Home FIVB ranking: misstep in the right direction
Article
Licensed
Unlicensed Requires Authentication

FIVB ranking: misstep in the right direction

  • Salma Tenni , Daniel Gomes de Pinho Zanco and Leszek Szczecinski EMAIL logo
Published/Copyright: April 30, 2025
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Journal of Quantitative Analysis in Sports
From the journal Journal of Quantitative Analysis in Sports Volume 21 Issue 3

Abstract

This work presents and evaluates the ranking algorithm that has been used by Fédération Internationale de Volleyball (FIVB) since 2020. The prominent feature of the FIVB ranking is the use of the probabilistic model, which explicitly calculates the probabilities of the future matches results using the estimated teams’ strengths. Such explicit modeling is new in the context of official sport rankings, especially for multi-level outcomes, and we study the optimality of its parameters using both analytical and numerical methods. We conclude that from the modeling perspective, the current thresholds fit well the data but adding the home-field advantage (HFA) would be beneficial. Regarding the algorithm itself, we explain the rationale behind the approximations currently used and show a simple method to find new parameters (numerical score) which improve the performance. We also show that the weighting of the match results is counterproductive.

Keywords: ranking; FIVB; volleyball; ordinal models
Corresponding author: Leszek Szczecinski, Institut National de la Recherche Scientifique, Montreal, Canada, E-mail: 

Funding source: Natural Sciences and Engineering Research Council of Canada

Appendix A: Notation in FIVB ranking

We show in Table 6, the relationship between our notation and the one used in the FIVB ranking description (FIVB 2024), where the following abbreviations are used

  1. WR: World ranking (here, a FIVB ranking)

  2. WRS: World ranking score (here, we call it skills θ m,t)

  3. SSV: Set score variant (we call it numerical score r y )

  4. EMR: Expected match result (here, the expected score r ̌ ( z ) , see (22))

  5. MWF: Match weighting factor (here, it corresponds to 10 ξ v t )

  6. Scaled difference between WRSs Δ = 8(WRS1 − WRS2)/1,000

Table 6:

Equivalence of this work’s notation and the one used in the description of the FIVB ranking.

Our notation FIVB notation
θm,t, θn,t WRS1, WRS2
c 0 F I V B , … , c 4 F I V B C1, …, C5
z t /s = (θm,t − θn,t)/s Δ = 8(WRS1 − WRS2)/1,000
s 1,000/8 = 125
P 0 ( z t ) , … , P 5 ( z t ) P1, …, P5
Φ(z) ∼N(0, 1)(z)
r y t F I V B SSV
r ̌ ( z t / s ) EMR
10 ξ v t MWF
− g F I V B ( z t / s ) = r y t F I V B − r ̌ ( z t / s ) WR value
θ m , t + 1 − θ m , t = − 10 ξ v t g F I V B ( z t / s ) WR points = WR values * MWF/8

With this notation, the update formula is given by

(59) WRS 1 ← WRS 1 + WR points

and corresponds to (20) which, focusing on the update of the skills of the home team m, may be written as

(60) θ m , t + 1 = θ m , t − μ s ξ v t g y t F I V B ( z t / s ) .

Appendix B: Optimization of the cross-validation metric

The simplest optimization of the cross-validation metric U( p ) may be done via the steepest descent

(61) p ̂ ← p ̂ − κ ∇ p U ( p ) ,

where, κ is the step-size, and to calculate the gradient ∇ p U( p ), we have to calculate derivatives of U( p ) with respect to a parameter q ∈ p . This can be done as follows:

(62) ∂ ∂ q U ( p ) = 1 T ∑ t = 1 T ∂ ∂ q ℓ y t v a l ( z ̂ t , \ t , p )
(63) ∂ ∂ q ℓ y t v a l ( z ̂ t , \ t ) = ∂ z ̂ t , \ t ∂ q ℓ ̇ v a l ( z ̂ t , \ t , p ) + ∂ ∂ q ℓ y t v a l ( z ̂ t , \ t , p )
(64) ∂ z ̂ t , \ t ∂ q = ∂ z ̂ t ∂ q + ∂ ∂ q ξ v t ℓ ̇ y t ( z ̂ t , p ) a t 1 − ξ v t ℓ ̈ y t ( z ̂ t , p ) a t .

In (64), we will need

(65) ∂ z ̂ t ∂ q = x t T ∂ θ ̂ ∂ q
(66) ∂ a t ∂ q = x t T ∂ H ̂ − 1 ∂ q x t = − x t T H ̂ − 1 ∂ H ̂ ∂ q H ̂ − 1 x t
(67) ∂ H ̂ ∂ q = ∑ t = 1 T x t ∂ ∂ q ξ v t ℓ ̈ y t ( z ̂ t , p ) x t T + I q = γ I
(68) ∂ ∂ q ℓ ̇ y t ( z ̂ t , p ) = ∂ z ̂ t ∂ q ℓ ̈ y t ( z ̂ t , p ) + ∂ ∂ q ℓ ̇ y t ( z ̂ t , p )
(69) ∂ ∂ q ℓ ̈ y t ( z ̂ t , p ) = ∂ z ̂ t ∂ q ℓ ⃛ y t ( z ̂ t , p ) + ∂ ∂ q ℓ ̈ y t ( z ̂ t , p )

where, in (66) we used (Petersen and Pedersen 2012, Eq. (40)), and ℓ ⃛ y ( z , p ) = ∂ 3 ∂ z 3 ℓ y ( z , p ) .

From implicit function theorem, see Lorraine et al. (2019, Th. 1), using θ ̂ ≡ θ ̂ ( p )

(70) 0 = ∂ ∂ q ∇ θ J ( θ ̂ ( p ) , p )
(71) 0 = ∇ θ 2 J ( θ ̂ ( p ) , p ) ∂ θ ̂ ( p ) ∂ q + ∂ ∂ q ∇ θ J ( θ ̂ , p )
(72) ∂ θ ̂ ∂ q = − H ̂ − 1 ∂ ∂ q ∇ θ J ( θ ̂ , p )
(73) ∂ ∂ q ∇ θ J ( θ ̂ , p ) = ∑ t = 1 T ∂ ∂ q ξ v t ℓ ̇ y t ( x t T θ ̂ , p ) x t + I γ = q θ ̂ .

By plugging z ̂ t , \ t ( p ) into (33) we obtain the function U( p ) which depends on p and we can calculate the gradient ∇ p U( p ).

Similarly, we can use the Newton method

(74) p ̂ ← p ̂ − ∇ p 2 U ( p ) − 1 ∇ p U ( p )

where, to calculate the Hessian, ∇ p 2 U ( p ) we need second order derivatives.

However, the expressions for the gradient and, especially, the Hessian, quickly become cumbersome; see Burn (2020). Thus, instead of explicit differentiation, we use the automatic differentiation available in JAX and JAXopt python-compliant packages (Blondel et al. 2021; Bradbury et al. 2018) with a particularly interesting feature which automatically finds the implicit differentiation required to find the derivative of θ ̂ with respect to hyperparameters in p as specified by (72).

We do not show more details to not overcomplicate the presentation, especially that they are not really required because the numerical optimization is used to confirm the observation we made using the analytical insight. In fact, the performance of the online algorithm shown in Section 4 uses the parameters (shown in Table 4) which are explicitly defined prior to the application of the algorithms.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors contributed to the analysis, data acquisition and treatment, writing and correction.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: Natural Sciences and Engineering Research Council of Canada.

  7. Data availability: Available at the repository https://github.com/brbalab/FIVB.

References

Agresti, A. (2013). Categortical data analysis. John Wiley & Sons, Hoboken, New Jersey.Search in Google Scholar

Aldous, D. (2017). Elo ratings and the sports model: a neglected topic in applied probability? Stat. Sci. 32: 616–629. https://doi.org/10.1214/17-STS628.Search in Google Scholar

Beirami, A., Razaviyayn, M., Shahrampour, S., and Tarokh, V. (2017). On optimal generalizability in parametric learning. In: Proceedings of the 31st international conference on neural information processing systems, NIPS’17. Curran Associates Inc., Red Hook, NY, USA, pp. 3458–3468.Search in Google Scholar

Blondel, M., Berthet, Q., Cuturi, M., Frostig, R., Hoyer, S., Llinares-López, F., Pedregosa, F., and Vert, J.-P. (2021). Efficient and modular implicit differentiation. arXiv preprint arXiv:2105.15183.Search in Google Scholar

Bradbury, J., Frostig, R., Hawkins, P., Johnson, M.J., Leary, C., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J., Wanderman-Milne, S., et al.. (2018). JAX: composable transformations of Python+NumPy programs, Available at: https://github.com/google/jax.Search in Google Scholar

Bradley, R.A. and Terry, M.E. (1952). Rank analysis of incomplete block designs: 1 the method of paired comparisons. Biometrika 39: 324–345. https://doi.org/10.2307/2334029.Search in Google Scholar

Burn, R. (2020). Optimizing approximate leave-one-out cross-validation to tune hyperparameters. ArXiv, abs/2011.10218, https://arxiv.org/abs/2011.10218.Search in Google Scholar

Csató, L. (2024). Club coefficients in the UEFA champions league: time for shift to an Elo-based formula. Int. J. Perform. Anal. Sport 24: 119–134. https://doi.org/10.1080/24748668.2023.2274221.Search in Google Scholar

Egidi, L. and Ntzoufras, I. (2019). A Bayesian quest for finding a unified model for predicting volleyball games. J. Roy. Stat. Soc. C Appl. Stat. 69, https://doi.org/10.1111/rssc.12436.Search in Google Scholar

Egidi, L., Pauli, F., and Torelli, N. (2018). Combining historical data and bookmakers’ odds in modelling football scores. Stat. Model. 18: 436–459. https://doi.org/10.1177/1471082X18798414.Search in Google Scholar

Elo, A.E. (1978). The rating of chess players, past and present. Arco Publishing Inc, New York, NY, USA.Search in Google Scholar

Fahrmeir, L. (1992). Posterior mode estimation by extended Kalman filtering for multivariate dynamic generalized linear models. J. Am. Stat. Assoc. 87: 501–509. https://doi.org/10.2307/2290283.Search in Google Scholar

FIVB (2024). Fivb volleyball world ranking, Available at: https://en.volleyballworld.com/volleyball/world-ranking/ranking-explained.Search in Google Scholar

Gabrio, A. (2020). Bayesian hierarchical models for the prediction of volleyball results. J. Appl. Stat. 48: 301–321. https://doi.org/10.1080/02664763.2020.1723506.Search in Google Scholar PubMed PubMed Central

Glenn, W.A. and David, H.A. (1960). Ties in paired-comparison experiments using a modified Thurstone-Mosteller model. Biometrics 16: 86–109. https://doi.org/10.2307/2527957.Search in Google Scholar

Glickman, M.E. (1993). Paired comparison models with time-varying parameters, Ph.D. thesis. Harvard University.10.21236/ADA272016Search in Google Scholar

Gomes de Pinho Zanco, D., Szczecinski, L., Vinicius Kuhn, E., and Seara, R. (2024). Stochastic analysis of the Elo rating algorithm in round-robin tournaments. Digit. Signal Process. 145: 104313. https://doi.org/10.1016/j.dsp.2023.104313.Search in Google Scholar

Hastie, T., Tibshirani, R., and Friedman, J. (2009). The elements of statistical learning. Springer Series in Statiscs, New York, NY, USA.10.1007/978-0-387-84858-7Search in Google Scholar

Hu, F. and Zidek, J.V. (2001). The relevance weighted likelihood with applications. Springer, New York, NY, pp. 211–235.10.1007/978-1-4613-0141-7_13Search in Google Scholar

Jabin, P.-E. and Junca, S. (2015). A continuous model for ratings. SIAM J. Appl. Math. 2: 420–442. https://doi.org/10.1137/140969324.Search in Google Scholar

Karlis, D. and Ntzoufras, I. (2008). Bayesian modelling of football outcomes: using the Skellam’s distribution for the goal difference. IMA J. Manag. Math. 20: 133–145. https://doi.org/10.1093/imaman/dpn026.Search in Google Scholar

Knorr-Held, L. (2000). Dynamic rating of sports teams. J. R. Stat. Soc. – Ser. D Statistician 49: 261–276. https://doi.org/10.1111/1467-9884.00236.Search in Google Scholar

Lasek, J. and Gagolewski, M. (2021). Interpretable sports team rating models based on the gradient descent algorithm. Int. J. Forecast. 37: 1061–1071. https://doi.org/10.1016/j.ijforecast.2020.11.008.Search in Google Scholar

Ley, C., Van de Wiele, T., and Van Eetvelde, H. (2019). Ranking soccer teams on the basis of their current strength: a comparison of maximum likelihood approaches. Stat. Model. 19: 55–73. https://doi.org/10.1177/1471082X18817650.Search in Google Scholar

Lorraine, J., Vicol, P., and Duvenaud, D. (2019). Optimizing millions of hyperparameters by implicit differentiation. CoRR, abs/1911.02590, https://arxiv.org/abs/1911.02590.Search in Google Scholar

Macrì Demartino, R., Egidi, L., and Torelli, N. (2024). Alternative ranking measures to predict international football results. Comput. Stat, https://doi.org/10.1007/s00180-024-01585-z.Search in Google Scholar

McCullagh, P. (1980). Regression models for ordinal data. J. Roy. Stat. Soc. B 42: 109–142. https://doi.org/10.1111/j.2517-6161.1980.tb01109.x.Search in Google Scholar

Ntzoufras, I., Palaskas, V., and Drikos, S. (2021). Bayesian models for prediction of the set-difference in volleyball. IMA J. Manag. Math. 32: 491–518. https://doi.org/10.1093/imaman/dpab007.Search in Google Scholar

Petersen, K.B. and Pedersen, M.S. (2012). The matrix cookbook, version 20121115, Available at: https://www2.compute.dtu.dk/pubdb/pubs/3274-full.html.Search in Google Scholar

Rad, K.R. and Maleki, A. (2020). A scalable estimate of the out-of-sample prediction error via approximate leave-one-out cross-validation. J. Roy. Stat. Soc. B Stat. Methodol. 82: 965–996. https://doi.org/10.1111/rssb.12374.Search in Google Scholar

Szczecinski, L. (2022). G-Elo: generalization of the Elo algorithm by modeling the discretized margin of victory. J. Quant. Anal. Sports 18: 1–14. https://doi.org/10.1515/jqas-2020-0115.Search in Google Scholar

Szczecinski, L. and Djebbi, A. (2020). Understanding draws in Elo rating algorithm. J. Quant. Anal. Sports 16: 211–220.10.1515/jqas-2019-0102Search in Google Scholar

Szczecinski, L. and Roatis, I.-I. (2022). FIFA ranking: evaluation and path forward. J. Sports Anal. 8: 231–250. https://doi.org/10.3233/jsa-200619.Search in Google Scholar

Szczecinski, L. and Tihon, R. (2023). Simplified Kalman filter for online rating: one-fits-all approach. J. Quant. Anal. Sports 19: 295–315. https://doi.org/10.1515/jqas-2021-0061.Search in Google Scholar

Tutz, G. (2012). Regression for categorical data. Cambridge University Press, New York, NY, USA.10.1017/CBO9780511842061Search in Google Scholar
Received: 2024-08-19
Accepted: 2025-01-15
Published Online: 2025-04-30
Published in Print: 2025-09-25

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. NHL aging curves using functional principal component analysis
  4. Aerodynamics, technology or pit strategy: why did overtaking in Formula 1 decline during the 1980s and 1990s? A micro-level analysis
  5. Analyzing and forecasting success in the Men’s Ice Hockey World (Junior) Championships using a dynamic ranking model
  6. FIVB ranking: misstep in the right direction
  7. Analyzing key factors influencing IPL cricket scores using explainability and multimodal data
https://doi.org/10.1515/jqas-2024-0128
Keywords for this article
ranking; FIVB; volleyball; ordinal models

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. NHL aging curves using functional principal component analysis
  4. Aerodynamics, technology or pit strategy: why did overtaking in Formula 1 decline during the 1980s and 1990s? A micro-level analysis
  5. Analyzing and forecasting success in the Men’s Ice Hockey World (Junior) Championships using a dynamic ranking model
  6. FIVB ranking: misstep in the right direction
  7. Analyzing key factors influencing IPL cricket scores using explainability and multimodal data
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 19.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jqas-2024-0128/pdf
Scroll to top button