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Extensive-form game abstraction with bounds

Abstraction has emerged as a key component in solving extensive-form games of incomplete information. However, lossless abstractions are typically too large to solve, so lossy abstraction is needed. All prior lossy abstraction algorithms for extensive-form games either 1) had no bounds on solution quality or 2) depended on specific equilibrium computation approaches, limited forms of abstraction, and only decreased the number of information sets rather than nodes in the game tree. We introduce a theoretical framework that can be used to give bounds on solution quality for any perfect-recall extensive-form game. The framework uses a new notion for mapping abstract strategies to the original game, and it leverages a new equilibrium refinement for analysis. Using this framework, we develop the first general lossy extensive-form game abstraction method with bounds. Experiments show that it finds a lossless abstraction when one is available and lossy abstractions when smaller abstractions are desired. While our framework can be used for lossy abstraction, it is also a powerful tool for lossless abstraction if we set the bound to zero. Prior abstraction algorithms typically operate level by level in the game tree. We introduce the extensive-form game tree isomorphism and action subset selection problems, both important problems for computing abstractions on a level-by-level basis. We show that the former is graph isomorphism complete, and the latter NP-complete. We also prove that level-by-level abstraction can be too myopic and thus fail to find even obvious lossless abstractions.