The winning strategy for Hironaka's polyhedra game is almost arbitrary (Q538063)

\textit{H. Hironaka} [in: F. Oort (ed.), Algebraic geometry, Oslo 1970. Proceedings of the 5th nordic summer-school in mathematics. Groningen: Wolters-Noordhoff Publishing, 291--313 (1972; Zbl 0248.14002)] introduced a two-player game involving Newton polyhedra; winning strategies for the first player are related to resolution of singularities. A modified version of the game has been presented by \textit{H. Hauser} [Bull. Am. Math. Soc., New Ser. 40, No. 3, 323--403 (2003; Zbl 1030.14007)]. Winning strategies for the first player have been given by \textit{M. Spivakovsky} [Prog. Math. 36, 419--432 (1983; Zbl 0531.14009)] and \textit{D. Zeillinger} [Enseign. Math. (2) 52, No.1--2, 143--158 (2006; Zbl 1245.14052)], respectively. In the present paper, the authors give a new winning strategy for the first player in Hauser's game. They furthermore show that if \(S\) is any sequence of moves generated by this strategy, then any sequence of moves containing \(S\) as a subsequence will result in victory for the first player. A similar result is also shown for Hironaka's game.