs in Sampling-Based Adversarial Planning. In: Bacchus, F., Domshlak, C., Edelkamp, S., Helmert, M. (eds.) 21st International Conference on Automated Planning and Scheduling, ICAPS 2011. AAAI (2011)
Sato, Y., Takahashi, D., Grimbergen, R.: A Shogi Program Based on Monte-Carlo Tree Search. ICGA Journal 33(2), 80–92 (2010)
Silver, D., Tesauro, G.: Monte-Carlo Simulation Balancing. In: Danyluk, A.P., Bottou, L., Littman, M.L. (eds.) 26th Annual International Conference on Machine Learning, ICML 2009. ACM International Conference Proceeding Series, vol. 382, pp. 945–952. ACM (2009)
Sturtevant, N.R.: An Analysis of UCT in Multi-Player Games. ICGA Journal 31(4), 195–208 (2008)
Winands, M.H.M., Björnsson, Y., Saito, J.-T.: Monte Carlo Tree Search in Lines of Action. IEEE Transactions on Computational Intelligence and AI in Games 2(4), 239–250 (2010)
Winands, M.H.M., Björnsson, Y.: Alpha-Beta-based Play-outs in Monte-Carlo Tree Search. In: Cho, S.-B., Lucas, S.M., Hingston, P. (eds.) 2011 IEEE Conference on Computational Intelligence and Games, CIG 2011, pp. 110–117. IEEE (2011)
Winands, M.H.M., Björnsson, Y., Saito, J.T.: Monte-Carlo Tree Search Solver. In: van den Herik, et al. (eds.) [11], pp. 25–36
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Baier, H., Winands, M.H.M. (2014). Monte-Carlo Tree Search and Minimax Hybrids with Heuristic Evaluation Functions. In: Cazenave, T., Winands, M.H.M., Björnsson, Y. (eds) Computer Games. CGW 2014. Communications in Computer and Information Science, vol 504. Springer, Cham. https://doi.org/10.1007/978-3-319-14923-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-14923-3_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14922-6
Online ISBN: 978-3-319-14923-3
eBook Packages: Computer ScienceComputer Science (R0)