Опубликована статья Александра Поддьякова
В журнале «Doklady Mathematics» опубликована статья Александра Поддьякова «Intransitively winning chess players’ positions».
Abstract
Ch ess play e r s’ p o siti ons in intransiti ve (roc k-p ap e r-s c is sor s) relat ions are considered. Intransit iv-
ity of c h ess playe r s’ p o sit ions means that: po sition A of Whit e i s prefe rable (it sh ould b e c ho sen if c hoice is
p os sible) to po sition B of Blac k, if A and B are on a chessb oard; p o sition B of Blac k is prefe rable t o p os it ion
C of White, if B and C are on the chessb oard; po s it ion C of White is prefe rable t o p o sition D of Blac k, if C
and D are on th e c hes sboard; but p o sition D of Blac k is prefe rable to po sit ion A of White, if A and D are on
the chessb oard. Intrans it ivity of winningness of chess playe r s’ p o s it ions i s conside red t o be a consequenc e of
co mple xity of th e chess en vir onment—in c o ntrast with s imple r gam es with trans it ive po sitions onl y . The
space of re lat ions betwe e n winningn es s of c hes s play er s’ p o sit ions is n on-Euclide an. The Ze rme lo-v on Neu-
mann theore m i s co mple ment ed by s t at e ments about p o ss ibility vs . imp os sibility of building pure winning
strategies bas ed on the as sumpt ion of transiti vity of play er s’ po s it ions. Qu est ions abo ut the p o ss ibility of
intransiti ve pla ye r s’ p o sitions in othe r p o sitional games are rai sed.
Ch ess play e r s’ p o siti ons in intransiti ve (roc k-p ap e r-s c is sor s) relat ions are considered. Intransit iv-
ity of c h ess playe r s’ p o sit ions means that: po sition A of Whit e i s prefe rable (it sh ould b e c ho sen if c hoice is
p os sible) to po sition B of Blac k, if A and B are on a chessb oard; p o sition B of Blac k is prefe rable t o p os it ion
C of White, if B and C are on the chessb oard; po s it ion C of White is prefe rable t o p o sition D of Blac k, if C
and D are on th e c hes sboard; but p o sition D of Blac k is prefe rable to po sit ion A of White, if A and D are on
the chessb oard. Intrans it ivity of winningness of chess playe r s’ p o s it ions i s conside red t o be a consequenc e of
co mple xity of th e chess en vir onment—in c o ntrast with s imple r gam es with trans it ive po sitions onl y . The
space of re lat ions betwe e n winningn es s of c hes s play er s’ p o sit ions is n on-Euclide an. The Ze rme lo-v on Neu-
mann theore m i s co mple ment ed by s t at e ments about p o ss ibility vs . imp os sibility of building pure winning
strategies bas ed on the as sumpt ion of transiti vity of play er s’ po s it ions. Qu est ions abo ut the p o ss ibility of
intransiti ve pla ye r s’ p o sitions in othe r p o sitional games are rai sed.
Ch ess play e r s’ p o siti ons in intransiti ve (roc k-p ap e r-s c is sor s) relat ions are considered. Intransit iv-
ity of c h ess playe r s’ p o sit ions means that: po sition A of Whit e i s prefe rable (it sh ould b e c ho sen if c hoice is
p os sible) to po sition B of Blac k, if A and B are on a chessb oard; p o sition B of Blac k is prefe rable t o p os it ion
C of White, if B and C are on the chessb oard; po s it ion C of White is prefe rable t o p o sition D of Blac k, if C
and D are on th e c hes sboard; but p o sition D of Blac k is prefe rable to po sit ion A of White, if A and D are on
the chessb oard. Intrans it ivity of winningness of chess playe r s’ p o s it ions i s conside red t o be a consequenc e of
co mple xity of th e chess en vir onment—in c o ntrast with s imple r gam es with trans it ive po sitions onl y . The
space of re lat ions betwe e n winningn es s of c hes s play er s’ p o sit ions is n on-Euclide an. The Ze rme lo-v on Neu-
mann theore m i s co mple ment ed by s t at e ments about p o ss ibility vs . imp os sibility of building pure winning
strategies bas ed on the as sumpt ion of transiti vity of play er s’ po s it ions. Qu est ions abo ut the p o ss ibility of
intransiti ve pla ye r s’ p o sitions in othe r p o sitional games are rai sed.
Ch ess play e r s’ p o siti ons in intransiti ve (roc k-p ap e r-s c is sor s) relat ions are considered. Intransit iv-
ity of c h ess playe r s’ p o sit ions means that: po sition A of Whit e i s prefe rable (it sh ould b e c ho sen if c hoice is
p os sible) to po sition B of Blac k, if A and B are on a chessb oard; p o sition B of Blac k is prefe rable t o p os it ion
C of White, if B and C are on the chessb oard; po s it ion C of White is prefe rable t o p o sition D of Blac k, if C
and D are on th e c hes sboard; but p o sition D of Blac k is prefe rable to po sit ion A of White, if A and D are on
the chessb oard. Intrans it ivity of winningness of chess playe r s’ p o s it ions i s conside red t o be a consequenc e of
co mple xity of th e chess en vir onment—in c o ntrast with s imple r gam es with trans it ive po sitions onl y . The
space of re lat ions betwe e n winningn es s of c hes s play er s’ p o sit ions is n on-Euclide an. The Ze rme lo-v on Neu-
mann theore m i s co mple ment ed by s t at e ments about p o ss ibility vs . imp os sibility of building pure winning
strategies bas ed on the as sumpt ion of transiti vity of play er s’ po s it ions. Qu est ions abo ut the p o ss ibility of
intransiti ve pla ye r s’ p o sitions in othe r p o sitional games are rai sed.
Ch ess play e r s’ p o siti ons in intransiti ve (roc k-p ap e r-s c is sor s) relat ions are considered. Intransit iv-
ity of c h ess playe r s’ p o sit ions means that: po sition A of Whit e i s prefe rable (it sh ould b e c ho sen if c hoice is
p os sible) to po sition B of Blac k, if A and B are on a chessb oard; p o sition B of Blac k is prefe rable t o p os it ion
C of White, if B and C are on the chessb oard; po s it ion C of White is prefe rable t o p o sition D of Blac k, if C
and D are on th e c hes sboard; but p o sition D of Blac k is prefe rable to po sit ion A of White, if A and D are on
the chessb oard. Intrans it ivity of winningness of chess playe r s’ p o s it ions i s conside red t o be a consequenc e of
co mple xity of th e chess en vir onment—in c o ntrast with s imple r gam es with trans it ive po sitions onl y . The
space of re lat ions betwe e n winningn es s of c hes s play er s’ p o sit ions is n on-Euclide an. The Ze rme lo-v on Neu-
mann theore m i s co mple ment ed by s t at e ments about p o ss ibility vs . imp os sibility of building pure winning
strategies bas ed on the as sumpt ion of transiti vity of play er s’ po s it ions. Qu est ions abo ut the p o ss ibility of
intransiti ve pla ye r s’ p o sitions in othe r p o sitional games are rai sed.
Chess players’ positions in intransitive (rock-paper-scissors) relations are considered. Intransitivity of chess players’ positions means that: position A of White is preferable (it should be chosen if choice is possible) to position B of Black, if A and B are on a chessboard; position B of Black is preferable to position C of White, if B and C are on the chessboard; position C of White is preferable to position D of Black, if C and D are on the chessboard; but position D of Black is preferable to position A of White, if A and D are on the chessboard. 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. The space of relations between winningness of chess players’ positions 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 players’ positions. Questions about the possibility of intransitive players’ positions in other positional games are raised.Poddiakov, A. (2024). Intransitively winning chess players’ positions. Doklady Mathematics, 110 (Suppl 2), S391–S398. https://doi.org/10.1134/S1064562424702417
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