Zhe Wang (zhewang@google.com), Petar Veličković (petarv@google.com), Karl
Tuyls<EMAIL_ADDRESS>
# TacticAI: an AI assistant for football tactics

Zhe Wang  Contributed equally to this work  Google DeepMind  Petar Veličković
Contributed equally to this work  Google DeepMind  Daniel Hennes  Contributed
equally to this work  Google DeepMind  Nenad Tomašev  Google DeepMind  Laurel
Prince  Google DeepMind  Michael Kaisers  Google DeepMind  Yoram Bachrach
Google DeepMind  Romuald Elie  Google DeepMind  Li Kevin Wenliang  Google
DeepMind  Federico Piccinini  Google DeepMind  William Spearman  Liverpool FC
Ian Graham  Work completed while at Liverpool FC  Jerome Connor  Google
DeepMind  Yi Yang  Google DeepMind  Adrià Recasens  Google DeepMind  Mina Khan
Google DeepMind  Nathalie Beauguerlange  Google DeepMind  Pablo Sprechmann
Google DeepMind  Pol Moreno  Google DeepMind  Nicolas Heess  Google DeepMind
Michael Bowling  University of Alberta  Demis Hassabis  Google DeepMind  Karl
Tuyls  Google DeepMind

###### Abstract

Identifying key patterns of tactics implemented by rival teams, and developing
effective responses, lies at the heart of modern football. However, doing so
algorithmically remains an open research challenge. To address this unmet
need, we propose TacticAI, an AI football tactics assistant developed and
evaluated in close collaboration with domain experts from Liverpool FC. We
focus on analysing corner kicks, as they offer coaches the most direct
opportunities for interventions and improvements. TacticAI incorporates both a
predictive and a generative component, allowing the coaches to effectively
sample and explore alternative player setups for each corner kick routine and
to select those with the highest predicted likelihood of success. We validate
TacticAI on a number of relevant benchmark tasks: predicting receivers and
shot attempts and recommending player position adjustments. The utility of
TacticAI is validated by a qualitative study conducted with football domain
experts at Liverpool FC. We show that TacticAI’s model suggestions are not
only indistinguishable from real tactics, but also favoured over existing
tactics 90% of the time, and that TacticAI offers an effective corner kick
retrieval system. TacticAI achieves these results despite the limited
availability of gold-standard data, achieving data efficiency through
geometric deep learning.

## Introduction

Association football, or simply _football_ or _soccer_ , is a widely popular
and highly professionalised sport, in which two teams compete to score goals
against each other. As each football team comprises of up to 11 active players
at all times and takes place on a very large pitch (also known as a soccer
field), scoring goals tends to require a significant degree of strategic team-
play. Under the rules codified in the _Laws of the Game_ [20], this
competition has nurtured an evolution of nuanced strategies and tactics,
culminating in modern professional football leagues. In today’s play, data-
driven insights are a key driver in determining the optimal player setups for
each game and developing counter-tactics to maximise the chances of success
[38].

When competing at the highest level the margins are incredibly tight, and it
is increasingly important to be able to capitalise on any opportunity for
creating an advantage on the pitch. To that end, top-tier clubs employ diverse
teams of coaches, analysts and experts, tasked with studying and devising
(counter-)tactics before each game. Several recent methods attempt to improve
tactical coaching and player decision-making through artificial intelligence
(AI) tools, using a wide variety of data types from videos to tracking sensors
and applying diverse algorithms ranging from simple logistic regression to
elaborate neural network architectures. Such methods have been employed to
help predict shot events from videos [16], forecast off-screen movement from
spatio-temporal data [27], determine whether a match is in-play or interrupted
[23], or identify player actions [2].

The execution of agreed-upon plans by players on the pitch is highly dynamic
and imperfect, depending on numerous factors including player fitness and
fatigue, variations in player movement and positioning, weather, the state of
the pitch, and the reaction of the opposing team. In contrast, _set pieces_
provide an opportunity to exert more control on the outcome, as the brief
interruption in play allows the players to reposition according to one of the
practiced and pre-agreed patterns, and make a deliberate attempt towards the
goal. Examples of such set pieces include free kicks, corner kicks, goal
kicks, throw-ins, and penalties [38].

Figure 1: A “bird’s eye” overview of TacticAI. (A), how corner kick situations
are converted to a graph representation. Each player is treated as a node in a
graph, with node, edge and graph features extracted as detailed in the main
text. Then, a graph neural network operates over this graph by performing
_message passing_ ; each node’s representation is updated using the messages
sent to it from its neighbouring nodes. (B), how TacticAI processes a given
corner kick. To ensure that TacticAI’s answers are robust in the face of
horizontal or vertical reflections, all possible combinations of reflections
are applied to the input corner, and these four views are then fed to the core
TacticAI model, where they are able to interact with each other to compute the
final player representations—each “internal blue arrow” corresponds to a
single message passing layer from (A). Once player representations are
computed, they can be used to predict the corner’s receiver, whether a shot
has been taken, as well as assistive adjustments to player positions and
velocities, which increase or decrease the probability of a shot being taken.

Among set pieces, _corner kicks_ are of particular importance, as an
improvement in corner kick execution may substantially modify game outcomes,
and they lend themselves to principled, tactical and detailed analysis. This
is because corner kicks tend to occur _frequently_ in football matches (with
$\sim$10 corners on average taking place in each match [33]), they are taken
from a fixed, _rigid_ position, and they offer an _immediate_ opportunity for
scoring a goal—no other set piece simultaneously satisfies all of the above.
In practice, corner kick routines are determined well ahead of each match,
taking into account the strengths and weaknesses of the opposing team and
their typical tactical deployment. It is for this reason that we focus on
corner kick analysis in particular, and propose TacticAI, an AI football
assistant for supporting the human expert with set piece analysis, and the
development and improvement of corner kick routines.

TacticAI is rooted in learning efficient representations of corner kick
tactics from raw, spatio-temporal player tracking data. It makes efficient use
of this data by representing each corner kick situation as a _graph_ —a
natural representation for modelling relationships between players (Figure
1-A, Table 2), and these player relationships may be of higher importance than
the absolute distances between them on the pitch [1]. Such a graph input is a
natural candidate for graph machine learning models [41], which we employ
within TacticAI to obtain high-dimensional latent player representations.

Uniquely, TacticAI takes advantage of _geometric deep learning_ [5] to
explicitly produce player representations that respect several _symmetries_ of
the football pitch (Figure 1-B). As an illustrative example, we can usually
safely assume that under a horizontal or vertical reflection of the pitch
state, the game situation is equivalent. Geometric deep learning ensures that
TacticAI’s player representations will be identically computed under such
reflections, such that this symmetry does not have to be learnt from data.
This proves to be a valuable addition, as high-quality tracking data is often
limited—with only a few hundred matches played each year in every league. We
provide an in-depth overview of how we employ geometric deep learning in
TacticAI in the Methods section.

From these representations, TacticAI is then able to answer various
_predictive_ questions about the outcomes of a corner—for example, which
player is most likely to make first contact with the ball, or whether a shot
will take place. TacticAI can also be used as a _retrieval_ system—for mining
similar corner kick situations based on the similarity of player
representations—and a generative _recommendation_ system, suggesting
adjustments to player positions and velocities to maximise or minimise
estimated shot probability. Through several experiments within a case study
with domain expert coaches and analysts from Liverpool FC, the results of
which we present in the next section, we obtain clear statistical evidence
that TacticAI readily provides useful, realistic and accurate tactical
suggestions.

## Results and Analysis

To demonstrate the diverse qualities of our approach, we design TacticAI with
three distinct predictive and generative components: _receiver prediction_ ,
_shot prediction_ , and tactic recommendation through _guided generation_ ,
which also correspond to the benchmark tasks for quantitatively evaluating
TacticAI. In addition to providing accurate quantitative insights for corner
kick analysis with its predictive components, the interplay between TacticAI’s
predictive and generative components allows coaches to sample alternative
player setups for each routine of interest, and directly evaluate the possible
outcomes of such alternatives.

We will first describe our quantitative analysis, which demonstrates that
TacticAI’s predictive components are accurate at predicting corner kick
receivers and shot situations on held-out test corners, and that the proposed
player adjustments do not strongly deviate from ground-truth situations.
However, such an analysis only gives an indirect insight into how _useful_
TacticAI would be once deployed. We tackle this question of utility head-on,
and conduct a comprehensive case study in collaboration with our partners at
Liverpool FC—where we directly ask human expert raters to judge the utility of
TacticAI’s predictions and player adjustments. The following sections expand
on the specific results and analysis we have performed.

In what follows, we will describe TacticAI’s components at a minimal level
necessary to understand our evaluation. We defer detailed descriptions of
TacticAI’s components to the Methods section.

Benchmark Task | Graph Learning Task | Node Features | Edge Features | Global Features
---|---|---|---|---
|  | Player positions | Teammate or opponent | None
| Player velocities |  |
Receiver Prediction | Node classification | Player weights |  |
|  | Player heights |  |
|  | Ball possession |  |
Shot Prediction | Graph classification | Same as above | Same as above | Receiver ID
Guided Generation | Node regression | Same as above | Same as above | Shot indicator
|  |  |  | Receiver ID

Table 1: Summary of the features used in the corresponding tasks.

### Benchmarking TacticAI

We evaluate the three components of TacticAI on a relevant benchmark dataset
of corner kicks. Our dataset consists of $7,176$ corner kicks from the
2020–2021 Premier League seasons, which we randomly shuffle and split into a
training (80%) and a test set (20%). As previously mentioned, TacticAI
operates on _graphs_. Accordingly, we represent each corner kick situation as
a graph, where each node corresponds to a player. The features associated with
each node encode the movements (velocities and positions) and simple profiles
(heights and weights) of on-pitch players at the timestamp when the
corresponding corner kick was being taken by the attacking kicker (see the
Methods section), and no information of ball movement was encoded. The graphs
are _fully connected_ ; that is, for every pair of players, we will include
the edge connecting them in the graph. Each of these edges encodes a binary
feature, indicating whether the two players are on opposing teams or not. For
each task, we generated the relevant dataset of node/edge/graph features and
corresponding labels (Table 1 and 2, see the Methods section). The components
were then trained separately with their corresponding corner kick graphs. In
particular, we only employ a minimal set of features to construct the corner
kick graphs, without encoding the movements of the ball nor explicitly
encoding the distances between players into the graphs. We used a consistent
training-test split for all benchmark tasks, as this made it possible to
benchmark not only the individual components, but also their interactions.

Feature | Feature Type | Explanation
---|---|---
Player positions | Node | XY-positions of 22 players on the pitch.
Player velocities | Node | XY-velocities of 22 players on the pitch.
Player weights | Node | Weights of 22 players.
Player heights | Node | Heights of 22 players.
Ball possession | Node | Binary indicator to indicate whether this player
|  | is possessing the ball.
Teammate or opponent | Edge | One-hot encoding to indicate the relationship
|  | between two players.
Receiver ID | Global | One-hot encoding to indicate the index of the receiver.
Shot indicator | Global | Binary indicator to indicate if there was a threatening
|  | shot attempt.
Table 2: Summary of the details of the features used to construct graphs.

### Accurate receiver and shot prediction through geometric deep learning

One of TacticAI’s key predictive models forecasts the receiver out of the 22
on-pitch players. The receiver is defined as the first player touching the
ball after the corner was taken. In our evaluation, all methods used the same
set of features (see the Receiver Prediction entry in Table 1 and the Methods
section). We leveraged the receiver prediction task to benchmark several
different TacticAI base models. Our best performing model—achieving $0.782\pm
0.039$ in top-3 test accuracy after $50,000$ training steps—was a deep graph
attention network [4, 43], leveraging geometric deep learning [5] through the
use of $D_{2}$ group convolutions [7]. We supplement this result with a
detailed ablation study, verifying that both our choice of base architecture
and group convolution yielded significant improvements on the receiver
prediction task (S.Table 2, see the Ablation Study in the Methods section).
Considering that corner kick receiver prediction is a highly challenging task
with many factors that are unseen by our model—including fatigue and fitness
levels, and actual ball trajectory—we consider TacticAI’s top-3 accuracy to
reflect a high level of predictive power, and keep the base TacticAI
architecture fixed for subsequent studies. In addition to this quantitative
evaluation with the evaluation dataset, we also evaluate the performance of
TacticAI’s receiver prediction component in a case study with human raters.
Please see the Case Study section for more details.

For shot prediction, we observe that reusing the base TacticAI architecture to
directly predict shot events—i.e., directly modelling the unconditional
probability $\mathbb{P}(\text{shot})$—proved challenging, only yielding a test
F1 score111The F1 score is the harmonic mean of the precision and recall, and
it is commonly used in binary classification problems over imbalanced
datasets. of $0.52\pm 0.03$. However, given that we already have a potent
receiver predictor, we decided to use its output to give us additional insight
into whether or not a shot had been taken. Hence, we opted to decompose the
probability of taking a shot as

$\mathbb{P}(\text{shot})=\sum_{i\in\text{players}}\mathbb{P}(\text{shot}|\text{receiver}=i)\mathbb{P}(\text{receiver}=i)$
(1)

where $\mathbb{P}(\text{receiver})$ are the probabilities computed by
TacticAI’s receiver prediction system, and
$\mathbb{P}(\text{shot}|\text{receiver})$ models the conditional shot
probability _after a specific player made first contact with the ball_. This
was implemented through providing an additional global feature to indicate the
receiver in the corresponding corner kick (Table 1) while the architecture
otherwise remained the same as that of receiver prediction (S.Figure 2, see
the Methods section). At training time, we feed the ground-truth receiver as
input to the model—at inference time, we attempt every possible receiver,
weighing their contributions using the probabilities given by TacticAI’s
receiver predictor, as per Equation 1. This two-phased approach yielded a
final test F1 score of $0.64\pm 0.02$ for shot prediction, which encodes
significantly more signal than the unconditional shot predictor, especially
considering the many unobservables associated with predicting shot events.

Figure 2: Corner kicks represented in the latent space shaped by TacticAI. We
visualise the latent representations of attacking and defending teams in
$1,024$ corner kicks using $t$-SNE. A latent team embedding in one corner kick
sample is the mean of the latent player representations on the same attacking
((A), (B) and (C)) or defending (D) team. Given the reference corner kick
sample (A), we retrieve another corner kick sample (B) with respect to the
closest distance of their representations in the latent space. We observe that
(A) and (B) are both out-swing corner kicks and share similar patterns of
their attacking tactics, which are highlighted with rectangles having the same
colours, although they bear differences with respect to the absolute positions
and velocities of the players. All the while, the latent representation of an
in-swing attack (C) is distant from both (A) and (B) in the latent space. The
red arrows are only used to demonstrate the difference between in- and out-
swing corner kicks, not the actual ball trajectories.

Moreover, we also observe that, even just through predicting the receivers,
without explicitly classifying any other salient features of corners, TacticAI
learned generalisable representations of the data. Specifically, team setups
with similar tactical patterns tend to cluster together in TacticAI’s latent
space (Figure 2). However, no clear clusters are observed in the raw input
space (S.Figure 1). This indicates that TacticAI can be leveraged as a useful
corner kick retrieval system, and we will present our evaluation of this
hypothesis in the Case Study section.

### Controlled tactic refinement using class-conditional generative models

Equipped with components that are able to potently relate corner kicks with
their various outcomes (e.g. receivers and shot events), we can explore the
use of TacticAI to suggest _adjustments_ of tactics, in order to amplify or
reduce the likelihood of certain outcomes.

Specifically, we aim to produce adjustments to the movements of players on one
of the two teams, including their positions and velocities, which would
maximise or minimise the probability of a shot event, conditioned on the
initial corner setup, consisting of the movements of players on both teams,
and their heights and weights. In particular, although in real-world scenarios
both teams may react simultaneously to the movements of each other, in our
study, we focus on moderate adjustments to player movements, which help to
detect players that are not responding to a tactic properly. Due to this
reason, we simplify the process of tactic refinement through generating the
adjustments for only one team while keeping the other fixed. The way we train
a model for this task is through an _auto-encoding_ objective: we feed the
ground-truth shot outcome (a binary indicator) as an additional graph-level
feature to TacticAI’s model (Table 1), and then have it learn to reconstruct a
probability distribution of the input player coordinates (Figure 1(B), also
see Methods). This autoencoder-based generative model is an individual
component which separates from TacticAI’s predictive systems. All three
systems share the encoder architecture (without sharing parameters), but use
different decoders (see the Methods section). At inference time, we can
instead feed in a _desired_ shot outcome for the given corner setup, and then
sample new positions and velocities for players on one team using this
probability distribution. This setup, in principle, allows for flexible
downstream use, as human coaches can optimise corner kick setups through
generating adjustments conditioned on the specific outcomes of their
interest—e.g., increasing shot probability for the attacking team, decreasing
it for the defending team (Figure 3) or amplifying the chance that a
particular striker receives the ball.

Figure 3: Example of refining a corner kick tactic with TacticAI. TacticAI
makes it possible for human coaches to redesign corner kick tactics in ways
that help maximise the probability of a positive outcome for either the
attacking or the defending team by identifying key players, as well as by
providing temporally coordinated tactic recommendations which take all players
into consideration. As demonstrated in this example, for a corner kick in
which there was a shot attempt in reality, TacticAI can generate a tactically-
adjusted setting in which the shot probability has been reduced, by adjusting
the positioning of the defenders. The suggested defender positions result in
reduced receiver probability for attacking players 2–5 (see bottom row), while
the receiver probability of Attacker 1, who is distant from the goalpost, has
been increased. The model is capable of generating multiple such scenarios.
Coaches can inspect the different options visually and additionally consult
TacticAI’s quantitative analysis of the presented tactics.

We first evaluate the generated adjustments quantitatively, by verifying how
distinguishable they are from real corner kicks using a classifier. To do
this, we synthesised a dataset consisting of 200 corner kick samples and their
corresponding conditionally-generated adjustments. Specifically, for corners
without a shot event, we generated adjustments for the attacking team by
setting the shot event feature to $1$, and vice-versa for the defending team
when a shot event did happen. We found that the real and generated samples
were not distinguishable by an MLP classifier, with an F1 score of $0.53\pm
0.05$, indicating random chance level accuracy. We also evaluated the realism
of the adjustments in a qualitative case study, which we will present in the
following section.

In addition, we leveraged our TacticAI shot predictor to estimate whether the
proposed adjustments were salient. We did this by analysing 100 corner kick
samples in which threatening shots occurred, and then, for each sample,
generated one defensive refinement through setting the shot event feature to
$0$. We observed that the average shot probability significantly decreased,
from $0.75\pm 0.14$ for ground-truth corners to $0.69\pm 0.16$ for adjustments
($z=2.62,p<0.001$). This observation was consistent when testing for attacking
team refinements (shot probability increased from $0.18\pm 0.16$ to $0.31\pm
0.26$ ($z=-4.46,p<0.001$)). Moving beyond this result, we also asked human
raters to assess the utility of TacticAI’s proposed adjustments within our
case study, which we detail next.

### Case study with expert raters

Although quantitative evaluation with well-defined benchmark datasets was
critical for the technical development of TacticAI, the ultimate test of
TacticAI as a football tactic _assistant_ is its practical downstream utility
being recognised by _professionals_ in the industry. To this end, we evaluated
TacticAI through a case study with our partners at Liverpool FC (LFC).
Specifically, we invited a group of five football experts: three data
scientists, one video analyst and one coaching assistant. Each of them
completed four tasks in the case study, which evaluated the utility of
TacticAI’s components from several perspectives. These include: 1) realism of
TacticAI’s generated adjustments, 2) plausibility of TacticAI’s receiver
predictions, 3) effectiveness of TacticAI’s embeddings for retrieving similar
corners, and 4) usefulness of TacticAI’s recommended adjustments. We provide
an overview of our study’s results here, and refer the interested reader to
S.Figures 3, 4 and 5 and the supplementary Case Study Design section (S1.2)
for additional details.

We first simultaneously evaluated the realism of the adjusted corner kicks
generated by TacticAI, and the plausibility of its receiver predictions. Going
through a collection of 50 corner kick samples, we first asked the raters to
classify whether a given sample was real or generated by TacticAI, then they
were asked to identify the most likely receivers in the corner kick sample
(S.Figure 3).

On the task of classifying real and generated samples, first we found that the
raters’ average F1 score of classifying the real vs. generated samples was
only $0.60\pm 0.04$, with individual F1 scores
($F_{1}^{A}=0.54,F_{1}^{B}=0.64,F_{1}^{C}=0.65,F_{1}^{D}=0.62,F_{1}^{E}=0.56$),
indicating that the raters were, in many situations, unable to distinguish
TacticAI’s adjustments from real corners.

Figure 4: Statistical analysis for the case study tasks. In task 1, we tested
the statistical difference between the real corner kick samples and the
synthetic ones generated by TacticAI from two aspects. (A) the distributions
of the ratings of the two types of samples, and (B) the distributions of the
top-3 accuracy of receiver prediction using those samples. No statistical
difference was observed in either cases ((A)($z=0.34,p>0.05$), and
(B)($z=0.968,p>0.05$)). On the other hand, for task 2 (receiver prediction),
we observed a statistically significant difference between the ratings of
different raters, with three clear clusters emerging (C). Specifically, Raters
A and E had similar ratings ($z=1.41,p>0.05$), and Raters B and D also rated
in similar ways ($z=2.53,p>0.05$), while Rater C responded differently from
all other raters. This suggests a good level of variety of the human raters
with respect to their perceptions of corner kicks. In task 3 of identifying
similar corners retrieved in terms of salient strategic setups, there was no
significant difference among the distributions of the ratings by different
raters (D), suggesting a high level of agreement on the usefulness of
TacticAI’s capability of retrieving similar corners ($F_{1,4}=1.01,p>0.1$).
Finally, in task 4, we compared the ratings of the strategic refinements by
the human raters (E), and the raters also agreed on the general effectiveness
of the refinements recommended by TacticAI ($F_{1,4}=0.45,p>0.05$).

The previous evaluation focused on analysing realism detection performance
_across raters_. We also conduct a study which analyses realism detection
_across samples_. Specifically, we assigned ratings for each sample—assigning
$+1$ to a sample if it was identified as real by a human rater, and $0$
otherwise—and computed the average rating for each sample across the five
raters. Importantly, by studying the distribution of ratings, we found that
there was no significant difference between the average ratings assigned to
real and generated corners ($z=0.34,p>0.05$) (Figure 4(A)). Hence, the real
and generated samples were assigned statistically indistinguishable average
ratings by human raters.

For the task of identifying receivers, we rated TacticAI’s predictions with
respect to a rater as $+1$ if at least one of the receivers identified by the
rater appeared in TacticAI’s top-3 predictions, and $0$ otherwise. The average
top-3 accuracy among the human raters was $0.79\pm 0.18$; specifically,
$0.82\pm 0.17$ for the real samples, and $0.77\pm 0.21$ for the generated
ones. These scores closely line up with the accuracy of TacticAI on predicting
receivers for held-out test corners, validating our quantitative study.
Further, after averaging the ratings for receiver prediction sample-wise, we
found no statistically significant difference between the average ratings of
predicting receivers over the real and generated samples ($z=0.968,p>0.05$)
(Figure 4(B)). This indicates that TacticAI was equally performant in
predicting the receivers of real corners and TacticAI-generated adjustments,
and hence may be leveraged for this purpose even in simulated scenarios.

There is a notably high variance in the average receiver prediction rating of
TacticAI. We hypothesise that this is due to the fact that different raters
may choose to focus on different salient features when evaluating the likely
receivers (or even the _amount_ of likely receivers). We set out to validate
this hypothesis by testing the pair-wise similarity of the predictions by the
human raters through running a one-away analysis of variance (ANOVA), followed
by a Tukey’s test. We found that the distributions of the five raters’
predictions were significantly different ($F_{1,4}=14.46,p<0.001$) forming
three clusters (Figure 4(C)). This result indicates that different human
raters—as suggested by their various titles at LFC—may often use very
different leads when suggesting plausible receivers. The fact that TacticAI
manages to retain a high top-3 accuracy in such a setting suggests that it was
able to capture the salient patterns of corner kick strategies, which broadly
align with human raters’ preferences. We will further test this hypothesis in
the third task—of identifying similar corners.

For the third task, we asked the human raters to judge 50 pairs of corners for
their similarity. Each pair consisted of a reference corner and a retrieved
corner, where the retrieved corner was chosen either as the nearest-neighbour
of the reference in terms of their TacticAI latent space representations,
or—as a feature-level heuristic—the cosine similarities of their raw features
(S.Figure 4) in our corner kick dataset. We score the raters’ judgement of a
pair as $+1$ if they considered the corners presented in the case to be
usefully similar, otherwise the pair is scored with $0$. We first computed,
for each rater, the recall with which they have judged a baseline- or
TacticAI-retrieved pair as usefully similar (see Task 3 of the Case Study
Design section (S1.2) in the Supplementary Materials). For TacticAI
retrievals, the average recall across all raters was $0.63\pm 0.09$, and for
the baseline system, the recall was $0.33\pm 0.10$. Secondly, we assess the
statistical difference between the results of the two methods by averaging the
ratings for each reference-retrieval pair, finding that the average rating of
TacticAI retrievals is significantly higher than the average rating of
baseline method retrievals ($z=2.34,p<0.05$). These two results suggest that
TacticAI significantly outperforms the feature-space baseline as a method for
mining similar corners. This indicates that TacticAI is able to extract
salient features from corners which are not trivial to extract from the input
data alone, reinforcing it as a potent tool for discovering opposing team
tactics from available data. Finally, we observed that this task exhibited a
high level of inter-rater agreement for TacticAI-retrieved pairs
($F_{1,4}=1.01,p>0.1$) (Figure 4(D)), suggesting that human raters were
largely in agreement with respect to their assessment of TacticAI’s
performance.

Finally, we evaluated TacticAI’s player adjustment recommendations for their
practical utility. Specifically, each rater was given 50 tactical refinements
together with the corresponding real corner kick setups—see S.Figure 5, and
the Case Study Design section in the Supplementary Materials S1.2. The raters
were then asked to rate each refinement as saliently improving the tactics
($+1$), saliently making them worse ($-1$), or offering no salient differences
($0$). We calculated the average rating assigned by each of the raters (giving
us a value in the range $[-1,1]$ for each rater). The average of these values
across all five raters was $0.7\pm 0.1$. Further, for $45$ of the $50$
situations ($90\%$), the human raters found TacticAI’s suggestion to be
favourable on average (by majority voting). Both of these results indicate
that TacticAI’s recommendations are salient and useful to a downstream
football club practitioner, and we set out to validate this with statistical
tests.

We performed statistical significance testing of the observed positive
ratings. First, for each of the 50 situations, we averaged its ratings across
all five raters, and then ran a $t$-test to assess whether the mean rating was
significantly larger than zero. Indeed, the statistical test indicated that
the tactical adjustments recommended by TacticAI were constructive overall
($t^{\text{avg}}_{49}=9.20,p<0.001$). Secondly, we verified that each of the
five raters individually found TacticAI’s recommendations to be constructive,
running a $t$-test on each of their ratings individually. For all of the five
raters, their average ratings were found to be above zero with statistical
significance
($t^{A}_{49}=5.84,p^{A}<0.001;t^{B}_{49}=7.30,p^{B}<0.001;t^{C}_{49}=7.00,p^{C}<0.001;t^{D}_{49}=6.04,p^{D}<0.001;t^{E}_{49}=7.88,p^{E}<0.001$).
In addition, their ratings also shared a high-level of inter-agreement
($F_{1,4}=0.45,p>0.05$) (Figure 4(E)), suggesting a level of practical
usefulness that is generally recognised by human experts, even though they
represent different backgrounds.

Figure 5: Examples of the tactic refinements recommended by TacticAI. These
examples are selected from our case study with human experts, to illustrate
the breadth of tactical adjustments that TacticAI suggests to teams defending
a corner. The density of the yellow circles coincides with the number of times
that the corresponding change is recognised as constructive by the human
experts. Instead of optimising the movement of one specific player, TacticAI
can recommend the improvements for multiple players in one generation step
through suggesting better positions to block the opposing players, or better
orientations to track them more efficiently. Some specific comments from
expert raters follow. In (A), according to raters, TacticAI suggests more
favourable positions for several defenders, and improved tracking runs for
several others—further, the goalkeeper is positioned more deeply, which is
also beneficial. In (B), TacticAI suggests that the defenders furthest away
from the corner make improved covering runs, which was unanimously deemed
useful, with several other defenders also positioned more favourably. In (C),
TacticAI recommends improved covering runs for a central group of defenders in
the penalty box, which was unanimously considered salient by our raters. And
in (D), TacticAI suggests substantially better tracking runs for two central
defenders, along with a better positioning for two other defenders in the goal
area.

Taking all of these results together, we find TacticAI to possess strong
components for prediction, retrieval, and tactical adjustments on corner
kicks. To illustrate the kinds of salient recommendations by TacticAI, in
Figure 5 we present four examples with high degree of inter-rater agreement.

## Discussion and Conclusions

We have demonstrated an AI assistant for football tactics, and provided
statistical evidence of its efficacy through a comprehensive case study with
expert human raters from Liverpool FC. First, TacticAI is able to accurately
predict the first receiver after a corner kick is taken as well as the
probability of a shot as the direct result of the corner. Second, TacticAI has
been shown to produce plausible tactical variations that improve outcomes in a
salient way, while being indistinguishable from real scenarios by domain
experts. And finally, the system’s latent player representations are a
powerful means to retrieve similar set piece tactics, allowing coaches to
analyse relevant tactics and counter-tactics that have been successful in the
past.

The broader scope of strategy modelling in football has previously been
addressed from various individual angles, such as pass prediction [18, 19,
30], shot prediction [16] or corner kick tactical classification [33].
However, to the best of our knowledge, our work stands out by combining and
evaluating _predictive and generative modelling_ of corner kicks for tactic
development. It also stands out in its method of applying geometric deep
learning, allowing for efficiently incorporating various symmetries of the
football pitch for improved data efficiency. Our method incorporates minimal
domain knowledge, and does not rely on intricate feature engineering—though
its factorised design naturally allows for more intricate feature engineering
approaches, when such features are available.

Our methodology requires the position and velocity estimates of all players at
the time of execution of the corner and subsequent events. Here, we derive
these from high-quality tracking and event data, with data availability from
tracking providers limited to top leagues. Player tracking based on broadcast
video would increase the reach and training data substantially, but would also
likely result in noisier model inputs. While the attention mechanism of GATs
would allow us to perform introspection of the most salient factors
contributing to the model outcome, our method does not explicitly model
exogenous (aleatoric) uncertainty, which would be valuable context for the
football analyst.

While the empirical study of our method’s efficacy has been focused on corner
kicks in association football, it readily generalises to other set pieces
(such as throw-ins, which similarly benefit from similarity retrieval, pass
and/or shot prediction) and other team sports with suspended play situations.
The learned representations and overall framing of TacticAI also lays the
ground for future research to integrate a natural language interface that
enables domain-grounded conversations with the assistant, with the aim to
retrieve particular situations of interest, make predictions for a given
tactical variant, compare and contrast, and guide through an interactive
process to derive tactical suggestions. It is thus our belief that TacticAI
lays the groundwork for the next-generation AI assistant for football.

## Methods

We devised TacticAI as a geometric deep learning pipeline, further expanded in
this section. We process labelled spatio-temporal football data into graph
representations, and train and evaluate on benchmarking tasks cast as
classification or regression. These steps are presented in sequence, followed
by details on the employed computational architecture.

#### Raw corner kick data

The raw dataset consisted of $9,693$ corner kicks collected from 2020–2021
Premier League seasons, which was provided by Liverpool FC. This dataset is
comprised of four separate data sources:

  * •

Spatio-temporal trajectory frames (“tracking data”), which tracked all on-
pitch players and the ball, for each match, at 25 frames per second. In
addition to player positions, their velocities are derived from position data
through filtering. For each corner kick, we only used the frame in which the
kick is being taken as input information.

  * •

Event stream data, which annotated the events or actions (e.g., passes, shots
and goals) that have occurred in the corresponding tracking frames.

  * •

Line-up data for the corresponding games, which recorded the players’
profiles, including their heights, weights and roles.

  * •

Game data, which contained the game days, stadium information, and pitch
length and width in meters.

#### Graph representation and construction

We assumed that we were provided with an input graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with a set of _nodes_ $\mathcal{V}$
and _edges_ $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. Within the
context of football games, we took $\mathcal{V}$ to be the set of twenty-two
players currently on the pitch for both teams, and we set
$\mathcal{E}=\mathcal{V}\times\mathcal{V}$; that is, we assumed all pairs of
players have the potential to interact. Further analyses, leveraging more
specific choices of $\mathcal{E}$, would be an interesting avenue for future
work.

Additionally, we assume that the graph is appropriately featurised.
Specifically, we provide a _node feature matrix_ , ${\bf
X}\in\mathbb{R}^{|\mathcal{V}|\times k}$, an _edge feature tensor_ , ${\bf
E}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{V}|\times l}$, and a _graph
feature vector_ , $\mathbf{g}\in\mathbb{R}^{m}$. The appropriate entries of
these objects provide us with the input features for each node, edge, and the
graph. For example, $\mathbf{x}_{u}\in\mathbb{R}^{k}$ would provide attributes
of an individual player $u\in V$, such as position, height and weight, and
$\mathbf{e}_{uv}\in\mathbb{R}^{l}$ would provide the attributes of a
particular pair of players $(u,v)\in\mathcal{E}$, such as their distance, and
whether they belong to the same team. The graph feature vector, $\mathbf{g}$,
can be used to store global attributes of interest to the corner kick, such as
the game time, current score, or ball position. For a simplified visualisation
of how a graph neural network would process such an input, refer to Figure
1(A).

To construct the input graphs, we first aligned the four data sources with
respect to their game IDs and timestamps and filtered out $2,517$ invalid
corner kicks, for which the alignment failed due to missing data, e.g.,
missing tracking frames or event labels. This filtering yielded $7,176$ valid
corner kicks for training and evaluation. We summarised the exact information
that was used to construct the input graphs in Table 2. In particular, other
than player heights (measured in centimeters (cm)) and weights (measured in
kilograms (kg)), the players were anonymous in the model. For the cases in
which the player profiles were missing, we set their heights and weights to
180cm and 75kg, respectively, as defaults. In total, we had 385 such
occurrences out of a total of $213,246\ (=22\times 9,693)$ during data
preprocessing. We downscaled the heights and weights by a factor of 100.
Moreover, for each corner kick, we zero-centered the positions of on-pitch
players and normalised them onto a $10\text{m}\times 10\text{m}$ pitch, and
their velocities were re-scaled accordingly. For the cases in which the pitch
dimensions were missing, we used a standard pitch dimension of
$110\text{m}\times 63\text{m}$ as default.

We summarised the grouping of the features in Table 1. The actual features
used in different benchmark tasks may differ, and we will describe this in
more details in the next section. To focus on modelling the high-level tactics
played by the attacking and defending teams, other than a binary indicator for
ball possession, no information of ball movement, neither positions nor
velocities, was used to construct the input graphs.

#### Benchmark tasks construction

TacticAI consists of three predictive and generative models, which also
corresponded to three benchmark tasks implemented in this study. Specifically,
1) Receiver prediction, 2) Threatening shot prediction, and 3) Guided
generation of team positions and velocities (Table 1). The graphs of all the
benchmark tasks used the same feature space of nodes and edges, differing only
in the global features.

For all three tasks, our models first transform the node features to a _latent
node feature matrix_ , ${\bf
H}=f_{\mathcal{G}}(\mathbf{X},\mathbf{E},\mathbf{g})$, from which we could
answer queries: either about individual players—in which case we learned a
relevant classifier or regressor over the $\mathbf{h}_{u}$ vectors (the rows
of ${\bf H}$)—or about the occurrence of a global event (e.g. shot taken)—in
which case we classified or regressed over the aggregated player vectors,
$\sum_{u}\mathbf{h}_{u}$. In both cases, the classifiers were trained using
stochastic gradient descent over an appropriately chosen loss function, such
as categorical cross-entropy for classifiers, and mean squared error for
regressors.

For different tasks, we extracted the corresponding ground-truth labels from
either the event stream data or the tracking data. Specifically, 1) We
modelled receiver prediction as a node classification task, and labelled the
first player to touch the ball after the the corner was taken as the target
node. This player could be either an attacking or defensive player. 2) Shot
prediction was modelled as graph classification. In particular, we considered
a next-ball-touch action by the attacking team as a shot if it was a direct
corner, a goal, an aerial, hit on the goalposts, a shot attempt saved by the
goalkeeper, or missing target. This yielded $1,736$ corners labelled as a shot
being taken, and $5,440$ corners labelled as a shot not being taken. 3) For
guided generation of player position and velocities, no additional label was
needed, as this model relied on a self-supervised reconstruction objective.

The entire dataset was split into training and evaluation sets with a $80:20$
ratio through random sampling, and the same splits were used for all tasks.

#### Graph neural networks

The central model of TacticAI is the _graph neural network_ (GNN) [41], which
computes latent representations on a graph by repeatedly combining them within
each node’s neighbourhood. Here we define a node’s neighbourhood,
$\mathcal{N}_{u}$, as the set of all first-order neighbours of node $u$, that
is, $\mathcal{N}_{u}=\\{v\ |\ (v,u)\in\mathcal{E}\\}$. A single GNN layer then
transforms the node features by passing messages between neighbouring nodes
[13], following the notation of related work [5], and the implementation of
the CLRS-30 benchmark baselines [42]:

$\mathbf{h}^{(t)}_{u}=\phi\left(\mathbf{h}^{(t-1)}_{u},\bigoplus\limits_{v\in\mathcal{N}_{u}}\psi\left(\mathbf{h}^{(t-1)}_{u},\mathbf{h}^{(t-1)}_{v},\mathbf{e}_{vu},\mathbf{g}\right)\right)$
(2)

where
$\psi:\mathbb{R}^{k}\times\mathbb{R}^{k}\times\mathbb{R}^{l}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{k^{\prime}}$
and
$\phi:\mathbb{R}^{k}\times\mathbb{R}^{k^{\prime}}\rightarrow\mathbb{R}^{k^{\prime}}$
are two learnable functions (e.g. multilayer perceptrons),
$\mathbf{h}_{u}^{(t)}$ are the features of node $u$ after $t$ GNN layers, and
$\bigoplus$ is any permutation-invariant aggregator, such as sum, max, or
average. By definition, we set $\mathbf{h}_{u}^{(0)}=\mathbf{x}_{u}$, and
iterate Equation 2 for $T$ steps, where $T$ is a hyperparameter. Then, we let
$\mathbf{H}=f_{\mathcal{G}}(\mathbf{X},\mathbf{E},\mathbf{g})=\mathbf{H}^{(T)}$
be the final node embeddings coming out of the GNN.

It is well known that Equation 2 is remarkably general; it can be used to
express popular models such as Transformers [39] as a special case, and it has
been argued that _all_ discrete deep learning models can be expressed in this
form [40, 3]. This makes GNNs a perfect framework for benchmarking various
approaches to modelling player-player interactions in the context of football.

Different choices of $\psi$, $\phi$ and $\bigoplus$ yield different
architectures. In our case, we utilise a message function that factorises into
an _attentional mechanism_ ,
$a:\mathbb{R}^{k}\times\mathbb{R}^{k}\times\mathbb{R}^{l}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$:

$\mathbf{h}^{(t)}_{u}=\phi\left(\mathbf{h}^{(t-1)}_{u},\bigoplus\limits_{v\in\mathcal{N}_{u}}a\left(\mathbf{h}^{(t-1)}_{u},\mathbf{h}^{(t-1)}_{v},\mathbf{e}_{vu},\mathbf{g}\right)\psi\left(\mathbf{h}^{(t-1)}_{v}\right)\right)$
(3)

yielding the graph attention network (GAT) architecture [43]. In our work,
specifically, we use a two-layer multilayer perceptron for the attentional
mechanism, as proposed by GATv2 [4]:

$a\left(\mathbf{h}^{(t-1)}_{u},\mathbf{h}^{(t-1)}_{v},\mathbf{e}_{vu},\mathbf{g}\right)=\operatornamewithlimits{softmax}_{v\in\mathcal{N}_{u}}\
\mathbf{a}^{\top}\mathrm{LeakyReLU}\left(\mathbf{W}_{1}\mathbf{h}_{u}^{(t-1)}+\mathbf{W}_{2}\mathbf{h}_{v}^{(t-1)}+\mathbf{W}_{e}\mathbf{e}_{vu}+\mathbf{W}_{g}\mathbf{g}\right)$
(4)

where $\mathbf{W}_{1},\mathbf{W}_{2}\in\mathbb{R}^{k\times h}$,
$\mathbf{W}_{e}\in\mathbb{R}^{l\times h}$,
$\mathbf{W}_{g}\in\mathbb{R}^{m\times h}$ and $\mathbf{a}\in\mathbb{R}^{h}$
are the learnable parameters of the attentional mechanism, and
$\mathrm{LeakyReLU}$ is the leaky rectified linear activation function. This
mechanism computes coefficients of interaction (a single scalar value) for
each pair of connected nodes $(u,v)$, which are then normalised across all
neighbours of $u$ using the $\mathrm{softmax}$ function.

Through early-stage experimentation we have ascertained that GATs are capable
of matching the performance of more generic choices of $\psi$ (such as the
MPNN [13]) while being more scalable. Hence, we focus our study on the GAT
model in this work. More details can be found in the Ablation Study section.

#### Geometric deep learning

In spite of the power of Equation 2, using it in its full generality is often
prone to overfitting, given the large number of parameters contained in $\psi$
and $\phi$. This problem is exacerbated in the football analytics domain,
where gold-standard data is generally very scarce—for example, in the English
Premier League, only a few hundred games are played every season.

In order to tackle this issue, we can exploit the immense _regularity_ of data
arising from football games. Strategically equivalent game states are also
called _transpositions_ , and symmetries such as arriving at the same chess
position through different move sequences have been exploited computationally
since the 1960s [17]. Similarly, game rotations and reflections may yield
equivalent strategic situations [32]. Using the blueprint of _geometric deep
learning_ (GDL) [5], we can design specialised GNN architectures that exploit
this regularity.

That is, geometric deep learning is a generic methodology for deriving
mathematical constraints on neural networks, such that they will behave
predictably when inputs are transformed in certain ways. In several important
cases, these constraints can be directly resolved, directly informing neural
network architecture design. For a comprehensive example for point clouds
under 3D rotational symmetry, see Fuchs _et al._ [12].

To elucidate several aspects of the GDL framework on a high level, let us
assume that there exists a group of input data transformations (_symmetries_),
$\mathfrak{G}$, under which the ground-truth label remains unchanged.
Specifically, if we let $y(\mathbf{X},\mathbf{E},\mathbf{g})$ be the label
given to the graph featurised with $\mathbf{X},\mathbf{E},\mathbf{g}$, then
for every transformation $\mathfrak{g}\in\mathfrak{G}$, the following property
holds:

$y(\mathfrak{g}(\mathbf{X}),\mathfrak{g}(\mathbf{E}),\mathfrak{g}(\mathbf{g}))=y(\mathbf{X},\mathbf{E},\mathbf{g})$
(5)

This condition is also referred to as $\mathfrak{G}$-invariance.

It is worth noting that GNNs may also be derived using a GDL perspective, if
we set the symmetry group $\mathfrak{G}$ to $S_{|V|}$, the permutation group
of $|V|$ objects. Owing to the design of Equation 2, its outputs will not be
dependent on the exact permutation of nodes in the input graph.

#### Frame averaging

A simple mechanism to enforce $\mathfrak{G}$-invariance, given any predictor
$f_{\mathcal{G}}(\mathbf{X},\mathbf{E},\mathbf{g})$, performs _frame
averaging_ across all $\mathfrak{G}$-transformed inputs:

$f^{\mathrm{inv}}_{\mathcal{G}}(\mathbf{X},\mathbf{E},\mathbf{g})=\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{g}\in\mathfrak{G}}f_{\mathcal{G}}(\mathfrak{g}(\mathbf{X}),\mathfrak{g}(\mathbf{E}),\mathfrak{g}(\mathbf{g}))$
(6)

This ensures that all $\mathfrak{G}$-transformed versions of a particular
input (also known as that input’s _orbit_) will have exactly the same output,
satisfying Equation 5. A variant of this approach has also been applied in the
AlphaGo architecture [36] to encode symmetries of a Go board.

In our specific implementation, we set
$\mathfrak{G}=D_{2}=\\{\mathrm{id},\leftrightarrow,\updownarrow,\leftrightarrow\updownarrow\\}$,
the dihedral group. Exploiting $D_{2}$-invariance allows us to encode
_quadrant symmetries_. Each element of the $D_{2}$ group encodes the presence
of vertical or horizontal reflections of the input football pitch. Under these
transformations, the pitch is assumed completely symmetric, and hence many
predictions, such as which player receives the corner kick, or takes a shot
from it, can be safely assumed unchanged. As an example for how to compute
transformed features in Equation 6, $\leftrightarrow(\mathbf{X})$ horizontally
reflects all positional features of players in $\mathbf{X}$ (e.g. the
coordinates of the player), and negates the $x$-axis component of their
velocity.

#### Group convolutions

While the frame averaging approach of Equation 6 is a powerful way to restrict
GNNs to respect input symmetries, it arguably misses an opportunity for the
different $\mathfrak{G}$-transformed views to interact while their
computations are being performed. For small groups such as $D_{2}$, a more
fine-grained approach can be assumed, operating over a single GNN layer in
Equation 2, which we will write shortly as
$\mathbf{H}^{(t)}=g_{\mathcal{G}}(\mathbf{H}^{(t-1)},\mathbf{E},\mathbf{g})$.
The condition that we need a symmetry-respecting GNN layer to respect is as
follows, for all transformations $\mathfrak{g}\in\mathfrak{G}$:

$g_{\mathcal{G}}(\mathfrak{g}(\mathbf{H}^{(t-1)}),\mathfrak{g}(\mathbf{E}),\mathfrak{g}(\mathbf{g}))=\mathfrak{g}(g_{\mathcal{G}}(\mathbf{H}^{(t-1)},\mathbf{E},\mathbf{g}))$
(7)

that is, it does not matter if we apply $\mathfrak{g}$ to the input or the
output of the function $g_{\mathcal{G}}$—the final answer is the same. This
condition is also referred to as $\mathfrak{G}$-equivariance, and it has
recently proved to be a potent paradigm for developing powerful GNNs over
biochemical data [12, 31].

To satisfy $D_{2}$-equivariance, we apply the _group convolution_ approach
[7]. Therein, views of the input are allowed to directly interact with their
$\mathfrak{G}$-transformed variants, in a manner very similar to grid
convolutions (which is, indeed, a special case of group convolutions, setting
$\mathfrak{G}$ to be the translation group). We use
$\mathbf{H}^{(t)}_{\mathfrak{g}}$ to denote the $\mathfrak{g}$-transformed
view of the latent node features at layer $t$. Omitting $\mathbf{E}$ and
$\mathbf{g}$ inputs for brevity, and using our previously designed layer
$g_{\mathcal{G}}$ as a building block, we can perform a group convolution as
follows:

$\mathbf{H}^{(t)}_{\mathfrak{g}}=g^{\mathrm{equiv}}_{\mathcal{G}}(\mathbf{H}^{(t-1)}_{\mathfrak{g}})=\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{h}\in\mathfrak{G}}g_{\mathcal{G}}\left(\mathbf{H}^{(t-1)}_{\mathfrak{h}}\|\mathbf{H}^{(t-1)}_{\mathfrak{g}^{-1}\mathfrak{h}}\right)$
(8)

Here, $\|$ is the concatenation operation, joining the two node feature
matrices column-wise; $\mathfrak{g}^{-1}$ is the inverse transformation to
$\mathfrak{g}$ (which must exist as $\mathfrak{G}$ is a group); and
$\mathfrak{g}^{-1}\mathfrak{h}$ is composition of the two transformations.

Effectively, Equation 8 implies our $D_{2}$-equivariant GNN needs to maintain
a node feature matrix $\mathbf{H}^{(t)}_{\mathfrak{g}}$ for every
$\mathfrak{G}$-transformation of the current input, and these views are
recombined by invoking $g_{\mathcal{G}}$ on all pairs related together by
applying a transformation $\mathfrak{h}$. Note that all reflections are self-
inverses, hence, in $D_{2}$, $\mathfrak{g}=\mathfrak{g}^{-1}$.

#### Network architectures

While the three benchmark tasks we are performing have minor differences in
the global features available to the model, the neural network models designed
for them all have the same _encoder-decoder_ architecture. The encoder has the
same structure in all tasks, while the decoder model is tailored to produce
appropriately-shaped outputs for each benchmark task.

Given an input graph, TacticAI’s model first generates all relevant
$D_{2}$-transformed versions of it, by appropriately reflecting the player
coordinates and velocities. We refer to the original input graph as the
_identity view_ , and the remaining three $D_{2}$-transformed graphs as
_reflected views_.

Once the views are prepared, we apply four group convolutional layers
(Equation 8) with a GATv2 base model (Equations 3–4). Each GATv2 layer has
eight attention heads, and computes four latent features overall per player.
Accordingly, once the four group convolutions are performed, we have a
representation of ${\bf H}\in\mathbb{R}^{4\times 22\times 4}$, where the first
dimension corresponds to the four views (${\bf H}_{\mathrm{id}},{\bf
H}_{\leftrightarrow},{\bf H}_{\updownarrow},{\bf
H}_{\leftrightarrow\updownarrow}\in\mathbb{R}^{22\times 4}$), the second
dimension corresponds to the players (eleven on each team), and the third
corresponds to the $4$-dimensional latent vector for each player node in this
particular view. How this representation is used by the decoder depends on the
specific downstream task:

  * •

For receiver prediction, which is a fully _invariant_ function (i.e.
reflections do not change the receiver), we perform simple frame averaging
across all views, arriving at

${\bf H}^{\mathrm{node}}=\frac{{\bf H}_{\mathrm{id}}+{\bf
H}_{\leftrightarrow}+{\bf H}_{\updownarrow}+{\bf
H}_{\leftrightarrow\updownarrow}}{4}$ (9)

and then learn a node-wise classifier over the rows of ${\bf
H}^{\mathrm{node}}\in\mathbb{R}^{22\times 4}$. We further decode
$\bf{H}^{\mathrm{node}}$ into a logit vector ${\bf O}\in\mathbb{R}^{22}$ with
a linear layer before computing the corresponding softmax cross entropy loss.

  * •

For shot prediction, which is once again fully _invariant_ (i.e. reflections
do not change the probability of a shot), we can further average the frame-
averaged features across all players to get a global graph representation:

${\bf
h}^{\mathrm{graph}}=\frac{1}{22}\sum_{u=1}^{22}\mathbf{h}^{\mathrm{node}}_{u}$
(10)

and then learn a binary classifier over
$\mathbf{h}^{\mathrm{graph}}\in\mathbb{R}^{4}$. Specifically, we decode the
hidden vector into a single logit with a linear layer, and the compute the
sigmoid binary cross entropy loss with the corresponding label.

  * •

For guided generation (position/velocity adjustments), we generate the player
positions and velocities with respect to a particular outcome of interest for
the human coaches, predicted over the rows of the hidden feature matrix. For
example, the model may adjust the defensive setup to decrease the shot
probability by the attacking team. The model output is now _equivariant_
rather than invariant—reflecting the pitch appropriately reflects the
predicted positions and velocity vectors. As such, we cannot perform frame
averaging, and take only the _identity view_ ’s features, ${\bf
H}_{\mathrm{id}}\in\mathbb{R}^{22\times 4}$. From this latent feature matrix,
we can then learn a conditional distribution from each row, which models the
positions or velocities of the corresponding player. To do this, we extend the
backbone encoder with a Variational Autoencoder (VAE [22]). Specifically, for
the $u$-th row of ${\bf H}_{\mathrm{id}}$, $\mathbf{h}_{u}$, we first map its
latent embedding to the parameters of a two-dimensional Gaussian distribution
$\mathcal{N}(\mu_{u}|\sigma_{u})$, and then sample the coordinates and
velocities from this distribution. At training time, we can efficiently
propagate gradients through this sampling operation using the
reparameterisation trick [22]: sample a random value
$\epsilon_{u}\sim\mathcal{N}(0,1)$ for each player from the unit Gaussian
distribution, and then treat $\mu_{u}+\sigma_{u}\epsilon_{u}$ as the sample
for this player. In what follows, we omit edge features for brevity. For each
corner kick sample $\mathbf{X}$ with the corresponding outcome $\mathbf{o}$
(e.g. a binary value indicating a shot event), we extend the standard VAE loss
[22] to our case of outcome-conditional guided generation as

$\mathcal{L}(\theta,\phi)=-\mathbb{E}_{\mathbf{h}_{u}\sim
q_{\theta}(\mathbf{h}_{u}|\mathbf{X},\mathbf{o})}[\log{p_{\phi}(\mathbf{x}_{u}|\mathbf{h}_{u},\mathbf{o})}]+\mathbb{KL}(q_{\theta}(\mathbf{h}_{u}|\mathbf{X},\mathbf{o})\|p(\mathbf{h}_{u}|\mathbf{o}))$
(11)

where $\mathbf{h}_{u}$ is the player embedding corresponding to the $u$-th row
of $\bf{H}_{\mathrm{id}}$, and $\mathbb{KL}$ is Kullback-Leibler (KL)
divergence. Specifically, the first term is the generation loss between the
real player input $\mathbf{x}_{u}$ and the reconstructed sample decoded from
$\mathbf{h}_{u}$ with the decoder $p_{\phi}$. Using the KL term, the
distribution of the latent embedding $\mathbf{h}_{u}$ is regularised towards
$p(\mathbf{h}_{u}|\mathbf{o})$, which is a multivariate Gaussian in our case.

A complete high-level summary of the generic encoder-decoder equivariant
architecture employed by TacticAI can be summarised in S.Figure 2. In the
following section, we will provide empirical evidence for justifying these
architectural decisions. This will be done through targeted ablation studies
on our predictive benchmarks (receiver prediction and shot prediction).

#### Ablation Study

As described briefly in the Results and Analysis section, we leveraged the
receiver prediction task as a way to evaluate various base model
architectures, and directly quantitatively assess the contributions of
geometric deep learning in this context. We already see that the raw corner
kick data can be better represented through geometric deep learning, yielding
separable clusters in the latent space which could correspond to different
attacking or defending tactics (Figure 2). In addition, we hypothesise that
these representations can also yield better performance on the task of
receiver prediction. Accordingly, we ablate several design choices using deep
learning on this task:

  * •

_Does a factorised graph representation help?_ To assess this, we compare
against a convolutional neural network (CNN [9]) baseline, which does not
leverage a graph representation.

  * •

_Does a graph structure help?_ To assess this, we compare against a Deep Sets
[44] baseline, which only models each node in isolation without considering
adjacency information—equivalently, setting each neighbourhood
$\mathcal{N}_{u}$ to a singleton set $\\{u\\}$.

  * •

_Are attentional GNNs a good strategy?_ To assess this, we compare against a
message passing neural network [14, MPNN] baseline, which uses the fully
potent GNN layer from Equation 2 instead of the GATv2.

  * •

_Does accounting for symmetries help?_ To assess this, we compare our
geometric GATv2 baseline against one which does not utilise $D_{2}$ group
convolutions but utilises $D_{2}$ frame averaging, and one which does not
explicitly utilise any aspect of $D_{2}$ symmetries at all.

Each of these models have been trained for a fixed budget of $50,000$ training
steps. The test top-$k$ receiver prediction accuracies of the trained models
are provided in S.Table 2. As already discussed in Results and Analysis, there
is a clear advantage to using a full graph structure, as well as directly
accounting for reflection symmetry. Further, the usage of the MPNN layer leads
to slight overfitting compared to the GATv2, illustrating how attentional GNNs
strike a good balance of expressivity and data efficiency for this task. Our
analysis highlights the quantitative benefits of both graph representation
learning and geometric deep learning for football analytics from tracking
data. We also provide a brief ablation study for the shot prediction task in
S.Table 3.

#### Training Details

We train each of TacticAI’s models in isolation, using NVIDIA Tesla P100 GPUs.
To minimise overfitting, each model’s learning objective is regularised with
an $L_{2}$ norm penalty with respect to the network parameters. During
training, we use the Adam stochastic gradient descent optimiser [21] over the
regularised loss. We summarise the hyperparameters of each TacticAI model in
S.Table 1.

## Data availability

The datasets generated and/or analysed during the current study are not
publicly available due to licensing restrictions. However, contact details of
the data providers are available from the corresponding authors on reasonable
request.

## Acknowledgements

We gratefully acknowledge the support of James French, Timothy Waskett, Hans
Leitert and Benjamin Hervey for their extensive efforts in analysing
TacticAI’s outputs. Further, we are thankful to Kevin McKee, Sherjil Ozair and
Beatrice Bevilacqua for useful technical discussions, and Marc Lanctôt and
Satinder Singh for reviewing the paper prior to submission.

## Author contributions

Z.W., D.H.11footnotemark: 1, L.P. and K.T. coordinated and organised the
research effort leading to this paper. P.V. and Z.W. developed the core
TacticAI models. Z.W., W.S. and I.G. prepared the Premier League corner kick
dataset used for training and evaluating these models. P.V., Z.W.,
D.H.11footnotemark: 1**footnotetext: D.H. stands for Daniel Hennes. and N.T.
designed the case study with human experts and performed the qualitative
evaluation and analysis of its outcomes. All authors contributed to writing
the paper, technical discussions and providing feedback on the final
manuscript.

## Supplementary Materials

Section S1.1 |  Literature Review
---|---
Section S1.2 |  Details of Case Study Design
Section S1.3 |  Supplementary Figures and Tables

##  S1 Supplementary Material

###  S1.1 Literature Review

In the following, we summarise the closest state of the art that pursues the
same objective of football corner kick tactics or pass prediction, or which
more broadly devises team-sport strategy modelling techniques that could be
transferred to the setting considered in this work. Sports analytics research
that addresses player transfers and match statistics is out of scope, as we
here focus on methods for in-game tactics.

Earlier work has generated a significant body of research on identifying key
properties of a game given diverse sources of data, and on using these to
provide tactical analysis and suggestions [8, 10, 15, 25, 28, 29]. One recent
example is a system for determining whether a match is in-play or interrupted,
called “in-game status”, by applying random forests and AdaBoost to process
time-continuous spatio-temporal data such as player positions [23], with
similar methods having been proposed for American football [26].

Other methods leverage game data to characterize players behaviour, such as
their passing style [6]. Such information can later feed downstream
applications that may take as input a representation of a player’s historical
behaviour. Similarly, pass prediction systems may use game data to predict the
probability of a pass in a given game state [18].

A more closely related research line analyzes soccer videos to determine shoot
event probabilities, which may serve as a proxy for player performance, by
examining spatio-temporal relations and prediction uncertainty [16]. The
method processes videos using a graph convolutional recurrent neural network
to capture latent features, and leverages Bayesian neural networks to predict
the shoot probabilities. Similar other methods apply machine learning and
explainable AI techniques for tactical analysis and for predicting the success
of defensive play [11].

Accurately identifying the salient patterns of corner kick tactics from player
trajectories is the key prerequisite for downstream tasks in football
analytics, including evaluating the effectiveness of the tactics, improving
their performance and designing counter-tactics. Through leveraging deep
learning, previous research demonstrated that the outcomes of in-game events,
including shots [37] and passes [9, 19], are predictable using the spatio-
temporal tracking data of player trajectories. Moreover, the patterns of
effective defensive tactics can also be recovered from tracking data [37].
Nevertheless, tracking data could be quite noisy because of the randomness and
complex dynamics in football games, rendering mining the tactic patterns from
it challenging.

There are also other examples of learning techniques for modelling team
sports. The Generative Relational Inference Network (GRIN) [24] also follows
the VAE framework to learn disentangled representations for each agent in the
scene. It encodes each entity into a latent representation, comprising an
intra-agent intention and an inter-agent relation, through a graph convolution
network (GCN). The inter-agent relations under go message passing through
graph attention layers, the result of which is combined with intra-agent
intention to form the reconstruction. Although there is no group-invariant
layers, experiments show that this method could discover disentangled factors
from basketball players, and interventions on these factors produced
interpretable variations in the predicted player trajectories.

Using NBA games spacio-temporal tracking data, DeepHoops [34] presents a Deep
Learning architecture that estimates the impact of all micro-actions in on the
outcome of Basket Ball plays. In particular, it allows to evaluate the
contribution of individual off-ball events to the success of a possession.

In [35] a tree-search method (Kernel Regression UCT) has been applied to
discover curling strategies in self-play. The authors show that counterfactual
justifications of actions taken by the model can be used to teach humans non-
trivial concepts of curling as demonstrated in a user-study.

###  S1.2 Case Study Design

As previously described in the Results section, we designed an in-depth case
study to evaluate the benefits of TacticAI’s predictive and generative
capabilities. We designed four specific tasks which we offered to five experts
affiliated with Liverpool FC. To avoid biasing our raters, we have kept all
details of these tasks strictly hidden from them prior to conducting the case
study. Further, for all corner situations shown we did not reveal the team or
player identities—only the players’ positions and velocities at the moment the
corner kick was taken, and whether they are on the attacking or defensive
team.

#### Task 1: Does TacticAI generate _realistic_ tactical adjustments?

To answer this question, we started from a dataset of 50 corners from the
Premier League, which have been held-out from TacticAI’s training set
(S.Figure 3). For a subset of those corners, we have applied TacticAI’s
generative head to propose alternate tactics for either minimising or
maximising shot probability, and used TacticAI’s suggestion in place of the
original corner. We then asked our raters to judge whether each of these
corners was real or generated. Through discussions with the raters after the
task was done, and observing their comments, we noted a variety of strategies
were employed to detect realism:

  * •

Most raters chose to decide realism based on detecting whether there was
something markedly unusual in the setup, for example a strategy that left
opposing players in clear space.

  * •

An alternate strategy, employed by one rater, was to decide realism based on
whether the situation could have plausibly happened, even if it had a clearly
suboptimal tactical setup. This approach ended up tagging most of the examples
as real.

  * •

Raters who specialise in corner kick analysis preferred a “retrieval” based
strategy, wherein they would rate a corner as real if they were able to recall
from memory that exact setup.

In spite of this variety of approaches, and as discussed in the Results
section, all of the raters were generally confused as to which samples were
real, and they unanimously highlighted how challenging they found this task.

#### Task 2: Does TacticAI predict _plausible_ receivers?

To answer this question, we reused the same 50 corners leveraged for Task 1,
and simultaneously asked our raters which of the attacking team’s players were
most likely to make first contact with the ball (S.Figure 3). The raters were
allowed to provide as little or as many players as they wanted to—and we
cross-referenced their proposals with TacticAI’s top three suggested
receivers. Generally, we found significant variability in the amount of
players proposed by different raters—some opted to suggest 3–5 receivers on
almost all situations studied, whereas others did not want to suggest more
than one receiver on most occasions. As discussed in the Results section, in
spite of this diversity of approach, the top-3 predictions made by TacticAI
largely agreed with the rater estimates, and there was no significant
difference between TacticAI’s accuracy over real and generated setups,
indicating that TacticAI could be robustly used to predictively analyse both
real and simulated corner kicks. And, as highlighted by several of the raters,
such a system would prove useful to a football analyst: a receiver predictor
can be used to indicate which players may need to be especially targeted
defensively, as well as to evaluate whether corners in already-played games
have unfolded in a manner which is expected.

#### Task 3: Can TacticAI be used to retrieve usefully _similar_ corners?

To answer this question, we leveraged 50 reference corners from the Premier
League, held-out from TacticAI’s training set (S.Figure 4). For each of these
corners, we retrieved one other Premier League corner that was closest to
it—either in TacticAI’s graph-level embedding space, or in terms of the
distance of the raw player features in the input space. This decision allowed
us to not only measure whether TacticAI’s embeddings can be used to mine for
similar corners, but also to assess whether they provide an edge over a
simpler mining heuristic which is purely position based. We asked our raters
whether they find the retrieved corner to be usefully similar to the reference
corner; that is, whether showing these two corners side by side would be
judged as useful. The raters once again assumed a diverse set of approaches
for deciding which features of the two corners were salient enough to be
useful. Some of the common considerations mentioned by the raters included:
whether the corner is in- or out-swinging, existence of a short corner option,
zonal vs. man-marking approach, counting the number of players in the 18-yard
box, positioning of the goalkeeper, and assessing whether the two corners
likely had the same attacking or defending team, or had come from the same
game. Similarly as for the previous tasks—from the discussion in the Results
section we can conclude that, despite the wide variety of salient features
considered, the raters generally found TacticAI’s retrievals to be superior to
the positional heuristic. They also highlighted the high utility of such a
retrieval system: it can be used by analysts and coaches to discover and
prepare for other teams’ common routines, as well as to discover new ideas and
variations on a particular corner that they might otherwise have overlooked.
It has also been stressed that the “top-1” retrieval study employed here is
likely too strict. While it was necessary to restrict attention to only one
retrieved corner here in order to keep the raters’ workload manageable, in
practice, an analyst may wish to analyse up to 10–20 retrieved corners for
every reference corner. Given that TacticAI already has a favourable score in
the top-1 regime, we anticipate its utility to only compound if deployed for
real-world use.

#### Task 4: Does TacticAI generate _useful_ tactical adjustments?

To answer this question, we once again gathered a dataset of 50 held-out
corners from the Premier League (S.Figure 5). For each of them, we have used
TacticAI’s guided generative model to propose adjustments to the defending
team’s player positions and velocities—leaving the attacking team
unchanged—such that the predicted shot probability is reduced. We chose to
focus on the defending team to match a real-world workload: typically, a
corner setup is dictated by the attacking team, after which the defensive team
needs to respond to this setup in an optimal way. For each of these corners,
we showed the raters the two versions (original and TacticAI-adjusted) side-
by-side. We stated that the corner on the left-hand side is a reference
corner, asking the raters to judge which of the defending team’s players are
in a better or worse position, considering the adjustments on the right-hand
side. We also asked the raters to judge whether the right-hand side corner is,
overall, a better or worse situation than the left-hand side for the defensive
team. To control for bias, we randomly select a subset of corner setups for
which we reverse the two versions, making the TacticAI suggestion the
reference corner. The raters focused on a variety of salient features in the
suggestions, including but not limited to: whether defenders are better at
tracking an attacker’s run, whether it is better for them to move or stand
still, and the positioning of the goalkeeper. There have been two situations
where an adjustment was judged as “possibly useful”, depending on the adjusted
player’s profile. For example, certain runs were only deemed likely to succeed
if performed by defenders of high physical capability or fitness. As the
TacticAI variant we trained here did not have access to player features beyond
height and weight, such nuances were out of scope for this study, and we did
not consider them to be salient observations. Overall, as outlined in the
Results section, the raters were overwhelmingly in favour of TacticAI’s
suggestions, demonstrating a high degree of inter-rater agreement. We
illustrate four of the most salient situations where TacticAI’s suggestions
were deemed significant in Figure 5; in one of these situations, ten defensive
players were considered usefully adjusted, which is nearly the entire
defending team! Further, this kind of position-adjustment system has been
deemed a highly useful tool for an analyst or coach to leverage by all of our
raters. Even the opportunity to view such suggestions side-by-side with a
reference corner was considered to be useful, as it prompted the rater to
immediately consider strategic variations. Since the guided generation system
was deliberately designed to offer subtle modifications to player coordinates
and velocities, the raters deemed such a system most useful to help detect
players that have likely not been following the instructions of their coaches,
and neglected the tactical play. This can then lead to targeted interventions
by the coaches—either to exploit a weakness posed by an opposing player that
tends to ignore instructions, or to improve coaching of the team’s player if
they are found to be neglecting the determined tactics. Coupled with the
positive outcome of Task 1, and the finding that the suggested adjustments are
generally hard to distinguish from real corner situations, we conclude Task
4’s question to be answered in the affirmative.

###  S1.3 Supplementary Tables and Figures

Supplementary Figure 1: $t$-SNE embeddings of raw input features. For the same
set of corner kick samples used in the $t$-SNE visualisation in Figure 2, we
visualise their raw input features with $t$-SNE. The $t$-SNE embeddings of the
attacking and defending setups are not clearly separable. Supplementary Figure
2: The encoder architecture of TacticAI’s predictive and generative component
models. For each input graph, we first generate three $D_{2}$-reflected views.
Secondly, we process them via a sequence of four GATv2 [4] group convolution
layers. The actual number of layers may be different in different tasks
(S.Table 1). Finally, we aggregate the encodings of the input graph (identity
encodings) and the encodings of its corresponding reflected views (equivariant
encodings) through a weighted sum to obtain the final graph encodings.
Specifically, for the receiver and shot prediction tasks (which are fully
invariant), we take the mean of the encodings, and for the guided generation
task (which is equivariant), we set the weight of the identity encodings to
$1.0$ and zero out the equivariant encodings. The graph encodings are further
processed with respect to the corresponding tasks (see Network Architectures
in the Methods section.) Supplementary Figure 3: Sample of Task 1 and 2 in the
case study. Each corner kick sample in this task is presented in a 2D “bird’s
eye” view, in which blue and red dots represent defending and attacking
players, respectively, and the length of the arrow attached to each dot
indicates their velocity. During the task, for each sample, we first ask the
human experts to identify whether the sample is a real corner kick or a
generated one by TacticAI. Next, they are asked to identify the most likely
receivers among the attacking team’s players. Supplementary Figure 4: Sample
of Task 3 in the case study. In this task, we present a pair of corner kicks
in each task sample, one of which represents the reference corner kick, and
the other represents a retrieved corner, taken as the closest sample in
distance, either in the TacticAI latent space or in the raw feature space. The
human raters are asked to rate whether the pairs are similar with respect
salient patterns. Supplementary Figure 5: Sample of Task 4 in the case study.
In this task, the human raters are given pairs of corner kick samples, each of
which consists of a reference corner kick sample and its adjustment generated
by TacticAI. Then, the raters are asked to identify whether there are salient
improvements in the adjustment and also to give specific account to those
improvements as well. For the adjustment in each pair, we rate it as $+1$ if
the rater confirms that overall adjustments are constructive, $0$ if no
distinct difference, or $-1$ if destructive.

Hyperparameter | Receiver Prediction | Shot Prediction | Guided Generation
---|---|---|---
Batch size | 256 | 128 | 128
Learning rate | 1e-4 | 1e-4 | 5e-5
$L^{2}$ weight decay | 1e-4 | 0. | 1e-4
Adam optimiser $\beta_{1}$ | 0.9 | 0.9 | 0.9
Adam optimiser $\beta_{2}$ | 0.999 | 0.999 | 0.999
Adam optimiser $\epsilon$ | 1e-8 | 1e-8 | 1e-8
# Graph attention layers | 4 | 2 | 2
Random seed | 42 | 42 | 42
Table 1: Hyperparameters for training TacticAI’s components. We list the hyperparameters with which we obtain TacticAI’s best performing component models. The models are selected according to their evaluation losses. In particular, for guided generation, attacking and defensive generations are trained as two models, and share a same set of hyperparameters. Model | Average Top-3 Accuracy
---|---
CNN [9] | $0.364\pm 0.031$
Deep Sets [44] | $0.713\pm 0.022$
MPNN [14] | $0.723\pm 0.017$
GATv2 [43, 4] | $0.748\pm 0.021$
GATv2 + $D_{2}$ frame averaging | $0.780\pm 0.011$
GATv2 + $D_{2}$ group convolution [7] | $\mathbf{0.782}\pm 0.039$
Table 2: Ablation results for receiver prediction. We use top-3 accuracy as the metric. First, we see that a graph representation with Deep Sets [44] outperforms a convolutional neural network(CNN [9]), which does not leverage a graph representation. Secondly, instead of modelling each player node in isolation without considering adjacency information with Deep Sets [44], we augment each pair of player nodes with an edge to indicate whether they are on the same team, and process the resulting fully connected graphs with various GNNs (MPNN [14] and GATv2 [4]). This yields another performance gain of $0.035$ in Top-3 prediction accuracy with GATv2 [4]. Finally, equipping the GATv2 with a $D_{2}$ group convolution yields our best performing model for receiver prediction. Model | Average $F_{1}$ Score
---|---
GATv2 [39, 4] (unconditional) | $0.521\pm 0.027$
GATv2 + receiver conditional | $0.677\pm 0.036$
GATv2 + receiver conditional + $D_{2}$ group convolution [7] | $\mathbf{0.712}\pm 0.011$
Table 3: Ablation results for shot prediction. We use the F1 score as the
metric, because the dataset used to develop the shot prediction component
model is imbalanced with a positive-to-negative ratio of 0.21. Instead of
directly predicting the unconditional probability of a successful shot
attempt, we predict the shot probability conditioned on the receiver of the
corner kick, which yields an improvement from $0.512\pm 0.027$ to $0.712\pm
0.011$ in F1 score.

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