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arxiv: 2605.13926 · v1 · pith:BCY3PZNTnew · submitted 2026-05-13 · 📊 stat.AP

Optimising football transfer strategy under budget constraints: A weighted multi-criteria approach

Tathagata Basu , Soudeep Deb , Rishideep Roy This is my paper

Pith reviewed 2026-05-15 02:43 UTC · model grok-4.3

classification 📊 stat.AP
keywords football transfersmulti-criteria optimizationlinear mixed-effects modelsplayer valuationbudget constraintsauction modelingsports analyticssquad optimization
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The pith

A club's optimal football transfers can be determined by feeding performance and price predictions into a weighted multi-criteria optimization that respects budget limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs linear mixed-effects models that use player traits, recent results, team setting, and league context to forecast both future performance ratings and likely transfer prices. These forecasts are then fed into a constrained optimization routine that selects buy and sell decisions to maximize a weighted combination of squad quality while staying inside a spending cap. The resulting transfer list is further analyzed by placing the decisions inside an auction model with random reserves to study bidding behavior when several clubs target the same player. A sympathetic reader would care because the method converts transfer planning from ad-hoc judgment into a repeatable calculation that can be rerun for different budgets or priorities. The approach is demonstrated on 2018-19 English Premier League data to show it produces plausible squad adjustments.

Core claim

Integrating predicted player ratings and estimated transfer prices from linear mixed-effects models into a weighted multi-criteria constrained optimisation framework determines a club's end-of-season transfer activities, with the decisions then embedded in an independent private-value auction model to examine market competition.

What carries the argument

The weighted multi-criteria constrained optimisation framework that balances predicted performance improvements against estimated costs subject to a total budget limit.

If this is right

  • Clubs can generate a ranked list of transfers that maximises weighted squad strength inside any given spending limit.
  • The auction layer shows how bidding competition changes the effective cost of a player when multiple teams pursue the same target.
  • The same optimisation can be rerun for different weightings to explore trade-offs between attacking strength, defensive stability, and age profile.
  • Transfer recommendations update automatically when new performance data or revised budgets become available.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Smaller clubs with tight budgets might gain the most from this systematic approach because it reduces the risk of wasting limited funds on marginal signings.
  • The framework could be extended to mid-season windows by updating the mixed-effects predictions with the latest match data.
  • If the auction component is realistic, it offers a way to simulate how a club's own bids influence the final price when rivals are also modeled.

Load-bearing premise

Linear mixed-effects models using player characteristics, recent performance, team context, and league effects will generate accurate enough forecasts of future ratings and market prices to support useful optimization choices.

What would settle it

Apply the full pipeline to a past season, execute the suggested transfers, and compare the actual next-season league results or squad value against the club's real outcomes to see whether the optimized path produces measurably better performance or value.

Figures

Figures reproduced from arXiv: 2605.13926 by Rishideep Roy, Soudeep Deb, Tathagata Basu.

Figure 1
Figure 1. Figure 1: Distribution of players in our dataset according to their nationality. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Observed (dashed) versus estimated (solid) club-level means for player ratings (top) and average transfer fees [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Summary of simulation studies for understanding transfer market dynamics. The left panel illustrates the [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
read the original abstract

The football transfer market is a complex, dynamic environment in which clubs compete to acquire players who strengthen their squads. While several frameworks estimate a player's worth, a comprehensive approach that captures both squad optimisation and transfer market dynamics remains limited. In this paper, we propose a quantitative framework for optimising football transfer strategy under budget constraints, integrated with a competitive bidding paradigm. Using data from professional football leagues, we construct player performance and transfer price models using linear mixed-effects frameworks that incorporate player characteristics, recent performance, team context, and league effects. The predicted ratings and estimated transfer prices are then integrated into a weighted multi-criteria constrained optimisation framework that determines a club's transfer activities at the end of the season. Finally, these optimal transfer decisions are embedded within an independent private-value auction model with a random reserve price to analyse market behaviour when multiple teams compete for the same player. We illustrate our approach using the 2018-19 season of the English Premier League to demonstrate its ability to capture transfer-market dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a quantitative framework for optimizing football transfer strategies under budget constraints. Linear mixed-effects models are used to predict player performance ratings and transfer prices from player characteristics, recent performance, team context, and league effects. These predictions are fed into a weighted multi-criteria constrained optimization to determine end-of-season transfers for a club, which are then embedded in an independent private-value auction model with random reserve price to study competitive bidding. The approach is illustrated on 2018-19 English Premier League data.

Significance. If the underlying predictions prove reliable, the integration of mixed-effects modeling, multi-criteria optimization, and auction analysis would offer a coherent pipeline for data-driven transfer decisions that accounts for both squad needs and market competition. The explicit incorporation of league and team random effects is a methodological strength that could generalize beyond the EPL illustration.

major comments (2)
  1. [Methods / model construction] The linear mixed-effects models for performance ratings and transfer prices (described in the methods section following the abstract) report no fitted coefficients, standard errors, validation metrics, cross-validation results, or out-of-sample performance. Because the subsequent weighted optimization and auction analysis rest directly on these point predictions, the absence of any predictive validation makes it impossible to judge whether the claimed optimal transfers are meaningful or merely artifacts of in-sample fitting.
  2. [Illustration / results] The 2018-19 EPL illustration (final section) presents optimal transfer decisions without any hold-out testing, sensitivity analysis to prediction error, or comparison against baseline strategies. This leaves the central claim—that the framework determines useful transfer activities—unsupported, as errors in the LME predictions would propagate directly into the constrained optimization without demonstrated robustness.
minor comments (2)
  1. [Optimization and auction sections] Notation for the weighted multi-criteria objective and the auction reserve-price distribution could be introduced with explicit equations rather than prose descriptions to improve reproducibility.
  2. [Abstract and methods] The abstract states that models incorporate 'league effects' but the manuscript does not clarify whether these are modeled as fixed or random effects; a short clarification would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We agree that strengthening the validation of the linear mixed-effects models and adding robustness checks to the illustration will improve the paper. We address each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Methods / model construction] The linear mixed-effects models for performance ratings and transfer prices (described in the methods section following the abstract) report no fitted coefficients, standard errors, validation metrics, cross-validation results, or out-of-sample performance. Because the subsequent weighted optimization and auction analysis rest directly on these point predictions, the absence of any predictive validation makes it impossible to judge whether the claimed optimal transfers are meaningful or merely artifacts of in-sample fitting.

    Authors: We acknowledge this limitation in the current version. The models were fitted using standard lme4 procedures with player, team, and league random effects, but we omitted the detailed coefficient tables and validation statistics to focus on the overall framework. In the revised manuscript we will add: (i) the full set of fixed-effect coefficients and standard errors for both the performance-rating and transfer-price models, (ii) marginal and conditional R² values, and (iii) k-fold cross-validation results (k=5) together with out-of-sample RMSE and MAE on held-out player-seasons. These additions will allow readers to assess the reliability of the point predictions that feed into the optimisation stage. revision: yes

  2. Referee: [Illustration / results] The 2018-19 EPL illustration (final section) presents optimal transfer decisions without any hold-out testing, sensitivity analysis to prediction error, or comparison against baseline strategies. This leaves the central claim—that the framework determines useful transfer activities—unsupported, as errors in the LME predictions would propagate directly into the constrained optimization without demonstrated robustness.

    Authors: We agree that the single-season illustration would be strengthened by explicit robustness checks. In the revision we will: (i) conduct a sensitivity analysis by adding Gaussian noise to the predicted ratings and prices at levels consistent with the cross-validation RMSE and re-running the optimisation to show how the selected transfer set changes; (ii) compare the framework’s recommended transfers against two simple baselines (highest predicted rating within budget; highest predicted rating-per-price within budget) and report the resulting squad-value improvement; and (iii) note the limitation that a true temporal hold-out is not feasible with only one season of data, while outlining how the approach could be validated on multi-season panels in future work. These additions will directly address the propagation of prediction error into the optimisation results. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes a sequential pipeline: fit linear mixed-effects models on historical player data (characteristics, performance, context, league effects) to generate predictions of ratings and prices, then feed those outputs into a separate weighted multi-criteria constrained optimization, and finally embed the decisions in an independent private-value auction model. No equations, definitions, or self-citations are provided that reduce the optimization step to the model fit by construction, rename fitted parameters as predictions, or rely on load-bearing self-citations for uniqueness. The derivation chain therefore remains self-contained against external data and does not collapse into tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are identifiable from the abstract alone; the approach relies on standard statistical assumptions whose details are not supplied.

pith-pipeline@v0.9.0 · 5476 in / 1194 out tokens · 39926 ms · 2026-05-15T02:43:43.112563+00:00 · methodology

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Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    2019 , publisher=

    Saikia, Hemanta and Bhattacharjee, Dibyojyoti and Mukherjee, Diganta and Saikia, Hemanta and Bhattacharjee, Dibyojyoti and Mukherjee, Diganta , journal=. 2019 , publisher=

    work page 2019

  2. [2]

    IEEE signal processing magazine , volume=

    Model-order selection: a review of information criterion rules , author=. IEEE signal processing magazine , volume=. 2004 , publisher=

    work page 2004

  3. [3]

    Proceedings of the 2024 9th International Conference on Machine Learning Technologies , pages=

    Rating prediction of football players using machine learning , author=. Proceedings of the 2024 9th International Conference on Machine Learning Technologies , pages=

    work page 2024

  4. [4]

    Humanities and Social Sciences , volume=

    Transfer market in football: analyzing the top 100 most expensive players , author=. Humanities and Social Sciences , volume=

    work page

  5. [5]

    2021 , publisher=

    Depken, Craig A and Globan, Tomislav , journal=. 2021 , publisher=

    work page 2021

  6. [6]

    Journal of Big Data , year=

    Klaiber, Marco and R. Journal of Big Data , year=

    work page

  7. [7]

    2022 , publisher=

    Poli, Raffaele and Besson, Roger and Ravenel, Lo. 2022 , publisher=

    work page 2022

  8. [8]

    2020 , organization=

    Kulkarni, Apurva and Kamath, Aditya Vidyadhar and Menon, Aadith and Dhatwalia, Prajwal and Rishabh, D and others , booktitle=. 2020 , organization=

    work page 2020

  9. [9]

    2012 , publisher=

    Arel, Barbara and Tomas III, Michael J , journal=. 2012 , publisher=

    work page 2012

  10. [10]

    Journal of Labor Economics , volume=

    Uncertainty, hiring, and subsequent performance: The NFL draft , author=. Journal of Labor Economics , volume=. 2003 , publisher=

    work page 2003

  11. [11]

    South African Statistical Journal , volume=

    Drafts versus auctions in the Indian Premier League: Theory and methods , author=. South African Statistical Journal , volume=. 2011 , publisher=

    work page 2011

  12. [12]

    Journal of Statistical Physics , volume=

    Analysis of the football transfer market network , author=. Journal of Statistical Physics , volume=. 2022 , publisher=

    work page 2022

  13. [13]

    Review of financial economics , volume=

    An option pricing framework for valuation of football players , author=. Review of financial economics , volume=. 2005 , publisher=

    work page 2005

  14. [14]

    Applied Economics Letters , volume=

    Determinants of football transfers , author=. Applied Economics Letters , volume=. 2015 , publisher=

    work page 2015

  15. [15]

    Professional Identities: Policy and Practice in Business and Bureaucracy , volume=

    Matters of the heart: The business of English Rugby League , author=. Professional Identities: Policy and Practice in Business and Bureaucracy , volume=. 2007 , publisher=

    work page 2007

  16. [16]

    Journal of Quantitative Analysis in Sports , volume=

    The football team composition problem: a stochastic programming approach , author=. Journal of Quantitative Analysis in Sports , volume=. 2017 , publisher=

    work page 2017

  17. [17]

    Environment Systems and Decisions , volume=

    Squad Optimizing: Integrating Tactics into Football Team Composition Problem via Integer Programming , author=. Environment Systems and Decisions , volume=. 2025 , publisher=

    work page 2025

  18. [18]

    Norsk IKT-konferanse for forskning og utdanning , year=

    Evaluating the efficiency of the association football transfer market using regression based player ratings , author=. Norsk IKT-konferanse for forskning og utdanning , year=

    work page

  19. [19]

    Top , volume=

    Maximizing performance with an eye on the finances: a chance-constrained model for football transfer market decisions , author=. Top , volume=. 2021 , publisher=

    work page 2021

  20. [20]

    , author=

    Football Player's Performance and Market Value. , author=

    work page

  21. [21]

    Sustainability , volume=

    Money talks: Team variables and player positions that most influence the market value of professional male footballers in Europe , author=. Sustainability , volume=. 2020 , publisher=

    work page 2020

  22. [22]

    International Journal of Economics and Management Studies , volume=

    High value football transfers: a win or loss? Correlation of high value football transfers, with their subsequent performances , author=. International Journal of Economics and Management Studies , volume=

    work page

  23. [23]

    Bulletin of Economic Research , volume=

    The labour market in association football: who gets transferred and for how much? , author=. Bulletin of Economic Research , volume=. 1999 , publisher=

    work page 1999

  24. [24]

    Applied Economics , volume=

    Bargaining in the transfer market: theory and evidence , author=. Applied Economics , volume=. 1993 , publisher=

    work page 1993

  25. [25]

    Journal of Sports Sciences , volume=

    Comparing player rating systems as a metric for assessing individual performance in soccer , author=. Journal of Sports Sciences , volume=. 2025 , publisher=

    work page 2025

  26. [26]

    Journal of Education, Health and Sport , volume=

    Short-and long-term effects of transfers on football clubs' sport performance , author=. Journal of Education, Health and Sport , volume=

    work page

  27. [27]

    European journal of law and economics , volume=

    Marc. European journal of law and economics , volume=. 2016 , publisher=

    work page 2016

  28. [28]

    Journal of Sports Economics , volume=

    Exploring the market of soccer player registrations: An empirical analysis of the difference between transfer fees and estimated players’ inherent value , author=. Journal of Sports Economics , volume=. 2022 , publisher=

    work page 2022

  29. [29]

    Statistics & Probability Letters , volume=

    Asymptotics of sums of lognormal random variables with Gaussian copula , author=. Statistics & Probability Letters , volume=. 2008 , publisher=

    work page 2008

  30. [30]

    Applied Soft Computing , volume=

    Multi-objective optimization and decision making approaches to cricket team selection , author=. Applied Soft Computing , volume=. 2013 , publisher=

    work page 2013

  31. [31]

    2011 , publisher=

    Singh, Sanjeet and Gupta, Shaurya and Gupta, Vibhor , journal=. 2011 , publisher=

    work page 2011

  32. [32]

    2022 , publisher=

    Malhotra, Gaurav , journal=. 2022 , publisher=

    work page 2022

  33. [33]

    2023 , publisher=

    Chittibabu, Vaseekaran and Sundararaman, Malolan , journal=. 2023 , publisher=

    work page 2023

  34. [34]

    International Game Theory Review , volume=

    Sequential Auctions with Multiple Synergies , author=. International Game Theory Review , volume=. 2023 , publisher=

    work page 2023

  35. [35]

    Australian & New Zealand Journal of Statistics , volume=

    Measuring the values of cricket players , author=. Australian & New Zealand Journal of Statistics , volume=. 2021 , publisher=

    work page 2021

  36. [36]

    Journal of Higher Education Athletics & Innovation , volume=

    The Unintended Consequences of the NCAA College Athletics Transfer Portal , author=. Journal of Higher Education Athletics & Innovation , volume=

    work page

  37. [37]

    Journal of Statistical Software , author=

    GA: A Package for Genetic Algorithms in R , volume=. Journal of Statistical Software , author=. 2013 , pages=. doi:10.18637/jss.v053.i04 , abstract=

    work page doi:10.18637/jss.v053.i04 2013

  38. [38]

    2024 , url =

    R: A Language and Environment for Statistical Computing , author =. 2024 , url =

    work page 2024

  39. [39]

    Marlow, N. A. , journal=. A normal limit theorem for power sums of independent random variables , year=

    work page

  40. [40]

    Journal of Sports Analytics , volume=

    When do soccer players peak? A note , author=. Journal of Sports Analytics , volume=. 2016 , publisher=

    work page 2016

  41. [41]

    Brockhoff and Rune H

    Alexandra Kuznetsova and Per B. Brockhoff and Rune H. B. Christensen , journal =. 2017 , volume =

    work page 2017

  42. [42]

    Journal of Physical Education and Sport , year =

    Adam Metelski , title =. Journal of Physical Education and Sport , year =

    work page

  43. [43]

    Beyond Crowd Judgments: Data-Driven Estimation of Market Value in Association Football , journal =

    Oliver M. Beyond Crowd Judgments: Data-Driven Estimation of Market Value in Association Football , journal =. 2017 , volume =

    work page 2017

  44. [44]

    McHale and Benjamin Holmes , title =

    Ian G. McHale and Benjamin Holmes , title =. European Journal of Operational Research , year =

    work page

  45. [45]

    Statistical Modeling of Football Players' Transfer Fees Worldwide , journal =

    Raffaele Poli and Roger Besson and Lo. Statistical Modeling of Football Players' Transfer Fees Worldwide , journal =. 2024 , volume =

    work page 2024

  46. [46]

    Managing Sport and Leisure , year =

    Christer Thrane , title =. Managing Sport and Leisure , year =

    work page

  47. [47]

    Determinants of Football Players' Valuation: A Systematic Review , journal =

    Maxence Franceschi and Jean. Determinants of Football Players' Valuation: A Systematic Review , journal =. 2024 , volume =

    work page 2024

  48. [48]

    A Decision Model to Value Football Player Investments under Uncertainty , journal =

    Florian Follert and Werner Glei. A Decision Model to Value Football Player Investments under Uncertainty , journal =. 2024 , volume =

    work page 2024

  49. [49]

    International Journal of Financial Studies , year =

    Danielle Khalife and Jad Yammine and Elias Chbat and Chamseddine Zaki and Nada Jabbour Al Maalouf , title =. International Journal of Financial Studies , year =

    work page

  50. [50]

    International Journal of Financial Studies , year =

    Jing Liu , title =. International Journal of Financial Studies , year =

    work page

  51. [51]

    Hill and James Skinner and Anna Grosman , title =

    Danny F. Hill and James Skinner and Anna Grosman , title =. Managing Sport and Leisure , year =

    work page

  52. [52]

    First Price Auctions in the Asymmetric N Bidder Case , urldate =

    Bernard Lebrun , journal =. First Price Auctions in the Asymmetric N Bidder Case , urldate =

    work page

  53. [53]

    Timber Sale Auctions with Random Reserve Prices , urldate =

    Tong Li and Isabelle Perrigne , journal =. Timber Sale Auctions with Random Reserve Prices , urldate =

    work page

  54. [54]

    , title =

    Kotowski, Maciej H. , title =. Theoretical Economics , volume =. doi:https://doi.org/10.3982/TE1824 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/TE1824 , abstract =

    work page doi:10.3982/te1824

  55. [55]

    Numerical simulations of asymmetric first-price auctions , journal =

    Gadi Fibich and Nir Gavish , keywords =. Numerical simulations of asymmetric first-price auctions , journal =. 2011 , issn =. doi:https://doi.org/10.1016/j.geb.2011.02.010 , url =

    work page doi:10.1016/j.geb.2011.02.010 2011

  56. [56]

    Myerson , journal =

    Roger B. Myerson , journal =. Optimal Auction Design , urldate =

    work page

  57. [57]

    Equilibrium in Sealed High Bid Auctions , urldate =

    Eric Maskin and John Riley , journal =. Equilibrium in Sealed High Bid Auctions , urldate =

    work page

  58. [58]

    Asymmetric Auctions , urldate =

    Eric Maskin and John Riley , journal =. Asymmetric Auctions , urldate =

    work page

  59. [59]

    2026 , note =

    nleqslv: Solve Systems of Nonlinear Equations , author =. 2026 , note =

    work page 2026

  60. [60]

    Ascher and J

    U. Ascher and J. Christiansen and R. D. Russell , journal =. A Collocation Solver for Mixed Order Systems of Boundary Value Problems , urldate =

    work page

  61. [61]

    Mathematics of Computation , year=

    A Class of Methods for Solving Nonlinear Simultaneous Equations , author=. Mathematics of Computation , year=

    work page

  62. [62]

    McHale and Javier López Peña , keywords =

    Tarak Kharrat and Ian G. McHale and Javier López Peña , keywords =. Plus–minus player ratings for soccer , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.ejor.2019.11.026 , url =

    work page doi:10.1016/j.ejor.2019.11.026 2020

  63. [63]

    Baker and Ian G

    Rose D. Baker and Ian G. McHale , keywords =. Time-varying ratings for international football teams , journal =. 2018 , issn =. doi:https://doi.org/10.1016/j.ejor.2017.11.042 , url =

    work page doi:10.1016/j.ejor.2017.11.042 2018

  64. [64]

    The wisdom of crowds and transfer market values , journal =

    Dennis Coates and Petr Parshakov , keywords =. The wisdom of crowds and transfer market values , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.ejor.2021.10.046 , url =

    work page doi:10.1016/j.ejor.2021.10.046 2022

  65. [65]

    Russell, Stuart and Norvig, Peter , edition = 3, publisher =

    work page

  66. [66]

    Optimization by Simulated Annealing,

    S. Kirkpatrick and C. D. Gelatt and M. P. Vecchi , title =. Science , volume =. 1983 , doi =. https://www.science.org/doi/pdf/10.1126/science.220.4598.671 , abstract =

    work page doi:10.1126/science.220.4598.671 1983