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Hannan consistency, or no external regret, is a~key concept for learning ingames. An action selection algorithm is Hannan consistent (HC) if itsperformance is eventually as good as selecting the~best fixed action inhindsight. If both players in a~zero-sum normal form game use a~Hannanconsistent algorithm, their average behavior converges to a~Nash equilibrium(NE) of the~game. A similar result is known about extensive form games, butthe~played strategies need to be Hannan consistent with respect tothe~counterfactual values, which are often difficult to obtain. We studyzero-sum extensive form games with simultaneous moves, but otherwise perfectinformation. These games generalize normal form games and they are a specialcase of extensive form games. We study whether applying HC algorithms in eachdecision point of these games directly to the~observed payoffs leads toconvergence to a~Nash equilibrium. This learning process corresponds to a~classof Monte Carlo Tree Search algorithms, which are popular for playingsimultaneous-move games but do not have any known performance guarantees. Weshow that using HC algorithms directly on the~observed payoffs is notsufficient to guarantee the~convergence. With an~additional averaging overjoint actions, the~convergence is guaranteed, but empirically slower. Wefurther define an~additional property of HC algorithms, which is sufficient toguarantee the~convergence without the~averaging and we empirically show thatcommonly used HC algorithms have this property.

Analysis of Hannan Consistent Selection for Monte Carlo Tree Search in Simultaneous Move Games