Intransitively winning chess players' positions

Abstract

Chess players' positions 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 chess players' positions is considered to be a consequence of complexity of the chess environment - in contrast with simpler games with transitive positions only. Perfect values of chess players' positions are impossible. Euclidian metric cannot be used to describe chess players' positions in space of winningness relations. 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 players' positions. Questions about the possibility of intransitive players' positions in other positional games are raised.