# 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.

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#### 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