  Positions of chess players in intransitive (rock-paper-scissors) relations
are considered. Namely, position A of White is preferable (it should be chosen
if choice is possible) to position B of Black, position B of Black is
preferable to position C of White, position C of White is preferable to
position D of Black, but position D of Black is preferable to position A of
White. Intransitivity of winningness of positions of chess players is
considered to be a consequence of complexity of the chess environment -- in
contrast with simpler games with transitive positions only. The space of
relations between winningness of positions of chess players is non-Euclidean.
The Zermelo-von Neumann theorem is complemented by statements about possibility
vs. impossibility of building pure winning strategies based on the assumption
of transitivity of positions of chess players. Questions about the possibility
of intransitive positions of players in other positional games are raised.
