The efficacy of tournament designs. (English) Zbl 1520.90168
Summary: Tournaments are a widely used mechanism to rank alternatives in a noisy environment. This paper investigates a fundamental issue of economics in tournament design: what is the best usage of limited resources, that is, how should the alternatives be compared pairwise to best approximate their true but latent ranking. We consider various formats including knockout tournaments, multi-stage championships consisting of round-robin groups followed by single elimination, and the Swiss-system. They are evaluated via Monte-Carlo simulations under six different assumptions on winning probabilities. Comparing the same pair of alternatives multiple times turns out to be an inefficacious policy. While seeding can increase the efficacy of the knockout and group-based designs, its influence remains marginal unless one has an unrealistically good estimation on the true ranking of the players. The Swiss-system is found to be the most accurate among all these tournament formats, especially in its ability to rank all participants. A possible explanation is that it does not eliminate a player after a single loss, while it takes the history of the comparisons into account. The results can be especially interesting for emerging esports, where the tournament designs are not yet solidified.
MSC:
| 90B99 | Operations research and management science |
| 05C20 | Directed graphs (digraphs), tournaments |
| 91A20 | Multistage and repeated games |
| 91B26 | Auctions, bargaining, bidding and selling, and other market models |
References:
| [1] | Adler, I.; Cao, Y.; Karp, R.; Peköz, E. A.; Ross, S. M., Random knockout tournaments, Oper. Res., 65, 6, 1589-1596 (2017) · Zbl 1405.91084 |
| [2] | Ales, L.; Cho, S.-H.; Körpeoğlu, E., Optimal award scheme in innovation tournaments, Oper. Res., 65, 3, 693-702 (2017) · Zbl 1407.91123 |
| [3] | Annis, D. H.; Wu, S. S., A comparison of potential playoff systems for NCAA IA football, Amer. Statist., 60, 2, 151-157 (2006) |
| [4] | Appleton, D. R., May the best man win?, J. R. Statist. Soc.: Ser. D (Statist.), 44, 4, 529-538 (1995) |
| [5] | Arlegi, R., How can an elimination tournament favor a weaker player?, Int. Trans. Oper. Res. (2021), (in press) |
| [6] | Arlegi, R.; Dimitrov, D., Fair elimination-type competitions, European J. Oper. Res., 287, 2, 528-535 (2020) · Zbl 1487.90469 |
| [7] | Atan, T.; Çavdaroğlu, B., Minimization of rest mismatches in round robin tournaments, Comput. Oper. Res., 99, 78-89 (2018) · Zbl 1458.90251 |
| [8] | Bergantiños, G.; Moreno-Ternero, J. D., Sharing the revenues from broadcasting sport events, Manage. Sci., 66, 6, 2417-2431 (2020) |
| [9] | Bimpikis, K.; Ehsani, S.; Mostagir, M., Designing dynamic contests, Oper. Res., 67, 2, 339-356 (2019) · Zbl 1455.91143 |
| [10] | Biró, P.; Fleiner, T.; Palincza, R. P., Designing chess pairing mechanisms, (Frank, A.; Recski, A.; Wiener, G., Proceedings of the 10th Japanese-Hungarian Symposium on Discrete Mathematics and its Applications (2017), Budapesti Műszaki s Gazdaságtudományi Egyetem: Budapesti Műszaki s Gazdaságtudományi Egyetem Budapest), 77-86 |
| [11] | Hearthstone tournament player handbook v2.5 (2020), 15 October. https://assets.blz-contentstack.com/v3/assets/bltc965041283bac56c/blt7180e70eed2edb7c/5f88cab321bd3d0cf67e06d6/Hearthstone_Tournament_Player_Handbook_v2.5.pdf |
| [12] | Brown, J.; Minor, D. B., Selecting the best? Spillover and shadows in elimination tournaments, Manage. Sci., 60, 12, 3087-3102 (2014) |
| [13] | Can, B., Weighted distances between preferences, J. Math. Econom., 51, 109-115 (2014) · Zbl 1296.91093 |
| [14] | Cea, S.; Durán, G.; Guajardo, M.; Sauré, D.; Siebert, J.; Zamorano, G., An analytics approach to the FIFA ranking procedure and the world cup final draw, Ann. Oper. Res., 286, 1-2, 119-146 (2020) · Zbl 1493.62633 |
| [15] | Chater, M.; Arrondel, L.; Gayant, J.-P.; Laslier, J.-F., Fixing match-fixing: Optimal schedules to promote competitiveness, European J. Oper. Res., 294, 2, 673-683 (2021) |
| [16] | Chen, R.; Hwang, F. K., Stronger players win more balanced knockout tournaments, Graphs Combin., 4, 1, 95-99 (1988) · Zbl 0666.62069 |
| [17] | Chen, S., Joachims, T., 2016. Predicting matchups and preferences in context. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 775-784. |
| [18] | Regulations for tournaments 2021 (2021), https://www.croquet.org.uk/?p=tournament/regulations |
| [19] | Csató, L., Ranking by pairwise comparisons for swiss-system tournaments, CEJOR Cent. Eur. J. Oper. Res., 21, 4, 783-803 (2013) · Zbl 1339.91033 |
| [20] | Csató, L., On the ranking of a Swiss system chess team tournament, Ann. Oper. Res., 254, 1-2, 17-36 (2017) · Zbl 1406.91024 |
| [21] | Csató, L., The incentive (in)compatibility of group-based qualification systems, Int. J. Gen. Syst., 49, 4, 374-399 (2020) |
| [22] | Csató, L., (Tournament Design: How Operations Research Can Improve Sports Rules. Tournament Design: How Operations Research Can Improve Sports Rules, Palgrave Pivots in Sports Economics (2021), Palgrave Macmillan: Palgrave Macmillan Cham, Switzerland) |
| [23] | Csató, L., A simulation comparison of tournament designs for the World Men’s handball championships, Int. Trans. Oper. Res., 28, 5, 2377-2401 (2021) · Zbl 07769665 |
| [24] | Csató, L., Quantifying incentive (in)compatibility: A case study from sports, European J. Oper. Res. (2022), (in press) · Zbl 1507.90096 |
| [25] | Dagaev, D.; Sonin, K., Winning by losing: Incentive incompatibility in multiple qualifiers, J. Sports Econ., 19, 8, 1122-1146 (2018) |
| [26] | Dagaev, D.; Suzdaltsev, A., Competitive intensity and quality maximizing seedings in knock-out tournaments, J. Combin. Optim., 35, 1, 170-188 (2018) · Zbl 1394.90478 |
| [27] | David, H. A., Tournaments and paired comparisons, Biometrika, 46, 1/2, 139-149 (1959) · Zbl 0087.14901 |
| [28] | Durán, G.; Guajardo, M.; Sauré, D., Scheduling the South American qualifiers to the 2018 FIFA world cup by integer programming, European J. Oper. Res., 262, 3, 1109-1115 (2017) · Zbl 1375.90184 |
| [29] | Edwards, C. T., Non-parametric procedure for knockout tournaments, J. Appl. Stat., 25, 3, 375-385 (1998) · Zbl 0930.62048 |
| [30] | Engist, O.; Merkus, E.; Schafmeister, F., The effect of seeding on tournament outcomes: Evidence from a regression-discontinuity design, J. Sports Econ., 22, 1, 115-136 (2021) |
| [31] | Handbook (2020), https://handbook.fide.com/ |
| [32] | Führlich, P.; Cseh, A.; Lenzner, P., Improving ranking quality and fairness in Swiss-system chess tournaments (2021), Manuscript. arXiv:2112.10522 |
| [33] | Ginsburgh, V. A.; Van Ours, J. C., Expert opinion and compensation: Evidence from a musical competition, Amer. Econ. Rev., 93, 1, 289-296 (2003) |
| [34] | Glenn, W. A., A comparison of the effectiveness of tournaments, Biometrika, 47, 3/4, 253-262 (1960) · Zbl 0094.15302 |
| [35] | Glickman, M. E., BayesIan locally optimal design of knockout tournaments, J. Statist. Plann. Inference, 138, 7, 2117-2127 (2008) · Zbl 1134.62015 |
| [36] | Goossens, D. R.; Beliën, J.; Spieksma, F. C.R., Comparing league formats with respect to match importance in Belgian football, Ann. Oper. Res., 194, 1, 223-240 (2012) · Zbl 1251.90150 |
| [37] | Goossens, D. R.; Spieksma, F. C.R., Soccer schedules in europe: an overview, J. Sched., 15, 5, 641-651 (2012) |
| [38] | Groh, C.; Moldovanu, B.; Sela, A.; Sunde, U., Optimal seedings in elimination tournaments, Econom. Theory, 49, 1, 59-80 (2012) · Zbl 1276.91058 |
| [39] | Guyon, J., Rethinking the FIFA World \(\operatorname{Cup}^{T M}\) final draw, J. Quant. Anal. Sports, 11, 3, 169-182 (2015) |
| [40] | Guyon, J., What a fairer 24 team UEFA Euro could look like, J. Sports Anal., 4, 4, 297-317 (2018) |
| [41] | Guyon, J., Risk of collusion: Will groups of 3 ruin the FIFA world cup?, J. Sports Anal., 6, 4, 259-279 (2020) |
| [42] | Harary, F.; Moser, L., The theory of round robin tournaments, Amer. Math. Monthly, 73, 3, 231-246 (1966) · Zbl 0142.41602 |
| [43] | Harris, C.; Vickers, J., Racing with uncertainty, Rev. Econom. Stud., 54, 1, 1-21 (1987) · Zbl 0602.90146 |
| [44] | Hartigan, J. A., Inference from a knockout tournament, Ann. Math. Stat., 39, 2, 583-592 (1968) · Zbl 0238.62029 |
| [45] | Hennessy, J.; Glickman, M., BayesIan optimal design of fixed knockout tournament brackets, J. Quant. Anal. Sports, 12, 1, 1-15 (2016) |
| [46] | Horen, J.; Riezman, R., Comparing draws for single elimination tournaments, Oper. Res., 33, 2, 249-262 (1985) · Zbl 0563.90066 |
| [47] | Hou, T.; Zhang, W., Optimal two-stage elimination contests for crowdsourcing, Transp. Res. E, 145, Article 102156 pp. (2021) |
| [48] | Hwang, F. K., New concepts in seeding knockout tournaments, Amer. Math. Monthly, 89, 4, 235-239 (1982) · Zbl 0486.05031 |
| [49] | Israel, R. B., Stronger players need not win more knockout tournaments, J. Amer. Statist. Assoc., 76, 376, 950-951 (1981) · Zbl 0489.62064 |
| [50] | Karpov, A., A new knockout tournament seeding method and its axiomatic justification, Oper. Res. Lett., 44, 6, 706-711 (2016) · Zbl 1408.90132 |
| [51] | Karpov, A., Generalized knockout tournament seedings, Int. J. Comput. Sci. Sport, 17, 2, 113-127 (2018) |
| [52] | Kendall, G.; Knust, S.; Ribeiro, C. C.; Urrutia, S., Scheduling in sports: An annotated bibliography, Comput. Oper. Res., 37, 1, 1-19 (2010) · Zbl 1171.90400 |
| [53] | Kendall, G.; Lenten, L. J.A., When sports rules go awry, European J. Oper. Res., 257, 2, 377-394 (2017) · Zbl 1394.90407 |
| [54] | Klumpp, T.; Polborn, M. K., Primaries and the new hampshire effect, J. Public Econ., 90, 6-7, 1073-1114 (2006) |
| [55] | Knuth, D. E.; Lossers, O. P., A random knockout tournament, SIAM Rev., 29, 1, 127-129 (1987) |
| [56] | Koning, R. H., Home advantage in professional tennis, J. Sports Sci., 29, 1, 19-27 (2011) |
| [57] | Kräkel, M., Optimal seedings in elimination tournaments revisited, Econ. Theory Bull., 2, 1, 77-91 (2014) |
| [58] | Krumer, A., Testing the effect of kick-off time in the UEFA Europa League, Eur. Sport Manage. Q., 20, 2, 225-238 (2020) |
| [59] | Kulhanek, T.; Ponomarenko, V., Surprises in knockout tournaments, Math. Mag., 93, 3, 193-199 (2020) · Zbl 1459.00007 |
| [60] | Laliena, P.; López, F. J., Fair draws for group rounds in sport tournaments, Int. Trans. Oper. Res., 26, 2, 439-457 (2019) · Zbl 07770933 |
| [61] | Lasek, J.; Gagolewski, M., The efficacy of league formats in ranking teams, Statist. Model., 18, 5-6, 411-435 (2018) · Zbl 07289516 |
| [62] | Lazear, E. P.; Rosen, S., Rank-order tournaments as optimum labor contracts, J. Polit. Econ., 89, 5, 841-864 (1981) |
| [63] | Marchand, E., On the comparison between standard and random knockout tournaments, J. R. Statist. Soc.: Ser. D (Statist.), 51, 2, 169-178 (2002) |
| [64] | McGarry, T.; Schutz, R. W., Efficacy of traditional sport tournament structures, J. Oper. Res. Soc., 48, 1, 65-74 (1997) |
| [65] | Mendonça, D.; Raghavachari, M., Comparing the efficacy of ranking methods for multiple round-robin tournaments, European J. Oper. Res., 123, 3, 593-605 (2000) · Zbl 1068.91507 |
| [66] | Orrison, A.; Schotter, A.; Weigelt, K., Multiperson tournaments: An experimental examination, Manage. Sci., 50, 2, 268-279 (2004) |
| [67] | Palacios-Huerta, I.; Volij, O., Field centipedes, Amer. Econ. Rev., 99, 4, 1619-1635 (2009) |
| [68] | Pauly, M., Can strategizing in round-robin subtournaments be avoided?, Soc. Choice Welf., 43, 1, 29-46 (2014) · Zbl 1302.91087 |
| [69] | Petróczy, D. G.; Csató, L., Revenue allocation in formula one: A pairwise comparison approach, Int. J. Gen. Syst., 50, 3, 243-261 (2021) |
| [70] | Prendergast, C., The provision of incentives in firms, J. Econ. Lit., 37, 1, 7-63 (1999) |
| [71] | Preston, I.; Szymanski, S., Cheating in contests, Oxf. Rev. Econ. Policy, 19, 4, 612-624 (2003) |
| [72] | Prince, M.; Cole Smith, J.; Geunes, J., Designing fair 8- and 16-team knockout tournaments, IMA J. Manag. Math., 24, 3, 321-336 (2013) · Zbl 1273.90129 |
| [73] | Ren, W.; Liu, J.; Shroff, N. B., On sample complexity upper and lower bounds for exact ranking from noisy comparisons (2019), Manuscript. arXiv:1909.03194 |
| [74] | Rosen, S., Prizes and incentives in elimination tournaments, Amer. Econ. Rev., 76, 4, 701-715 (1986) |
| [75] | Rubinstein, A., Ranking the participants in a tournament, SIAM J. Appl. Math., 38, 1, 108-111 (1980) · Zbl 0442.05028 |
| [76] | Ryvkin, D., The selection efficiency of tournaments, European J. Oper. Res., 206, 3, 667-675 (2010) · Zbl 1188.91050 |
| [77] | Ryvkin, D.; Ortmann, A., The predictive power of three prominent tournament formats, Manage. Sci., 54, 3, 492-504 (2008) |
| [78] | Scarf, P. A.; Yusof, M. M., A numerical study of tournament structure and seeding policy for the soccer world cup finals, Stat. Neerl., 65, 1, 43-57 (2011) · Zbl 07881864 |
| [79] | Scarf, P.; Yusof, M. M.; Bilbao, M., A numerical study of designs for sporting contests, European J. Oper. Res., 198, 1, 190-198 (2009) |
| [80] | Schwenk, A. J., What is the correct way to seed a knockout tournament?, Amer. Math. Monthly, 107, 2, 140-150 (2000) · Zbl 0981.05054 |
| [81] | Searls, D. T., On the probability of winning with different tournament procedures, J. Amer. Statist. Assoc., 58, 304, 1064-1081 (1963) · Zbl 0118.35709 |
| [82] | Szymanski, S., The economic design of sporting contests, J. Econ. Lit., 41, 4, 1137-1187 (2003) |
| [83] | Taylor, C. R., Digging for golden carrots: An analysis of research tournaments, Amer. Econ. Rev., 85, 4, 872-890 (1995) |
| [84] | New format for Champions League post-2024: everything you need to know (2021), 25 May. https://www.uefa.com/uefachampionsleague/news/0268-12157d69ce2d-9f011c70f6fa-1000-new-format-for-champions-league-post2024-everything-you-need-to-know/ |
| [85] | Van Bulck, D.; Goossens, D., Handling fairness issues in time-relaxed tournaments with availability constraints, Comput. Oper. Res., 115, Article 104856 pp. (2020) · Zbl 1458.90359 |
| [86] | Vong, A. I.K., Strategic manipulation in tournament games, Games Econom. Behav., 102, 562-567 (2017) · Zbl 1409.91036 |
| [87] | Vu, T.; Shoham, Y., Fair seeding in knockout tournaments, ACM Trans. Intell. Syst. Technol. (TIST), 3, 1, 9 (2011) |
| [88] | Wright, M., OR analysis of sporting rules - A survey, European J. Oper. Res., 232, 1, 1-8 (2014) · Zbl 1305.90006 |
| [89] | Yücesan, E., An efficient ranking and selection approach to boost the effectiveness of innovation contests, IIE Trans., 45, 7, 751-762 (2013) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
