Geodesic
Intransitively winning chess players' positions
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 14 (2022) no. 3, pp. 75-100.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: game theory, positional games, checkers, intransitivity, intransitively winning chess players' positions, perfect values, Euclidian metric, Zermelo-von Neumann theorem.
Mots-clés : chess 
@article{MGTA_2022_14_3_a3,
     author = {Alexander N. Poddiakov},
     title = {Intransitively winning chess players' positions},
     journal = {Matemati\v{c}eska\^a teori\^a igr i e\"e prilo\v{z}eni\^a},
     pages = {75--100},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MGTA_2022_14_3_a3/}
}
TY  - JOUR
AU  - Alexander N. Poddiakov
TI  - Intransitively winning chess players' positions
JO  - Matematičeskaâ teoriâ igr i eë priloženiâ
PY  - 2022
SP  - 75
EP  - 100
VL  - 14
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MGTA_2022_14_3_a3/
LA  - ru
ID  - MGTA_2022_14_3_a3
ER  - 
%0 Journal Article
%A Alexander N. Poddiakov
%T Intransitively winning chess players' positions
%J Matematičeskaâ teoriâ igr i eë priloženiâ
%D 2022
%P 75-100
%V 14
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MGTA_2022_14_3_a3/
%G ru
%F MGTA_2022_14_3_a3
Alexander N. Poddiakov. Intransitively winning chess players' positions. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 14 (2022) no. 3, pp. 75-100. http://geodesic.mathdoc.fr/item/MGTA_2022_14_3_a3/

[1] Bogdanov I.I., “Netranzitivnye ruletki”, Matematicheskoe prosveschenie. Cer. 3, 14, 2010, 240–255

[2] Vasyukova E.E., “Evristiki myshleniya”, Psikhologiya kognitivnykh protsessov, 2020, no. 9, 129–142

[3] Gardner M., Krestiki-noliki, Mir, M., 1988 | MR

[4] Gardner M., Puteshestvie vo vremeni, Mir, M., 1990 | MR

[5] Gorbunova A.V., Lebedev A.V., “Effekty stokhasticheskoi netranzitivnosti v sistemakh massovogo obsluzhivaniya”, Upravlenie bolshimi sistemami, 85 (2020), 23–50 | MR

[6] Ilkov L., “Paradoksy komandnykh sorevnovanii”, Kvant, 2009, no. 1, 44–46

[7] Kasparov G., Chelovek i kompyuter. Vzglyad v buduschee, Alpina Pablisher, M., 2018

[8] Kolchin V.F., “Igr teoriya”, Bolshaya Rossiiskaya entsiklopediya https://bigenc.ru/mathematics/text/1999808

[9] Lebedev A.V., “Problema netranzitivnosti dlya trekh nepreryvnykh sluchainykh velichin”, Avtomatika i telemekhanika, 2019, no. 6, 91–103 | Zbl

[10] Levidov M.Yu., Steinits, Lasker, Russian Chess house, M., 2008

[11] Poddyakov A.N., “Netranzitivnost prevoskhodstva i ee ispolzovanie dlya obmana i trenirovki myshleniya”, Psikhologo-ekonomicheskie issledovaniya, 3:4 (2016), 43–50

[12] Poddyakov A.N., “Paket kompyuternykh igr dlya izucheniya i formirovaniya kombinatornogo logicheskogo myshleniya detei”, Poznanie. Obschestvo. Razvitie, ed. D.V. Ushakov, In-t psikhologii RAN, M., 1996, 126–135

[13] Poddyakov A.N., “Printsip netranzitivnosti prevoskhodstva v raznykh paradigmakh”, Voprosy psikhologii, 2019, no. 2, 3–16

[14] Popov G.L., Netranzitivnost – kladez dlya shakhmatnykh kompozitorov, 2021 http://superproblem.ru/doc/columns/expert/2021/Non-transitivity.pdf

[15] Filatov A.Yu., “Netranzitivnye pozitsii v shakhmatakh”, Nauka i zhizn, 2017, no. 7, 117–120

[16] Sheinerman E., Putevoditel dlya vlyublennykh v matematiku, Alpina non-fikshn, M., 2018

[17] Shteingauz G., Zadachi i razmyshleniya, Mir, M., 1974

[18] Akin E., “Generalized intransitive dice: mimicking an arbitrary tournament”, Journal of dynamics and games, 8:1 (2019), 1–20 | DOI | MR

[19] Atkinson G., Chess and Machine Intuition, IntellectTM, Exeter, 1998

[20] Beardon T., Transitivity http://nrich.maths.org/1345

[21] Bednay D., Bozoki S., “Constructions for nontransitive dice sets”, Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and its Applications (Veszprem, Hungary, June 4-7, 2013), 15–23

[22] Bozoki S., “Nontransitive dice sets realizing the Paley tournaments for solving Schutte's tournament problem”, Miskolc Mathematical Notes, 15:1 (2014), 39–50 | DOI | MR | Zbl

[23] Buhler J., Graham R., Hales A., “Maximally nontransitive dice”, American Mathematical Monthly, 125:5 (2018), 387–399 | DOI | MR | Zbl

[24] Conrey B., Gabbard J., Grant K., Liu A., Morrison K., “Intransitive dice”, Mathematics Magazine, 89:2 (2016), 133–143 | DOI | MR | Zbl

[25] Csakany B., “A form of the Zermelo-von Neumann theorem under minimal assumptions”, Acta Cybernetica, 15 (2002), 321–325 | MR | Zbl

[26] Frequently asked questions - and their answers https://www.figg.org/old_figg/www.figg.org/varie/faq.txt

[27] Fishburn P.C., “Nontransitive preferences in decision theory”, Journal of risk and uncertainty, 1991, no. 4, 113–134 | DOI | Zbl

[28] Gardner M., “The paradox of the nontransitive dice and the elusive principle of indifference”, Scientific American, 223 (1970), 110–114 | DOI | MR

[29] Gardner M., “On the paradoxical situations that arise from nontransitive relations”, Scientific American, 231 (1974), 120–125 | DOI

[30] Grime J., “The bizarre world of nontransitive dice: games for two or more players”, The College Mathematics Journal, 48:1 (2017), 2–9 | DOI | MR | Zbl

[31] Hulko A., Whitmeyer M.A., “Game of nontransitive dice”, Mathematics Magazine, 92:5 (2019), 368–373 | DOI | MR | Zbl

[32] Lake R., Schaeffer J., Lu P., Solving large retrograde analysis problems using a network of workstations, 1993 https://webdocs.cs.ualberta.ca/ ̃ jonathan/publications/ai_publications/databases.pdf

[33] Peterson C., The effect of endgame tablebases on modern chess engines, California Polytechnic State University, San Luis Obispo, 2018

[34] Schaeffer J., Burch N., Bjornsson Y., Kishimoto A., Muller M., Lake R., Luand P., Sutphen S., “Checkers is solved”, Science, 317:5844 (2007), 1518–1522 | DOI | MR | Zbl

[35] Strogatz S. H., Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Boulder, CO, 2015 | MR | Zbl

[36] Sturtevant N. R., “Challenges and progress on using large lossy endgame databases in Chinese Checkers”, Computer Games, CGW 2015, GIGA 2015, Communications in Computer and Information Science, 614, eds. Cazenave T. et al., 2015, 3–15 | DOI

[37] Trybula S., “On the paradox of three random variables”, Applicationes Mathematicae, 5:4 (1961), 321–332 | DOI | MR

[38] Unzueta D., AlphaGo: How AI mastered the game of Go https://towardsdatascience.com/alphago-how-ai-mastered-the-game-of-go-b1355937c98d

[39] Weisstein E.W., Pathological, MathWorld – A Wolfram Web Resource https://mathworld.wolfram.com/Pathological.html

[40] West L.J., Hankin R., “Exact tests for two-way contingency tables with structural zeros”, Journal of statistical software, 28:11 (2008), 1–19 | DOI