Source-linked AI summary
An application of incomplete pairwise comparison matrices for ranking top tennis players
Sándor Bozóki, László Csató, József Temesi
TL;DR
The paper asks how to rank leading tennis players when some never played each other, using their head-to-head results in incomplete pairwise comparison matrices. It calculates player weights from those matrices and reports Nadal, Federer, and Sampras as the top three. The analysis also identifies limitations in how match timing and playing surfaces are treated.
Problem
Some selected top players never met, leaving their direct pairwise comparisons unavailable for a long-term ranking.
Source: §1 · p. 2Method
The paper builds incomplete pairwise comparison matrices from head-to-head results for 25 leading ATP players and calculates weights using the Eigenvector and Logarithmic Least Squares methods.
Source: §1 · pp. 1–2, §3 · p. 10Results
Nadal, Federer, and Sampras were the top three players in the reported rankings, which span players who never met on court.
Source: §3 · p. 11Takeaways & Limitations
The results show that pairwise comparisons can produce rankings spanning four decades even when some players never met.
Source: §3 · p. 11The analysis gives every match identical weight, does not account for a player's career path or playing surface, and cannot correct matrix inconsistency in the given dataset.
Source: §2 · p. 4, §4 · p. 11
Abstract
from arXiv · showhide
Pairwise comparison is an important tool in multi-attribute decision making. Pairwise comparison matrices (PCM) have been applied for ranking criteria and for scoring alternatives according to a given criterion. Our paper presents a special application of incomplete PCMs: ranking of professional tennis players based on their results against each other. The selected 25 players have been on the top of the ATP rankings for a shorter or longer period in the last 40 years. Some of them have never met on the court. One of the aims of the paper is to provide ranking of the selected players, however, the analysis of incomplete pairwise comparison matrices is also in the focus. The eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete PCMs. In our results the top three players of four decades were Nadal, Federer and Sampras. Some questions have been raised on the properties of incomplete PCMs and remains open for further investigation.
1 Introduction
The paper applies incomplete pairwise comparison matrices to the historical ranking of leading tennis players, while also discussing the matrices’ theoretical properties. Its focus is on comparing players through their results against one another, including players who never met on court.
1 Introduction
- The study ranks 25 players who led ATP rankings for varying periods over the last 40 years using their results against each other.Source: §1 · p. 1
- Players who never met cannot be compared directly, and the paper relies on official ATP records that provide reliable, complete data from 1973.Source: §1 · pp. 1–2
- The highest-weight player can be regarded as best only within the selected sample, and the result varies across ranking lists.Source: §1 · p. 2
- Earlier tennis rankings used match networks and PageRank-like methods; reported findings include higher predictive power than official rankings.Source: §1 · p. 2
- Related work also includes Bradley–Terry match forecasts and cross-efficiency rankings using Data Envelopment Analysis.Source: §1 · pp. 2–3
- The paper discusses theoretical results and applications of incomplete PCMs, building on their use in multi-attribute decision making to rank criteria or score alternatives.Source: §1 · pp. 1, 3
2 Theory and methods
The section formulates tennis results as pairwise comparisons for estimating player weights and presents methods for handling incomplete matrices. It also states when the estimates are unique and why consistency correction is unavailable for the given data.
2 Theory and methods
- Match-win ratios and their reciprocals form the tennis comparison matrix, whose weights are normalized and estimated using the Eigenvector Method or Logarithmic Least Squares Method.For players Pi and Pj, the comparison is their wins over one another; the diagonal entries equal 1.Source: §2 · pp. 3–4
- Intransitivity can make inconsistency natural in the tennis data, but the given data set does not permit correction of matrix elements.The authors therefore cannot apply consistency correction to the comparisons.Source: §2 · p. 4
- For incomplete LLSM, a connected comparison graph is necessary and sufficient for a unique solution, which can be calculated by solving linear equations.The graph's vertices are objects and its edges represent known comparisons.Source: §2 · p. 4
- LLSM estimates weights directly; missing comparisons can then be recovered as ratios of the corresponding optimal weights.The paper focuses on weights rather than filling in missing matrix elements.Source: §2 · p. 5
- The incomplete eigenvector method first replaces missing comparisons with positive variables, then applies the eigenvector method to the completed matrix.The completion is chosen by minimizing the maximal eigenvalue, equivalently the inconsistency index; uniqueness also holds exactly when the comparison graph is connected.Source: §2 · p. 5
3 Calculation and results
The paper constructs incomplete pairwise comparison matrices from head-to-head results for selected top ATP players and computes rankings using two estimation methods. Nadal, Federer, and Sampras lead both reported rankings, while match-count normalization can change the results.
3 Calculation and results
- The sample contains 25 players who reached ATP No. 1 from 1973, but the authors note there is no ideal choice of players, eras, or comparison periods.They also raise whether rankings should be unified or separated by playing surface.Source: §3 · p. 5
- Pairwise values use win-to-loss ratios and reciprocals, while missing matches leave comparisons unavailable and undefeated pairs receive artificial values under two correction methods.The first correction assigns values based on match-count ranges; the second sets pij to xij + 2 when yij = 0.Source: §3 · pp. 6–7
- The transformation adjusts pairwise ratios by each pair’s match count relative to the maximum, addressing totals ranging from fewer than five to more than 30 matches.For Agassi–Becker, the ratio 2.5 becomes 1.3895 after scaling by 14/39; the transformation approaches 1 for pairs with few matches.Source: §3 · p. 7
- The comparison graph is connected, so the authors could compute weights from the incomplete matrix using the logarithmic least-squares and eigenvector methods.A spanning tree comprises Agassi’s 20 adjacent edges plus Nadal–Djokovic, Newcombe–Nastase, Nastase–Connors, and Connors–Borg.Source: §3 · pp. 7, 10, Figure 2 · p. 9
- Nadal, Federer, and Sampras lead both reported rankings, while Courier, Muster, and Connors occupy the last three places.The authors conclude that pairwise comparisons can rank players across four decades, including players who never met on court.Source: Table 2 · p. 10, §3 · p. 11
- The eigenvector and logarithmic least-squares rankings are practically the same, and correction-method effects are not significant.The authors report that correlations support these conclusions and that normalization reduces the minor impact of correction methods.Source: §3 · p. 10, Table 3 · p. 11
- Match-count transformation may change rankings significantly, leaving experts to decide whether equal ratios from different match totals should be represented differently.The authors recommend normalization when experts believe a 4:2 record and a 20:10 record warrant different representations.Source: §3 · pp. 10–11, Table 3 · p. 11
4 Conclusions and open questions
The paper applies incomplete historical match matrices to ranking while noting that every match receives equal weight, regardless of career timing or surface. It also asks how matrix density and comparison distribution may affect rankings.
4 Conclusions and open questions
- Every match has identical weight, without accounting for a player's career path or differences between playing surfaces.Source: §4 · p. 11
- The paper asks which properties of matrix T affect the resulting ranking.Source: §4 · p. 11
- 341/625 ≈0.5456 is the density of T, reflecting 341 known elements in the 25 × 25 matrix.Source: §4 · p. 11
- Vertex-degree distribution is another structural indicator of T; Agassi has the maximum degree, 20.Source: §4 · pp. 11–12, Figure 3 · p. 12