Playing odds and evens with finite automata

05/09/2020

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by   Vladislav Makarov, et al.

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This paper is concerned with asymptotic behaviour of a repeated game of "odds and evens", with strategies of both players represented by finite automata. It is proved that, for every n, there is an automaton with 2^n ·poly(n) states which defeats every n-state automaton, in the sense that it wins all rounds except for finitely many. Moreover, every such automaton has at least 2^n · (1 - o(1)) states, meaning that the upper bound is tight up to polynomial factors. This is a significant improvement over a classic result of Ben-Porath in the special case of "odds and evens". Moreover, I conjecture that the approach can be generalised to arbitrary zero-sum games.