Eleme Turnuvalarında Seribaşı Eşlemelerinin Sayılması ve Sayımlanması

Yıl 2021, Cilt: 14 Sayı: 2, 137 - 145, 22.12.2021

Can Atılgan , Mehmet Emin Dalkılıç

https://doi.org/10.54525/tbbmd.1023558

Öz

Eleme turnuvaları, yarışma tasarımlarının muhtemelen en yaygın olarak kullanılanıdır. Bu turnuvalarda, her turda tüm yarışmacılar bir eleme karşılaşması oynamak üzere eşlenirler ve kaybeden turnuva dışında kalır. Seribaşı eşlemesi olarak adlandırılan ilk tur maçlarının seçimi, turnuvanın geri kalanını da doğrudan etkiler. Seribaşı eşlemelerini çeşitli açılardan değerlendiren ve inceleyen, dikkate değer sayıda çalışma olmasına karşın, bildiğimiz kadarıyla seribaşı eşlemelerini sayımlayan bir algoritma sunulmamıştır. Ayrıca, literatürde farklı seribaşı eşlemelerinin sayısının bulunması da yalnızca kısaca tartışılmıştır. Bu makalede farklı seribaşı eşlemelerinin nasıl sayılacağı tartışılmış ve eleme turnuvalarında seribaşı eşlemelerinin sayımlanması için etkin bir algoritma önerilmiştir.

Anahtar Kelimeler

Hesaplamalı Turnuva Tasarımı, Eleme Turnuvası, Seribaşı Eşleme, Sayma, Sayımlama

Counting and Enumerating Seedings in Knockout Tournaments

Öz

Knockout tournaments are probably the most common of all competition designs. In those tournaments, all competitors are paired to play an elimination match at each round, where the loser is immediately knocked out of the tournament. The setting of the first round of matches, named as seeding, directly influences the rest of the tournament. While there is a considerable amount of work that evaluate and examine different seedings from various perspectives, an algorithm that enumerates seedings has not been given to the best of our knowledge. Also, counting the number of unique seedings has been only briefly discussed in the literature. This paper discusses how to count the number of unique seedings and proposes an efficient algorithm for enumerating seedings in a knockout tournament.

Anahtar Kelimeler

Computational Tournament Design, Knockout Tournament, Seeding, Counting, Enumeration

Kaynakça

• F. K. Hwang, New concepts in seeding knockout tournaments, The American Mathematical Monthly, vol. 89, no. 4, pp. 235–239, 1982.
• J. Horen, R. Riezman, 1985, Comparing draws for single elimination tournaments, Operations Research, vol. 33, no. 2, pp. 249–262, 1985.
• D. R. Appleton, May the best man win?, The Statistician, vol. 44, no. 4, pp. 529–538, 1995.
• A. J. Schwenk, What is the correct way to seed a knockout tournament?, The American Mathematical Monthly, vol. 107, no. 2, pp. 140–150, 2000.
• J. Phillips, S. B. Caudill, F. G. Mixon Jr., Tournament Seeding Efficiency and Home Court Advantage: College Basketball’s National Invitation Tournament, International Journal of Statistics and Probability, vol. 4, no. 3, 2015.
• D. Dagaev, R. Vladimir, Seeding the UEFA Champions League Participants: Evaluation of the Reform, Higher School of Economics Research Paper No. WP BRP 129/EC/2016, 2016.
• L. Csató, The UEFA Champions League seeding is not strategy-proof since the 2015/16 season, Annals of Operations Research vol. 292, pp. 161–169, 2020.
• L. Csató, Optimal Tournament Design: Lessons From the Men’s Handball Champions League, Journal of Sports Economics, doi: 10.1177/1527002520944442, 2020.
• F. Della Groce, G. Dragotto, R. Scatamacchia, On fairness and diversification in WTA and ATP tennis tournaments generation, Annals of Operations Research, doi: 10.1007/s10479-020-03517-8, 2020.
• T. Vu, Y. Shoham, “Fair seeding in knockout tournaments”, vol. 3, no. 1, Article no. 9, 2011.
• C. Groh, B. Moldovanu, A. Sela, U. Sunde, Optimal seedings in elimination tournaments, Economic Theory, vol. 49, pp. 59–80, 2012.
• M. Kräkel, Optimal seedings in elimination tournaments revisited, Economic Theory Bulletin, vol. 2, no. 1, pp. 77–91, 2014.
• A. Karpov, A new knockout tournament seeding method and its axiomatic justification, Operations Research Letters, vol. 44, no. 6, pp. 706–711, 2016.
• D. Dagaev, A. Suzdaltsev, Competitive intensity and quality maximizing seedings in knock-out tournaments, Journal of Combinatorial Optimization, 35, pp. 170–188, 2018.
• H. Aziz, S. Gaspers, S. Mackenzie, N. Mattei, P. Stursberg, T. Walsh, Fixing balanced knockout and double elimination tournaments, Artificial Intelligence, vol. 262, pp. 1–14, 2018.
• R. Arlegi, D. Dimitrov, Fair elimination-type competitions, European Journal of Operational Research, vol. 287, no. 2, pp. 528–535, 2020.
• R. Arlegi, How can an elimination tournament favor a weaker player?, International Transactions in Operational Research, doi: 10.1111/itor.12955, 2021.
• N. J. A. Sloane, Sequence: A001147, The on-line encyclopedia of integer sequences, http://oeis.org, son erişim: 09.11.2021.
• C. Atilgan, M. E. Dalkilic, Enumerating Possible Seedings in a Knockout Tournament, 2020 5th International Conference on Computer Science and Engineering (UBMK), Diyarbakir, Turkey, Sept. 2020, doi: 10.1109/UBMK50275.2020.9219510.