  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 $2^n \cdot
\mathrm{poly}(n)$ 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 $2^n \cdot (1 - o(1))$ 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.
