Computing Maxmin Strategies in Extensive-form Zero-sum Games with Imperfect Recall

Branislav Bosansky, Jiri Cermak, Karel Horak, Michal Pechoucek

Abstract

Extensive-form games with imperfect recall are an important game-theoretic model that allows a compact representation of strategies in dynamic strategic interactions. Practical use of imperfect recall games is limited due to negative theoretical results: a Nash equilibrium does not have to exist, computing maxmin strategies is NP-hard, and they may require irrational numbers. We present the first algorithm for approximating maxmin strategies in two-player zero-sum imperfect recall games without absentmindedness. We modify the well-known sequence-form linear program to model strategies in imperfect recall games resulting in a bilinear program and use a recent technique to approximate the bilinear terms. Our main algorithm is a branch-and-bound search that provably reaches the desired approximation after an exponential number of steps in the size of the game. Experimental evaluation shows that the proposed algorithm can approximate maxmin strategies of randomly generated imperfect recall games of sizes beyond toy-problems within few minutes.

References

  1. Bos?anskÉ, B., Kiekintveld, C., LisÉ, V., and Pe?chouc?ek, M. (2014). An Exact Double-Oracle Algorithm for Zero-Sum Extensive-Form Games with Imperfect Information. Journal of Artificial Intelligence Research , 51:829-866.
  2. Bowling, M., Burch, N., Johanson, M., and Tammelin, O. (2015). Heads-up limit hold'em poker is solved. Science, 347(6218):145-149.
  3. Gilpin, A. and Sandholm, T. (2007). Lossless Abstraction of Imperfect Information Games. Journal of the ACM, 54(5).
  4. Hoda, S., Gilpin, A., Pen˜a, J., and Sandholm, T. (2010). Smoothing Techniques for Computing Nash Equilibria of Sequential Games. Mathematics of Operations Research, 35(2):494-512.
  5. Kaneko, M. and Kline, J. J. (1995). Behavior Strategies, Mixed Strategies and Perfect Recall. International Journal of Game Theory, 24:127-145.
  6. Kline, J. J. (2002). Minimum Memory for Equivalence between Ex Ante Optimality and Time-Consistency. Games and Economic Behavior, 38:278-305.
  7. Koller, D. and Megiddo, N. (1992). The Complexity of Two-Person Zero-Sum Games in Extensive Form. Games and Economic Behavior, 4:528-552.
  8. Koller, D., Megiddo, N., and von Stengel, B. (1996). Efficient Computation of Equilibria for Extensive TwoPerson Games. Games and Economic Behavior, 14(2):247-259.
  9. Koller, D. and Milch, B. (2003). Multi-agent influence diagrams for representing and solving games. Games and Economic Behavior, 45(1):181-221.
  10. Kolodziej, S., Castro, P. M., and Grossmann, I. E. (2013). Global optimization of bilinear programs with a multiparametric disaggregation technique. Journal of Global Optimization, 57(4):1039-1063.
  11. Kroer, C. and Sandholm, T. (2014). Extensive-Form Game Abstraction with Bounds. In ACM conference on Economics and computation.
  12. Kroer, C. and Sandholm, T. (2016). Imperfect-Recall Abstractions with Bounds in Games. In EC.
  13. Kuhn, H. W. (1953). Extensive Games and the Problem of Information. Contributions to the Theory of Games, II:193-216.
  14. Lanctot, M., Gibson, R., Burch, N., Zinkevich, M., and Bowling, M. (2012). No-Regret Learning in Extensive-Form Games with Imperfect Recall. In ICML.
  15. Nash, J. F. (1950). Equilibrium Points in n-person Games. Proc. Nat. Acad. Sci. USA, 36(1):48-49.
  16. Piccione, M. and Rubinstein, A. (1997). On the Interpretation of Decision Problems with Imperfect Recall. Games and Economic Behavior, 20:3-24.
  17. von Stengel, B. (1996). Efficient Computation of Behavior Strategies. Games and Economic Behavior, 14:220- 246.
  18. Wichardt, P. C. (2008). Existence of nash equilibria in finite extensive form games with imperfect recall: A counterexample. Games and Economic Behavior, 63(1):366-369.
  19. Zinkevich, M., Johanson, M., Bowling, M., and Piccione, C. (2008). Regret Minimization in Games with Incomplete Information. In NIPS.
Download


Paper Citation


in Harvard Style

Bosansky B., Cermak J., Horak K. and Pechoucek M. (2017). Computing Maxmin Strategies in Extensive-form Zero-sum Games with Imperfect Recall . In Proceedings of the 9th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART, ISBN 978-989-758-220-2, pages 63-74. DOI: 10.5220/0006121200630074


in Bibtex Style

@conference{icaart17,
author={Branislav Bosansky and Jiri Cermak and Karel Horak and Michal Pechoucek},
title={Computing Maxmin Strategies in Extensive-form Zero-sum Games with Imperfect Recall},
booktitle={Proceedings of the 9th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART,},
year={2017},
pages={63-74},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006121200630074},
isbn={978-989-758-220-2},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 9th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART,
TI - Computing Maxmin Strategies in Extensive-form Zero-sum Games with Imperfect Recall
SN - 978-989-758-220-2
AU - Bosansky B.
AU - Cermak J.
AU - Horak K.
AU - Pechoucek M.
PY - 2017
SP - 63
EP - 74
DO - 10.5220/0006121200630074