The reviewed record of science sign in
Pith

arxiv: 2607.05845 · v1 · pith:KY2APWYT · submitted 2026-07-07 · physics.soc-ph

A behavioral principle underlying attacker-defender interactions in soccer

Issei Yamazaki , Hirotaka Goto , Kojiro Otoguro , Hiraku Nishimori , Masashi Shiraishi , Takuma Narizuka This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 19:39 UTCgrok-4.5pith:KY2APWYTrecord.jsonopen to challenge →

classification physics.soc-ph PACS 89.65.Ef87.23.Ge05.45.-a45.50.Dd
keywords soccerattacker-defender interactionsrelative-speed minimizationpursuit-evasionone-on-one open playplayer tracking databehavioral principleinvasion team sports
0
0 comments X

The pith

Defenders minimize relative speed to the attacker; attackers move first to block that plan

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

This paper asks what short-term goals actually guide player movements in open-play soccer when an attacker with the ball faces a defender. By building a simple mathematical model of one-on-one interactions and testing it on tracking data from 306 professional matches, the authors argue that a single behavioral principle organizes these encounters: the defender tries to reduce their future speed relative to the attacker, while the attacker starts moving so as to spoil that plan before it works. Relative-speed minimization accounts for the observed patterns of approach angles, interceptions, and successful dribbles without needing elaborate team tactics or long-horizon strategy. Because the principle depends little on soccer-specific rules, the same logic may describe pursuit-evasion in other invasion sports and beyond. A sympathetic reader cares because it replaces vague talk of "pressure" or "space" with a concrete, testable objective that both sides share.

Core claim

A single behavioral principle—relative-speed minimization by the defender, with the attacker initiating movements to preempt that objective—provides a consistent and unified account of empirical one-on-one attacker-defender interactions in open-play soccer, as shown by synthesizing a mathematical model with analysis of 306 professional games.

What carries the argument

The relative-speed-minimization principle: the defender continuously chooses accelerations that reduce the projected future relative velocity to the attacker, while the attacker anticipates and initiates motion that raises that same projected relative speed. The mathematical model of short-horizon one-on-one pursuit-evasion makes this objective explicit and generates the geometric predictions tested against the tracking data.

Load-bearing premise

That short-horizon one-on-one open-play moments, with the attacker dribbling the ball, are the right unit of analysis and that team tactics or longer-term goals can be set aside without changing the inferred principle.

What would settle it

If high-resolution tracking from professional matches shows that defender accelerations systematically increase rather than decrease projected relative speed to the attacker, or that successful attackers do not move in ways that raise that projected relative speed, the principle fails.

Figures

Figures reproduced from arXiv: 2607.05845 by Hiraku Nishimori, Hirotaka Goto, Issei Yamazaki, Kojiro Otoguro, Masashi Shiraishi, Takuma Narizuka.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagrams in the angle space ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Soccer is widely popular for its simple rules and complex yet coordinated play that unfolds on the pitch. Nevertheless, the fundamental mechanisms governing such play are not well understood: what shapes player interactions on the pitch? What short-term goals guide players' decisions about their movements over the next few seconds? We address these questions by focusing on one-on-one settings in open play, in which the attacker, in possession of the ball and typically dribbling, faces a defender aiming to stop or delay the attacker's actions over a short period. Here we develop a mathematical model of attacker-defender interactions and analyze 306 professional soccer games. Synthesizing the large-scale dataset with an analysis of the model reveals a simple behavioral principle that may underlie these interactions: the defender seeks to minimize their future relative speed to the attacker, whereas the attacker initiates their movements to preempt the defender's objective. This principle, relative-speed minimization, provides a consistent and unified account of the empirical data. Since our framework depends little on soccer-specific details, this principle may govern diverse pursuit-evasion scenarios as well as other invasion team sports.

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

3 major / 0 minor

Summary. The manuscript develops a mathematical model of short-horizon one-on-one attacker–defender interactions in open-play soccer and synthesizes it with tracking data from 306 professional matches. It claims that a single behavioral principle—defender minimization of future relative speed to the attacker, with the attacker initiating movements that preempt that objective—provides a consistent, unified account of observed trajectories, and that the principle may extend to other pursuit–evasion and invasion-sport settings.

Significance. If the identification holds, the work would supply a parsimonious, largely soccer-agnostic mechanism for short-horizon 1v1 dynamics and a reusable model–data pipeline for invasion sports. Strengths include a large professional dataset (306 games), an explicit dynamical model, and a clear falsifiable framing of defender and attacker objectives. The significance hinges on whether relative-speed minimization is uniquely preferred over close alternatives (distance, intercept-time, pure pursuit, constant bearing) and whether attacker “preemption” is better predicted by a nested defender model than by attacker-centric goals alone.

major comments (3)
  1. The central claim requires that relative-speed minimization uniquely accounts for trajectories. The manuscript must report rigorous model comparison against plausible alternatives that produce similar short-horizon paths (minimize distance to attacker, minimize time-to-intercept, pure pursuit, constant-bearing). Consistency with one objective is not identification; without rejection of close alternatives the “principle” remains a descriptive fit rather than the governing mechanism.
  2. The attacker “preemption” half is load-bearing: attacker movements must be shown to be better predicted by a nested model of the defender’s relative-speed objective than by attacker-centric goals alone (progress toward goal, create space). Nested likelihood or out-of-sample prediction comparisons are needed; otherwise the dual-agent claim is under-supported.
  3. Selection of 1v1 episodes and the precise definition of “future relative speed” (horizon, assumed dynamics, free parameters) condition uniqueness. The paper should state data-selection rules, horizon choice, and any free parameters explicitly, and show that the ranking of objectives is robust to those choices rather than absorbed by a single-parameter family.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments correctly identify that our central claim is an identification claim, not merely a descriptive fit, and that the manuscript as submitted does not yet supply the comparisons and robustness checks needed to support uniqueness of relative-speed minimization or the nested character of attacker preemption. We agree with this framing and will revise accordingly: we will add explicit model comparisons against the listed alternative defender objectives, nested and out-of-sample tests of the attacker side against attacker-centric goals, and a fully specified account of episode selection, horizon, dynamics, and free parameters together with robustness of the ranking. We believe these revisions address the significance concerns and convert the present consistency evidence into a properly identified behavioral principle.

read point-by-point responses

  1. Referee: The central claim requires that relative-speed minimization uniquely accounts for trajectories. The manuscript must report rigorous model comparison against plausible alternatives that produce similar short-horizon paths (minimize distance to attacker, minimize time-to-intercept, pure pursuit, constant-bearing). Consistency with one objective is not identification; without rejection of close alternatives the “principle” remains a descriptive fit rather than the governing mechanism.

    Authors: We agree. The submitted manuscript shows that defender trajectories are consistent with minimization of future relative speed and that this objective unifies several empirical regularities, but it does not formally reject close alternatives. Consistency alone is not identification. In revision we will implement a common short-horizon control framework in which the defender’s instantaneous heading (and, where identifiable, speed) is chosen to minimize each candidate objective—future relative speed, distance to the attacker, time-to-intercept, pure pursuit, and constant bearing—under the same assumed dynamics and horizon. We will report likelihoods (or equivalent trajectory-error scores) on held-out 1v1 segments, pairwise model comparisons, and residual diagnostics that show whether relative-speed minimization is preferred and whether the alternatives leave systematic structure that relative-speed minimization removes. Where two objectives are observationally close on short horizons we will state that explicitly rather than claim uniqueness by assertion. This directly converts the present descriptive fit into a comparative identification result. revision: yes

  2. Referee: The attacker “preemption” half is load-bearing: attacker movements must be shown to be better predicted by a nested model of the defender’s relative-speed objective than by attacker-centric goals alone (progress toward goal, create space). Nested likelihood or out-of-sample prediction comparisons are needed; otherwise the dual-agent claim is under-supported.

    Authors: We agree that the dual-agent claim is load-bearing and that the submitted evidence for preemption is weaker than for the defender side. The manuscript argues that attackers initiate movements that spoil the defender’s relative-speed objective, but it does not yet pit a nested defender model against purely attacker-centric objectives (progress toward goal, space creation, or simple ball-progress heuristics) in a predictive comparison. In revision we will (i) define an attacker policy that chooses short-horizon actions to minimize the defender’s attainable relative-speed objective (nested), (ii) define parallel attacker-centric policies that ignore the defender’s objective, and (iii) compare nested versus attacker-centric models by in-sample likelihood and out-of-sample trajectory prediction on held-out episodes. We will report whether nesting improves prediction beyond attacker-centric goals alone, and where it does not we will qualify the preemption claim. This supplies the nested-likelihood / predictive test the referee requests. revision: yes

  3. Referee: Selection of 1v1 episodes and the precise definition of “future relative speed” (horizon, assumed dynamics, free parameters) condition uniqueness. The paper should state data-selection rules, horizon choice, and any free parameters explicitly, and show that the ranking of objectives is robust to those choices rather than absorbed by a single-parameter family.

    Authors: We agree that uniqueness can be conditioned on selection rules and on the operational definition of future relative speed. The submitted text does not state these choices with enough precision or demonstrate robustness of the objective ranking. In revision we will (i) give explicit, reproducible rules for extracting 1v1 open-play episodes (spatial isolation criteria, possession continuity, duration bounds, exclusion of set pieces and multi-defender presses), (ii) define future relative speed with a stated prediction horizon, assumed kinematics (e.g., constant-velocity or bounded-acceleration continuation), and any free parameters (horizon length, speed bounds, discounting), and (iii) re-estimate the defender and attacker comparisons of Comments 1–2 across a grid of horizons, kinematic assumptions, and selection thresholds, reporting whether relative-speed minimization remains preferred and whether preemption retains predictive value. If the ranking collapses under plausible alternatives we will report that failure rather than absorb it into a single-parameter family. These additions make the identification claim conditional on transparent, tested choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity: relative-speed minimization is an independently stated objective whose predictions are tested against empirical trajectories rather than forced by definition or self-citation.

full rationale

The paper proposes a behavioral principle (defender minimizes future relative speed; attacker preempts that objective) and synthesizes a mathematical model with analysis of 306 professional games. The derivation is not self-definitional: the objective is stated as a hypothesis and then checked for consistency with observed one-on-one trajectories, not defined from those trajectories. There is no evidence in the abstract or framing that a fitted parameter is relabeled as a prediction of a closely related quantity, nor that uniqueness is imported solely via self-citation of an unverified theorem. Self-citation, if any, is not load-bearing for the central claim. Residual scientific risks (whether alternatives such as distance or intercept-time minimization were rigorously rejected; whether the short-horizon 1v1 unit of analysis is the right frame) are identification/correctness concerns, not circularity. Under the stated rules—only flag when a specific reduction can be quoted—the honest finding is no significant circularity (score 0). The reader’s residual concern that the objective might have been chosen post hoc is noted but cannot be elevated to circularity without a quoted construction that forces the result.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

Abstract-only review: free parameters, axioms, and invented entities cannot be exhaustively extracted from equations or methods. The ledger records what the abstract itself commits to: a mathematical model of attacker-defender interactions, the one-on-one open-play framing, and the relative-speed objective as the organizing principle. No fitted constants or new physical entities are named in the abstract.

free parameters (1)
  • model parameters (unspecified)
    Any pursuit-evasion model of this type typically has time horizons, speed/acceleration bounds, or weights; the abstract does not state whether they are fixed a priori or fitted to the 306-game data.
axioms (3)
  • domain assumption One-on-one open-play attacker-defender interactions are a sufficient unit for identifying the short-term behavioral principle governing player movements.
    Abstract frames the study around this setting and sets aside broader team tactics as outside the short-horizon focus.
  • domain assumption Defender and attacker objectives can be captured by a simple mathematical model of relative motion over a few seconds.
    Abstract states a mathematical model of attacker-defender interactions is developed and then synthesized with data.
  • ad hoc to paper Relative-speed minimization (defender) and preemption of that objective (attacker) are the operative short-term goals.
    This is the paper's proposed principle; treated as the central modeling choice rather than a standard math axiom.

pith-pipeline@v0.9.1-grok · 6316 in / 2185 out tokens · 31061 ms · 2026-07-08T19:39:18.396660+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    American Philatelic Society, The Miracle of Bern: West Germany and the 1954 World Cup (2022), accessed March 2, 2026

    work page 1954

  2. [2]

    Stephens, No hiding place for lazy soccer defenders, Physics15, s144 (2022)

    M. Stephens, No hiding place for lazy soccer defenders, Physics15, s144 (2022)

    work page 2022

  3. [3]

    R. S. Mendes, L. C. Malacarne, and C. Anteneodo, Statis- tics of football dynamics, Eur. Phys. J. B57, 357 (2007)

    work page 2007

  4. [4]

    Clauset, M

    A. Clauset, M. Kogan, and S. Redner, Safe leads and lead changes in competitive team sports, Phys. Rev. E 91, 062815 (2015)

    work page 2015

  5. [5]

    Marcelino, J

    R. Marcelino, J. Sampaio, G. Amichay, B. Gon¸ calves, I. D. Couzin, and M. Nagy, Collective movement analysis reveals coordination tactics of team players in football matches, Chaos Solitons Fractals138, 109831 (2020)

    work page 2020

  6. [6]

    Yamamoto, P´ olya urn model for analysis of football passes, Phys

    K. Yamamoto, P´ olya urn model for analysis of football passes, Phys. Rev. E112, L062303 (2025)

    work page 2025

  7. [7]

    Gudmundsson and M

    J. Gudmundsson and M. Horton, Spatio-temporal anal- ysis of team sports, ACM Comput. Surv.50, 1 (2017)

    work page 2017

  8. [8]

    Fujii, T

    K. Fujii, T. Kawasaki, Y. Inaba, and Y. Kawahara, Pre- diction and classification in equation-free collective mo- tion dynamics, PLoS Comput. Biol.14, e1006545 (2018)

    work page 2018

  9. [9]

    Koudela, T

    H. Koudela, T. M. Schaerf, T. Lathlean, A. Murphy, and M. Welch, Investigating the emergence of collective states within rugby sevens gameplay, J. Sports Sci.43, 48 (2025)

    work page 2025

  10. [10]

    Merritt and A

    S. Merritt and A. Clauset, Scoring dynamics across pro- fessional team sports: tempo, balance and predictability, EPJ Data Sci.3, 4 (2014)

    work page 2014

  11. [11]

    Ara´ ujo and K

    D. Ara´ ujo and K. Davids, Team synergies in sport: the- ory and measures, Front. Psychol.7, 1449 (2016)

    work page 2016

  12. [12]

    Sarlis and C

    V. Sarlis and C. Tjortjis, Sports analytics—evaluation of basketball players and team performance, Inf. Syst.93, 101562 (2020)

    work page 2020

  13. [13]

    Bai and X

    Z. Bai and X. Bai, Sports big data: management, anal- ysis, applications, and challenges, Complexity2021, 6676297 (2021)

    work page 2021

  14. [14]

    Yamamoto and T

    K. Yamamoto and T. Narizuka, Preferential model for the evolution of pass networks in ball sports, Phys. Rev. E103, 032302 (2021)

    work page 2021

  15. [15]

    Chacoma and O

    A. Chacoma and O. V. Billoni, Simple mechanism rules the dynamics of volleyball, J. Phys. Complex.3, 035006 (2022)

    work page 2022

  16. [16]

    Chacoma and O

    A. Chacoma and O. V. Billoni, Probabilistic model for padel games dynamics, Chaos Solitons Fractals174, 113784 (2023)

    work page 2023

  17. [17]

    Nguyen, R

    Q. Nguyen, R. Jiang, M. Ellingwood, and R. Yurko, Frac- tional tackles: leveraging player tracking data for within- play tackling evaluation in american football, Sci. Rep. 15, 2148 (2025)

    work page 2025

  18. [18]

    Li, Machine learning-based analysis of defensive strate- gies in basketball using player movement data, Sci

    J. Li, Machine learning-based analysis of defensive strate- gies in basketball using player movement data, Sci. Rep. 15, 13887 (2025)

    work page 2025

  19. [19]

    Parker, G

    S. Parker, G. Duthie, and S. Robertson, A framework for player movement analysis in team sports, Front. Sports Act. Living6, 1375513 (2024)

    work page 2024

  20. [20]

    Cheng, A

    Y.-S. Cheng, A. Y.-C. Chang, and K. Doya, Information- theoretical analysis of team dynamics in football matches, Entropy27, 224 (2025)

    work page 2025

  21. [21]

    Nakahara, K

    H. Nakahara, K. Tsutsui, K. Takeda, and K. Fujii, Ac- tion valuation of on- and off-ball soccer players based on multi-agent deep reinforcement learning, IEEE Access 11, 131237 (2023)

    work page 2023

  22. [22]

    Fujii,Machine learning in sports: open approach for next play analytics(Springer Nature, 2025)

    K. Fujii,Machine learning in sports: open approach for next play analytics(Springer Nature, 2025)

    work page 2025

  23. [23]

    Ichinose, D

    G. Ichinose, D. Miyagawa, J. Ito, and N. Masuda, Win- ning by hiding behind others: An analysis of speed skat- 9 ing data, PLoS One15, e0237470 (2020)

    work page 2020

  24. [24]

    Yeung and K

    C. Yeung and K. Fujii, A strategic framework for opti- mal decisions in football 1-vs-1 shot-taking situations: an integrated approach of machine learning, theory-based modeling, and game theory, Complex Intell. Syst.10, 5989 (2024)

    work page 2024

  25. [25]

    J. H. Fewell, D. Armbruster, J. Ingraham, A. Petersen, and J. S. Waters, Basketball teams as strategic networks, PLoS One7, e47445 (2012)

    work page 2012

  26. [26]

    Ribeiro, P

    J. Ribeiro, P. Silva, R. Duarte, K. Davids, and J. Gar- ganta, Team sports performance analysed through the lens of social network theory: implications for research and practice, Sports Med.47, 1689 (2017)

    work page 2017

  27. [27]

    Heuer, C

    A. Heuer, C. Mueller, and O. Rubner, Soccer: Is scoring goals a predictable poissonian process?, EPL89, 38007 (2010)

    work page 2010

  28. [28]

    Chacoma, N

    A. Chacoma, N. Almeira, J. I. Perotti, and O. V. Billoni, Stochastic model for football’s collective dynamics, Phys. Rev. E104, 024110 (2021)

    work page 2021

  29. [29]

    Chacoma, O

    A. Chacoma, O. V. Billoni, and M. N. Kuperman, Com- plexity emerges in measures of the marking dynamics in football games, Phys. Rev. E106, 044308 (2022)

    work page 2022

  30. [30]

    Narizuka, K

    T. Narizuka, K. Yamamoto, and Y. Yamazaki, Statistical properties of position-dependent ball-passing networks in football games, Physica A412, 157 (2014)

    work page 2014

  31. [31]

    Ichinose, T

    G. Ichinose, T. Tsuchiya, and S. Watanabe, Robust- ness of football passing networks against continuous node and link removals, Chaos Solitons Fractals147, 110973 (2021)

    work page 2021

  32. [32]

    Morishita, Y

    T. Morishita, Y. Aruga, M. Nakayama, A. Kijima, and H. Shima, Tactical analysis of football games by vector calculus of last-pass performance, Physica A666, 130507 (2025)

    work page 2025

  33. [33]

    Spearman, Beyond expected goals, inProceedings of the 12th MIT Sloan Sports Analytics Conference(2018) pp

    W. Spearman, Beyond expected goals, inProceedings of the 12th MIT Sloan Sports Analytics Conference(2018) pp. 1–17

    work page 2018

  34. [34]

    Narizuka and Y

    T. Narizuka and Y. Yamazaki, Clustering algorithm for formations in football games, Sci. Rep.9, 13172 (2019)

    work page 2019

  35. [35]

    Narizuka, Y

    T. Narizuka, Y. Yamazaki, and K. Takizawa, Space eval- uation in football games via field weighting based on tracking data, Sci. Rep.11, 5509 (2021)

    work page 2021

  36. [36]

    Fernandez and L

    J. Fernandez and L. Bornn, Wide open spaces: A sta- tistical technique for measuring space creation in profes- sional soccer, inSloan sports analytics conference, Vol. 2018 (2018)

    work page 2018

  37. [37]

    J. B. Keller, A theory of competitive running, Phys. To- day26, 42 (1973)

    work page 1973

  38. [38]

    B. R. Fajen and W. H. Warren, Behavioral dynamics of steering, obstacle avoidance, and route selection., J. Exp. Psychol. Hum. Percept. Perform.29, 343 (2003)

    work page 2003

  39. [39]

    Fujimura and K

    A. Fujimura and K. Sugihara, Geometric analysis and quantitative evaluation of sport teamwork, Syst. Com- put. Jpn.36, 49 (2005)

    work page 2005

  40. [40]

    M. A. L´ opez-Felip and H. S. Harrison, The touch & go dribbling model, Chaos Solitons Fractals140, 110186 (2020)

    work page 2020

  41. [41]

    Narizuka, K

    T. Narizuka, K. Takizawa, and Y. Yamazaki, Validation of a motion model for soccer players’ sprint by means of tracking data, Sci. Rep.13, 865 (2023)

    work page 2023

  42. [42]

    Chiappori, S

    P.-A. Chiappori, S. Levitt, and T. Groseclose, Test- ing mixed-strategy equilibria when players are hetero- geneous: The case of penalty kicks in soccer, Am. Econ. Rev.92, 1138 (2002)

    work page 2002

  43. [43]

    Z. Wang, P. Veliˇ ckovi´ c, D. Hennes, N. Tomaˇ sev, L. Prince, M. Kaisers, Y. Bachrach, R. Elie, L. K. Wen- liang, F. Piccinini,et al., TacticAI: an AI assistant for football tactics, Nat. Commun.15, 1906 (2024)

    work page 1906

  44. [44]

    Narizuka and Y

    T. Narizuka and Y. Yamazaki, Statistical properties for directional alignment and chasing of players in football games, EPL116, 68001 (2016)

    work page 2016

  45. [45]

    Pappalardo, P

    L. Pappalardo, P. Cintia, A. Rossi, E. Massucco, P. Fer- ragina, D. Pedreschi, and F. Giannotti, A public data set of spatio-temporal match events in soccer competitions, Sci. Data6, 236 (2019)

    work page 2019

  46. [46]

    Bassek, R

    M. Bassek, R. Rein, H. Weber, and D. Memmert, An integrated dataset of spatiotemporal and event data in elite soccer, Sci. Data12, 195 (2025)

    work page 2025

  47. [47]

    F. R. Goes, L. A. Meerhoff, M. J. O. Bueno, D. M. Rodrigues, F. A. Moura, M. S. Brink, M. T. Elferink- Gemser, A. J. Knobbe, S. A. Cunha, R. S. Torres,et al., Unlocking the potential of big data to support tactical performance analysis in professional soccer: A system- atic review, Eur. J. Sport Sci.21, 481 (2021)

    work page 2021

  48. [48]

    Brink, S

    L. Brink, S. K. Ha, J. Snowdon, F. Vidal-Codina, B. Rauch, F. Wang, D. Wu, M. A. L´ opez-Felip, C. Clanet, and A. E. Hosoi, Measuring skill via player dynamics in football dribbling, Sci. Rep.13, 19004 (2023)

    work page 2023

  49. [49]

    Kamimura and T

    A. Kamimura and T. Ohira, Group chase and escape, New J. Phys.12, 053013 (2010)

    work page 2010

  50. [50]

    P. J. Nahin,Chases and escapes: the mathematics of pur- suit and evasion(Princeton University Press, 2012)

    work page 2012

  51. [51]

    Tsutsui, M

    K. Tsutsui, M. Shinya, and K. Kudo, Underlying struc- ture in the dynamics of chase and escape interactions, Sci. Rep.9, 15051 (2019)

    work page 2019

  52. [52]

    B. R. Fajen and W. H. Warren, Behavioral dynamics of intercepting a moving target, Exp. Brain Res.180, 303 (2007)

    work page 2007

  53. [53]

    Domenici, J

    P. Domenici, J. M. Blagburn, and J. P. Bacon, Animal escapology I: theoretical issues and emerging trends in escape trajectories, J. Exp. Biol.214, 2463 (2011)

    work page 2011

  54. [54]

    Kawabata, H

    Y. Kawabata, H. Akada, K. Shimatani, G. N. Nishihara, H. Kimura, N. Nishiumi, and P. Domenici, Multiple pre- ferred escape trajectories are explained by a geometric model incorporating prey’s turn and predator attack end- point, eLife12, e77699 (2023)

    work page 2023

  55. [55]

    Y. Jiao, B. Colvert, Y. Man, M. J. McHenry, and E. Kanso, Evaluating evasion strategies in zebrafish lar- vae, Proc. Natl. Acad. Sci. U.S.A.120, e2218909120 (2023)

    work page 2023

  56. [56]

    A. W. Szopa-Comley and C. C. Ioannou, Responsive robotic prey reveal how predators adapt to predictabil- ity in escape tactics, Proc. Natl. Acad. Sci. U.S.A.119, e2117858119 (2022)

    work page 2022

  57. [57]

    Andrienko, N

    G. Andrienko, N. Andrienko, G. Budziak, J. Dykes, G. Fuchs, T. von Landesberger, and H. Weber, Visual analysis of pressure in football, Data Min. Knowl. Dis- cov.31, 1793 (2017)

    work page 2017

  58. [58]

    Yokoyama, H

    K. Yokoyama, H. Shima, K. Fujii, N. Tabuchi, and Y. Ya- mamoto, Social forces for team coordination in ball pos- session game, Phys. Rev. E97, 022410 (2018)

    work page 2018

  59. [59]

    Radicchi, Universality, limits and predictability of gold-medal performances at the olympic games, PLoS One7, e40335 (2012)

    F. Radicchi, Universality, limits and predictability of gold-medal performances at the olympic games, PLoS One7, e40335 (2012)

    work page 2012

  60. [60]

    Di Marzio, A

    M. Di Marzio, A. Panzera, and C. C. Taylor, Kernel den- sity estimation on the torus, J. Stat. Plann. Inference 141, 2156 (2011). 10 IV. METHODS A. Data Our dataset was provided by DataStadium Inc., in- cluding both event and tracking data from 306 games in the 2023 season of the J1 League (the premier division of the professional soccer league in Japan)....

    work page 2011

  61. [61]

    The event duration exceeds 0.5 s

    work page

  62. [62]

    The attacker and defender both have net displacements exceeding 5 m between the start and end of the event

    work page

  63. [63]

    The attacker is not a goalkeeper

    work page

  64. [64]

    The angle betweenr=r d −r a and the vector from the attacker’s position to the opponent’s goal is less than 60◦ at the start of the event

    work page

  65. [65]

    The nearest opposing player remains unchanged throughout the event. These criteria are motivated as follows: interactions between the attacker and defender should occur over sufficiently large temporal and spatial scales so that trajectory fitting is feasible and meaningful (Conditions 1 and 2). Goalkeepers may behave differently and therefore are not inc...

    work page 2016

  66. [66]

    Brink, S

    L. Brink, S. K. Ha, J. Snowdon, F. Vidal-Codina, B. Rauch, F. Wang, D. Wu, M. A. L´ opez-Felip, C. Clanet, and A. E. Hosoi, Sci. Rep.13, 19004 (2023)

    work page 2023

  67. [67]

    Gomez and J.-P

    S. Gomez and J.-P. Hennart,Advances in optimization and numerical analysis, Vol. 275 (Springer Science & Business Media, 2013)

    work page 2013