# Understanding team collapse via probabilistic graphical models

Iasonas Nikolaou  Boston UniversityBostonUSA<EMAIL_ADDRESS>,  Konstantinos
Pelechrinis  University of PittsburghPittsburghUSA<EMAIL_ADDRESS>and
Evimaria Terzi  Boston UniversityBostonUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

In this work, we develop a graphical model to capture team dynamics. We
analyze the model and show how to learn its parameters from data. Using our
model we study the phenomenon of _team collapse_ from a computational
perspective. We use simulations and real-world experiments to find the main
causes of team collapse. We also provide the principles of building resilient
teams, i.e., teams that avoid collapsing. Finally, we use our model to analyze
the structure of NBA teams and dive deeper into games of interest.

team collapse, team dynamics, probabilistic graphical models

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: ACM KDD Conference 2024; August 25-29, 2024;
Barcelona, Spain††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Information systems Data mining††ccs: Computing methodologies Learning in
probabilistic graphical models††ccs: Theory of computation Social networks

##  1\. Introduction

What are the reasons that a (sports) team severely under-performs and
eventually _collapses_? Are some teams more prone to collapse than others? Are
there some guiding principles for building resilient teams?

In this paper, we make a significant step towards addressing these questions.
We achieve this by introducing a probabilistic graphical model that assumes
the following: at any time $t$ a player’s _hidden_ (mental) state depends on
the player’s own _hidden_ state at $t-1$ and the observed performances of all
team members at $t-1$. Consequently, a _team collapse_ happens at time $t$ if
more than a certain percentage of the team’s players are in a “low” hidden
(mental) state at $t$.

Our model parameters enable us to encode different types of dependencies among
team members. For example, there may be players whose performance is only
affected by their own hidden/observed state and is unaffected by the
performance of others. We call such players _pillars_. On the other hand, non-
pillar players are more susceptible to be influenced by their teammates’
observed performance. An interesting question we explore is to what extend
having more or less pillars affects the resilience of a team and influences
its collapse probability. Moreover, is it just the number of pillars important
or their dependency patterns as well? We believe that being able to answer
questions like this can inform roaster-building decisions.

From the computational point of view, we show that we can use the Expectation-
Maximization (EM) framework to learn the parameters of our model. In
particular, we borrow several ideas from probabilistic graphical models
(Koller and Friedman, 2009) and Hidden Markov Models (HMMs) (Bishop, 2006) in
order to implement the expectation and the maximization steps of EM. In
practice, we show that despite the large number of parameters of our model, EM
can learn those parameters effectively in a small number of iterations.

Using real data from National Basketball Association (NBA) games (season
2021-22), we deploy our EM algorithm to learn the parameters of our model for
all NBA teams. We identify some interesting dependencies among players and
demonstrate that pillar players do not always coincide with players widely
acknowledged as “stars”. Moreover, our framework allows us to identify team
collapses in various games and further analyze them in order to find the
reasons for which these collapses happened. We believe that these results
provide useful insights on team dynamics and can be used from team
psychologists and General Managers in order to safeguard teams from collapses.

Overall, our model allows us to capture complex team dynamics and can be used
as a tool by experts in team sports to analyze their teams’ compositions,
simulate the performance of different compositions and use these insights to
create teams that are more resilient to team collapses.

Although related to other probabilistic models proposed in the literature (see
related work for an extended discussion), the exact details of our model and
the corresponding computational problem associated with the learning of its
parameters are novel. This is because, to the best of our knowledge, our model
is the first probabilistic graphical model that is tailored to capture how
teammate dependencies give rise to team collapses.

Related work: During the last decade, the ability to collect multiple player-
tracking data has led to a growth of computational approaches for evaluating
players, teams and team strategies in a variety of sports (Cervone et al.,
2016; Chen et al., 2021; D’Amour et al., 2015; Decroos et al., 2019; Miller et
al., 2014; Nguyen et al., 2023; Lucey et al., 2016; Seidl et al., 2018;
Sicilia et al., 2019; Yue et al., 2014; Yurko et al., 2020). From the
application point of view, our work falls into this general area and it
relates to work on sport analytics and athlete choking. From the computational
point of view, our work is related to existing work on probabilistic models
and interconnected Hidden Markov Models (HMMs). We summarize these connections
below.

##### Team and player performance ratings:

One of the main tasks that the field of sports analytics has dealt with is
that of team and player performance evaluation. This typically is achieved
through models that learn team and player ratings, which are then used in
match prediction and gambling or to help teams identify players to acquire.
The simplest team ratings are based on regression methods (Langville and
Meyer, 2012; W Bassett, 2007; Bradley and Terry, 1952; Winston et al., 2022).
More sophisticated ones use network analysis of the “who-wins-whom” network
(Park and Newman, 2005; Pelechrinis et al., 2016), while others use latent
space models to learn an embedding for each team that will help obtain a team
ranking (Chen and Joachims, 2016a, b; Pelechrinis, 2018). Similarly, for
evaluating players the traditional approach for obtaining player ratings is
using regression based approaches (Macdonald, 2011; Thomas et al., 2013;
Sabin, 2021) to model the score differential during a game possession as a
function of the players on the court/pitch.

These ratings ignore dependencies between the players and make the implicit
assumptions that player ratings are additive, i.e., a team is the sum of the
individual players. As such, it is hard for these type of models to help us
understand and explain situations where a team significantly over or under
performs its expectation within a game. In our work, we explicitly consider
the dependencies between players and we develop a framework that can explain
how these dependencies can lead to a team collapsing.

##### Athlete’s “choking”

: The common belief is that super-star athletes shine when the stakes are
high, while athletes in lower tiers severely under perform under pressure,
i.e., they “choke” or collapse. While there has been some literature on
players performance and decision-making under pressure, this has traditionally
been in the field of sports psychology (Baumeister and Showers, 1986; Pineda-
Hernández, 2022; Parkin and Walsh, 2017). The vast majority of the literature
has been on the complementary phenomenon of “clutch” players, i.e., players
that deliver when the stakes are high. The majority of this literature
indicates that individual clutch performance is highly noisy (Solomonov et
al., 2015; Wallace et al., 2013; Birnbaum, 2008). One of the few computational
studies on player performance under pressure is that from Bransen et al.
(Bransen et al., 2019) who develop metrics to evaluate the performance of
individual soccer players under situations with high mental pressure. In a
similar direction, Dohmen (Dohmen, 2008) focused on studying penalty shootouts
and found that players of the home team are more likely to choke; the same
study identified no evidence in their data that high stakes induce choking. A
detailed overview of mechanisms and potential moderators of choking in sport
is further provided by Hill et al. (Hill et al., 2010).

All the above work considers individual players’ choking. Our work enhances
this literature by studying the team effects of individual performance, as
well as, the possible influence they exert on the performance of their
teammates and how this translates to a team’s ability to withstand, or not,
periods of subpar performance of individual players.

##### Computational work on interacting HMMs

Our model is inspired by HMMs (Koller and Friedman, 2009) and in particular
work on “coupled” HMMs (Dong et al., 2012; Pan et al., 2012) where the nodes
of a network are HMMs themselves influenced by the other nodes via the edges
of the network. The most related to ours is the work of Pan et al. (Pan et
al., 2012). Motivated by applications of influence in social media and disease
propagation, Pan et al. propose a model very similar to ours, where the hidden
state of every individual at time step $t$ depends on their own hidden state
at $t-1$ and the others’ hidden states at $t-1$. Contrary to this, our model
assumes that an individual’s hidden state at $t$ depends on their own hidden
state at $t-1$ and the others’ _observed_ states at $t-1$. From the
application point of view this difference is important: players in a team
don’t know the hidden state of their teammates; they only know their
teammates’ observed performance. Thus, our model is better suited for our
application. The computational consequences of this difference are also
important: we are able to compute (and sample) from the posterior probability
distribution during the E-step of our EM algorithm, while they cannot as they
have to deploy variational inference techniques.

$H_{1}^{1}$$H_{1}^{2}$$O_{1}^{2}$$O_{1}^{1}$$H_{2}^{1}$$H_{2}^{2}$$O_{2}^{2}$$O_{2}^{1}$$H_{3}^{1}$$H_{3}^{2}$$O_{3}^{2}$$O_{3}^{1}$
Figure 1. Graphical model representation of our model for $n=2$ entities and
$T=3$ timesteps

##  2\. The model

We start this section by giving some notational conventions we follow
throughout the paper. Then, we give a high-level description of our model, and
provide example instantiations that enable us to better convey its
expressivity.

###  2.1. Notational conventions

For the rest of the paper, we establish the following notation: we use
uppercase letters to denote random variables, e.g. $X$, $O$, $H$ and lowercase
letters to denote non-random quantities, e.g. $x$, $o$, $h$. When the random
variables are vectors we denote them with bold uppercase letters $\mathbf{X}$,
$\mathbf{O}$ and $\mathbf{H}$ and the corresponding vector non-random
quantities $\mathbf{x}$, $\mathbf{o}$ and $\mathbf{h}$. For a vector random
(resp. non-random) variable $\mathbf{X}$ (resp. $\mathbf{x}$), we denote its
$i$-th coordinate with $X^{i}$ (resp. ${x}^{i})$. We also use uppercase
calligraphic letters to denote (non-random) matrices, e.g.
$\mathcal{M},\mathcal{N}$. All matrices in this paper are non-random
quantities. In some cases, we abuse notation, and we also use use uppercase
calligraphic letters to denote sets, e.g. $\mathcal{H}$, and $|\cdot|$ to
denote their cardinality. We use $\text{Pr}(X=x)$ to denote the probability of
random variable $X$ taking the value $x$. For brevity, when clear from
context, we abuse the notation and use $\text{Pr}(X)$ instead of
$\text{Pr}(X=x)$.

###  2.2. Model description

Our team model aims to capture the influence that the team entities/players
have on each other. The model is inspired by Hidden Markov Models and their
interaction as described by Pan et al.(Pan et al., 2012). Despite this initial
inspiration, our model deviates from the aforementioned work in some key ways
which have significant application and computational consequences, discussed
in the last paragraph of the Related work.

Entities: The model assumes of $n$ entities. At time step $t=1,\ldots,T$, each
entity $i$ is associated with a _hidden_ (not observed) state
$H_{t}^{i}\in\mathcal{H}$, where $\mathcal{H}$ is a discrete finite set of
hidden states with cardinality $n_{h}$. At $t$, entity $i$ emits an _observed_
signal $O_{t}^{i}\in\mathcal{O}$, where $\mathcal{O}$ is a discrete finite set
of _observed states_ with cardinality $n_{o}$. We assume that we have
observations for $T$ timesteps.

In our case, the entities correspond to the _players_ , members of a team.
Since we consider basketball teams we have $n=5$.

The hidden states can be thought of as the psychological/mental states of each
player. In our experiments, we define $3$ hidden states
$\mathcal{H}=\\{-1,0,1\\}$ where $-1$ is the lowest (worst), $0$ is the
average and $3$ is the highest (best) possible mental state of a player. The
observed states capture the performance of each player during a game. We again
experiment with $3$ observed states $\mathcal{O}=\\{-1,0,1\\}$; $0$
corresponds to a player performing within their average performance, $1$
(resp. $-1$) corresponds to a player’s performance above (resp. below) his/her
expected performance. We give the details of how we set those in real data in
Section 5.

Model parameters: The key assumption of our model is that the hidden state
$H_{t}^{i}$ of an entity $i$ at time $t$ depends on the observed states of all
entities at time $t-1$, denoted by $\mathbf{O}_{t-1}$, as well as
$H_{t-1}^{i}$ (i.e., the hidden state of $i$ at time $t-1$). Figure 1 shows
the graphical model representation of our model for $n=3$ and $T=3$.

Central for our model is the computation of the conditional probability

$\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{t-1},H_{t-1}^{i}).$

In general, computing this probability is hard due to the exponential state
space. Thus, we adopt the following simplifying assumption:

$\displaystyle\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{t-1},H_{t-1}^{i})=$
$\displaystyle\sum_{j=1}^{n}\underbrace{R_{ij}}_{\text{tie
strength}}\times\underbrace{\text{Infl}(H_{t}^{i}\mid
O_{t-1}^{j})}_{\text{influence }j\rightarrow
i}\;+\underbrace{r_{i}\times\text{Infl}(H_{t}^{i}\mid
H_{t-1}^{i})}_{\text{previous hidden state influence}},$

where $R_{ij}$’s and $r_{i}$ are convex coefficients; i.e.,
$\sum_{j=1}^{n}R_{ij}+r_{i}=1$ and $R_{ij},r_{i}\geq 0$. For brevity, we also
use vector $\mathcal{R}^{i}=(r_{i}\mid R_{i1},\dots,R_{in})$ to store the
convex coefficients that appear in Eq. (2.2). Finally, we use $\mathcal{R}$ to
denote the $n\times(n+1)$ matrix that contains all $n$ $\mathcal{R}^{i}$
vectors as its rows.

Note that $\mathcal{R}$ is an important parameter as it encodes the structure
of influence within the team. In Section 5 this is the matrix we are
inspecting in order to analyze the structure of the teams we consider. We
refer to $\mathcal{R}$ as the _team structure_.

$\text{Infl}(H_{t}^{i}=h\mid O_{t-1}^{j}=o)$ corresponds to the influence that
$O_{t-1}^{j}=o$ has on $H_{t}^{i}=h$. We store these influence values in
$n_{o}\times n_{h}$ matrices $\mathcal{M}^{ij}=\text{Infl}(H_{t}^{i}\mid
O_{t-1}^{j})$. Similarly, $\text{Infl}(H_{t}^{i}=h\mid
H_{t-1}^{i}=h^{\prime})$ corresponds to the influence that
$H_{t-1}^{i}=h^{\prime}$ has on $H_{t}^{i}=h$. For notational convenience, we
define an $n_{h}\times n_{h}$ matrix $\mathcal{N}^{i}$ storing these influence
values; i.e., $\mathcal{N}^{i}=\text{Infl}(H_{t}^{i}\mid H_{t-1}^{i})$. We
assume that both matrices $\mathcal{M}^{ij}$ and $\mathcal{N}^{i}$ are row-
stochastic.

One may be tempted to interpret these influence values as probabilities and
assume that $\text{Infl}(H_{t}^{i}\mid O_{t-1}^{j})=\text{Pr}(H_{t}^{i}\mid
O_{t-1}^{j})$ and $\text{Infl}(H_{t}^{i}\mid
H_{t-1}^{i})=\text{Pr}(H_{t}^{i}\mid H_{t-1}^{i})$. In Appendix A.2 we show
that this is not true and that the $\text{Infl}(\cdot)$ values are not
necessarily probabilities. In order to establish that our model is well-
defined we show the following lemma, with proof in Appendix A.1:

###### Lemma 0.

$\text{Pr}(\cdot\mid\mathbf{O}_{t-1},H_{t-1}^{i})$ induces a probability law.

Finally, we need to specify the _emission probabilities_
$\text{Pr}(O_{t}^{i}|H_{t}^{i})$ that connect the hidden state of an entity to
its observed state. We store those in matrix:
$\mathcal{E}^{i}=\text{Pr}(O_{t}^{i}|H_{t}^{i})$.

Matrices $\mathcal{M}^{ij}$, $\mathcal{N}^{i}$, $\mathcal{R}$ and
$\mathcal{E}^{i}$ are _the parameters_ of our model denoted by
$\boldsymbol{\Theta}=\\{\mathcal{M}^{ij},\mathcal{N}^{i},\mathcal{R},\mathcal{E}^{i}\mid
i,j\in[n]\\}$. These parameters fully describe our model and each
instantiation of the parameters corresponds to a different _team profile_.

In many practical applications it is reasonable to assume that matrices
$\mathcal{M}^{ij}$ are the same for all $j\in[n]$. In such cases we write
$\mathcal{M}^{i}$, instead of $\mathcal{M}^{ij}$. Simplifying the model even
further we may have all entities to share the same $\mathcal{M}$ matrix. In
practice, it is also reasonable to assume that there is a single
$\mathcal{N}$, and $\mathcal{E}$ shared among all entities. In our experiments
we make such simplifying assumptions assuming that its parameters are
$\boldsymbol{\Theta}=\\{\mathcal{M},\mathcal{N},\mathcal{R},\mathcal{E}\\}$.

###  2.3. Team collapse

Given the above model, we can now formally define the _hidden_ and the
_observed team states_ and the notions of _player choking_ and _team
collapse_.

###### Definition 0.

The _hidden (resp. observed) team state_ is the average hidden (resp.
observed) state of the team’s players.

###### Definition 0.

A player $i$ is in a _chocking state_ at time $t$, if $H_{t}^{i}=-1$, i.e.,
the player is in the lowest possible hidden state.

###### Definition 0.

A team is in an $\alpha$-_collapse state_ at time $t$, if $\alpha$ fraction of
the team’s players are in a _chocking_ state at time $t$; for
$\alpha\in[0,1]$.

We use these definitions of player choking and team collapse when analysing
the expressivity of our model in Section 3 and when doing experiments with
data from NBA teams in Section 5.

##  3\. Exploring team profiles

In this section, we experimentally evaluate how prone are different team
profiles to team collapse. For this, we focus on a few basic team profiles,
i.e., different instantiations of the model parameters
$\Theta=\\{\mathcal{M},\mathcal{N},\mathcal{R},\mathcal{E}\\}$111We focus on
the simplified version of the model.. Then, we generate data using the
instances of $\Theta$ we consider and study different properties of the
corresponding team profile. These properties include the _average hidden
(resp. observed) team state_ (Def. 2) and the probability that a team gets
into a collapse state (Def. 4).

The goal of this section is not only to study the properties of the particular
profiles we are considering, but also demonstrate how our model can be used by
experts in team sports to analyze different team compositions and their
robustness to team collapses.

###  3.1. Team profiles

Below we describe the design principles for the team profiles we experiment
with in the main body of the paper; a more extensive list of team profiles as
well as some specific instances of the parameters are also shown in Appendix
B.

Setting $\mathcal{M}$: Matrix $\mathcal{M}$ captures the influence of
everyone’s observed states to the hidden state of a player. When designing
$\mathcal{M}$, we assume that
$\mathcal{M}(o=\ell,h=\ell)>\mathcal{M}(o=\ell,h=\ell^{\prime})$ for all
$\ell\neq\ell^{\prime}$ and $\ell,\ell^{\prime}\in\\{-1,0,1\\}$. That is, when
observing a state that is similar to their hidden state, the players do not
change their hidden state. Moreover, $\mathcal{M}(o=\ell,h=\ell^{\prime})$ is
monotonically decreasing with the difference $|\ell-\ell^{\prime}|$.
Intuitively, this means that a player’s hidden state gets affected less by
observed states that are very different from his/hers current hidden state.
For the experiments we show here we assume that $\mathcal{M}(o=\ell,h=\ell)$
is the same for all values of $\ell\in\\{-1,0,1\\}$. In this scenario any
observed state of any player influences the others to stay in the
corresponding hidden state. The particular matrix we used for our simulations
in shown in Appendix B.1.

Setting $\mathcal{N}$: Matrix $\mathcal{N}$ captures the influence of a
player’s hidden state at $t-1$ to their own hidden state at $t$. Our
assumption is that each player is inclined to stay in the same hidden state;
i.e., $\mathcal{N}(h,h)>\mathcal{N}(h,h^{\prime})$, for all $h\neq h^{\prime}$
and $h,h^{\prime}\in\mathcal{H}$.

Setting $\mathcal{R}$: Parameters $\mathcal{R}^{i}=(r_{i}\mid
R_{i1},\cdots,R_{in})$ for all $i\in[n]$ assign weights to the influence
matrices $\mathcal{M}$ and $\mathcal{N}$ (see Eq. (2.2)). We consider
different scenarios for $\mathcal{R}^{i}$:

  * $\bullet$

$\mathcal{R}_{\text{H}}$: All players depend on their previous hidden state.
That is, $r_{i}=1,\forall i\in[n]$ and $R_{ij}=0,\forall i,j\in[n]$. We denote
the profile that corresponds to this $\mathcal{R}$ with H.

  * $\bullet$

$\mathcal{R}_{\text{HO}}$: All players depend only on their previous hidden
and observed states. That is, $r_{i}>R_{ii}>0,\forall i\in[n]$ and
$R_{ij}=0,\forall i,j\in[n]$. For our experiments we use $r_{i}=0.7$ and
$R_{ii}=0.3$, for all $i\in[n]$.

  * $\bullet$

$\mathcal{R}_{k}$: There are $k$ _pillar players_ $P$ that affect all others;
i.e., $r_{i}=R_{ii}=0$ and $R_{ij}=1/|P|$ for all $i\notin P$ and $j\in P$ and
$r_{i}=1$ for $i\in P$.

  * $\bullet$

$\mathcal{R}_{\text{kH}}$: There are $k$ pillar players $P$ that affect all
others, but all non-pillar players are also affected by their hidden states.
That is, $r_{i}=1$ for all $i\in P$, $r_{i}>0$ for all $i\notin P$ and
$R_{ij}=(1-r_{i})1/|P|$ for all $i\notin P$ and $j\in P$. In our experiments
we set $r_{i}=0.5$ for all $i\notin P$.

  * $\bullet$

$\mathcal{R}_{\text{kD}}$: In this case, there are again $k$ pillar players
$P$ (as in $\mathcal{R}_{k}$, but this time they depend on each other. That
is, for every $i\in P$ $0<r_{i}<1$ and $R_{ij}=(1-r_{i})1/|P|$ for every
$i,j\in P$. In our experiments we again set $r_{i}=0.5$.

Setting $\mathcal{E}$: Matrix $\mathcal{E}$ controls the emission
probabilities. We assume that the observed states are strongly correlated with
hidden states. That is,
$\mathcal{E}(o=\ell,h=\ell)>\mathcal{E}(o=\ell,h=\ell^{\prime})$ all
$\ell,\ell^{\prime}\in\\{-1,0,1\\}$ and $\ell\neq\ell^{\prime}$. For example,
if a player is in a poor mental state, we expect their performance (observed
state) to also be poor low.

Since we do not vary $\mathcal{M}$, $\mathcal{N}$ and $\mathcal{E}$, the
instantiations of $\mathcal{R}$ give rise to different team profiles. Thus, we
have the following mappings between the $\mathcal{R}$ instances and team-
profile names: $\mathcal{R}_{\text{H}}\rightarrow\texttt{H}$,
$\mathcal{R}_{\text{HO}}\rightarrow\texttt{HO}$,
$\mathcal{R}_{k}\rightarrow\texttt{kPillars}(\texttt{kP})$,
$\mathcal{R}_{\text{kH}}\rightarrow\texttt{kPillars+H}(\texttt{kP + H})$ and
$\mathcal{R}_{\text{kD}}\rightarrow\texttt{kPillars+D}(\texttt{kP + D})$.

###  3.2. Simulations

In order to better understand our models we create data that correspond to $7$
distinct team profiles; for a larger set of team profiles see Appendix B.2.
Using these profiles we generate $1000$ datasets for each profile, where each
dataset has $n=5$, $T=100$ and every player $i\in[n]$ starts from hidden state
$H_{1}^{i}=0$.

In Appendix B.3 we show the _hidden_ and _observed team states_ for the
profiles we mention above and others we consider in the appendix. The summary
of the results is the following: in almost all cases the hidden and the
observed team states follow closely one another. All teams’ states converge to
the expected performance (0). Some differences are observed between profiles
with respect to the 25-75 percentiles of the values we get in the simulations.

Probability of team collapse. In order to empirically estimate the probability
of a team collapse for a given team, we count all the times the team collapsed
in the simulations and divide by the total number of samples ($1000\times
100=10^{5})$. In these experiments, we consider that $\alpha=1$ (see Dfn. 4);
i.e. a team collapses if all its players have choked. In Fig. 10 we show the
team-collapse probability for the different team profiles we considered.

Figure 2. Team collapse probability vs team profile

First, we observe that when all players are completely unaffected by other
players (profiles H and HO), then the team rarely collapses. In this case, the
only way for a collapse to occur is all players independently under-performing
at the same time. In general, adding dependence on the hidden state of the
players, i.e., making the players more self-dependent results in a more
resilient team.

We also observe that the when there is one pillar in the team (profiles
1Pillar and 1Pillar+H), the team has a single point of failure. That is, when
the pillar happens to under-perform, the rest of the players are negatively
affected and transition to their lowest mental (hidden) state. Subsequently,
these players also under-perform until the performance of the pillar player
improves. Naturally, adding more pillars (profiles kPillars, ) makes the team
more resilient to collapse since not all pillars under-perform at the same
time.

Furthermore, dependence between pillar players (e.g., profile 2Pillars+D)
results in less resilient teams. This is because, dependent pillar players
have a tendency to under-perform at the same time. Once this happens, the rest
of the players transition to hidden state $-1$ and start under-performing.

Finally, we emphasize that a team that doesn’t under-perform isn’t necessarily
a good team. Observed performance is measured relatively to the average
performance of the team. Thus, a consistently bad team might never under-
perform, but still perform poorly relatively to other teams.

###  3.3. Remarks on team profiles

We summarize here our observations and state some principles we experimentally
found to be true and can provide useful insights to experts reasoning about
team composition and resilience of teams to collapses.

###### Remark 1.

(Self-dependence.) Self-dependent players, i.e., players that primarily depend
on their own hidden and observed states, improve the resilience of teams.

We say that a player is a _pillar_ if other players heavily depend on him/her,
while he/she is self-dependent. For example, a player $i$ is a pillar player
if $r_{i}\approx 1$ and $R_{ji}\approx 1,\forall j\in[n]$.

###### Remark 2.

(Pillars.) More pillars are better than less pillars.

###### Remark 3.

(Dependence between pillars.) Dependence between pillars makes teams more
vulnerable to team collapse.

These observations have several implications in terms of team and roster
construction. More specifically, self-dependent players are also typically
players that can do several things at a high level. This is in contrast to
“specialists” that bring to the team a very well-delineated skillset (e.g.,
individual defense and three point shooting) and depend on their teammates for
other aspects of the game (e.g., shot generation). While lately there are many
teams that prioritize the acquisition of specialists who possess specific
skillset at an elite level, our observations indicate that having more self-
dependent players makes a team more resilient to a possible collapse.

Furthermore, as one might have expected, a team with self-dependent players
that can also influence/help their teammates’ performance (i.e., pillar
players) is in general more resilient. However, the choice of these pillar
players should take into consideration their dependence. According to our
third remark, if a team includes pillars that are dependent, the resiliency to
team collapse reduces. This dependency can be related to the position the two
pillars cover on the team. For instance, if a team’s pillar A is a center and
its other pillar B is a guard there can be directional dependency from the B
to A, just because B controls the ball more and is the one responsible for
distributing it to other players. This dependency does not have to be strong,
but its effects can trickle down to the rest of the players. Hence, from a
roster-construction perspective, assuming a team already has one pillar,
understanding the dependency that will be introduced when adding a second one
is crucial. Going back to the example above, the best pillar pairing for
player B might be another guard instead of a center.

##  4\. Learning the model

In this section, we describe a learning algorithm. Given the observed states
of the players of a team over a period of time, it estimates the parameters of
our model and the corresponding hidden states of the players.

Throughout the description we follow the following notation: we use
$\mathbf{O}$ to denote the sequence of observations
$\mathbf{O}_{1},\cdots,\mathbf{O}_{T}$, where
$\mathbf{O}_{t}=(O_{t}^{1},\dots,O_{t}^{n})$ are the observed signals of all
entities at time $t$. Accordingly, we use $\mathbf{H}$ to denote the sequence
of hidden states $\mathbf{H}_{1},\cdots,\mathbf{H}_{t}$, where
$\mathbf{H}_{t}=(H_{t}^{1},\dots,H_{t}^{n})$ are the hidden states of all
entities at time $t$. Finally, we use $\mathbf{O}_{1:t}$ to denote
$\mathbf{O}_{1},\dots,\mathbf{O}_{t}$. Thus, given $\mathbf{O}$, we want to
learn the parameters
$\boldsymbol{\Theta}=\\{\mathcal{M}^{ij},\mathcal{N}^{i},\mathcal{R},\mathcal{E}^{i}\mid
i,j\in[n]\\}$ of the model and the sequence of hidden states $\mathbf{H}$.

###  4.1. Learning the parameters of the model

We learn the parameters of the model by applying the _Expectation-
Maximization_ (EM) algorithm. EM alternates between two steps: the expectation
step (E-step) and the maximization step (M-step). In our case, these steps
are:

E-step: In this step, we need to calculate the _expected complete log-
likelihood_ :

(2) $\displaystyle
Q(\boldsymbol{\Theta}\mid\boldsymbol{\Theta}^{\text{old}})=\mathbb{E}_{\mathbf{H}\sim\text{Pr}(\mathbf{H}\mid\mathbf{O},\boldsymbol{\Theta}^{\text{old}})}\left[\log\text{Pr}(\mathbf{H},\mathbf{O}\mid\boldsymbol{\Theta})\right],$

Usually, we approximate the above expectation by drawing samples from the
_posterior distribution_
$\text{Pr}(\mathbf{H}\mid\mathbf{O},\boldsymbol{\Theta})$ and calculating the
average of the function.

M-step: In this step, we need to maximize the _expected complete log-
likelihood_ w.r.t. the parameters of the model.

EM alternates between the E and the M steps until convergence.

####  4.1.1. E-step

In the E-step we need to sample from the posterior distribution
$\text{Pr}(\mathbf{H}\mid\mathbf{O},\boldsymbol{\Theta})$, assuming that the
model parameters $\boldsymbol{\Theta}$ are known. Since it is clear from the
context that all probabilities will be conditioned on $\boldsymbol{\Theta}$,
we omit the parameters in the condition and write the posterior as
$\text{Pr}(\mathbf{H}\mid\mathbf{O})$

The key idea for being able to sample from the posterior is that we need to
write $\text{Pr}(\mathbf{H}\mid\mathbf{O})$ as a product where each random
variable $\mathbf{H}_{t}$ is only conditioned on observed quantities or on
random variables $\mathbf{H}_{t^{\prime}}$ with $t^{\prime}<t$. This is
summarized in the following Lemma, which is the main result of this section:

###### Lemma 0.

It holds that

(3)
$\displaystyle\text{Pr}(\mathbf{H}\mid\mathbf{O})=\prod_{i=1}^{n}\prod_{t=1}^{T}\frac{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{1:t-1})$

The proof of the lemma is based on a set of observations and it is quite
lengthy, so it is given in Appendix D.1. This lemma allows us to sample
$H_{t}^{i}$ from a distribution that is proportional to

$\frac{\text{Pr}(H_{t}^{i}=h\mid\mathbf{O}_{1:t})\text{Pr}(H_{t}^{i}=h\mid
H_{t-1}^{i},\mathbf{O}_{t-1})}{\text{Pr}(H_{t}^{i}=h\mid\mathbf{O}_{1:t-1})},$

as dictated by Lemma 1.

####  4.1.2. M-step

In the M-step we want to maximize the the expected complete log-likelihood
w.r.t. the parameters of the model. To see how this can be approached we first
write the complete likelihood:

$\displaystyle\text{Pr}(\mathbf{H},\mathbf{O})$
$\displaystyle=\text{Pr}(\mathbf{H}_{1})\prod_{t=2}^{T}\text{Pr}(\mathbf{H}_{t}\mid\mathbf{H}_{t-1},\mathbf{O}_{t-1})\prod_{t=1}^{T}\text{Pr}(\mathbf{O}_{t}\mid\mathbf{H}_{t})$
$\displaystyle=\text{Pr}(\mathbf{H}_{1})\prod_{t=2}^{T}\prod_{i=1}^{n}\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{t-1})\prod_{t=1}^{T}\prod_{i=1}^{n}\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})$

Taking the logarithm on both sides we get the complete log-likelihood:

(4) $\displaystyle\log\text{Pr}(\mathbf{H},\mathbf{O})$
$\displaystyle=\log\text{Pr}(\mathbf{H}_{1})$
$\displaystyle+\sum_{t=2}^{T}\sum_{i=1}^{n}\log\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{t-1})$
$\displaystyle+\sum_{t=1}^{T}\sum_{i=1}^{n}\log\text{Pr}(O_{t}^{i}\mid
H_{t}^{i}).$

Thus in the M-step we have to solve the following problem:

(5)
$\displaystyle\boldsymbol{\Theta}^{\text{new}}=\operatorname*{arg\,max}_{\boldsymbol{\Theta}}Q(\boldsymbol{\Theta}\mid\boldsymbol{\Theta}^{\text{old}}),$

where $Q()$ is given by Eq. (2) and the complete log-likelihood is computed by
Eq. 4. Note that $Q(\boldsymbol{\Theta}\mid\boldsymbol{\Theta}^{\text{old}})$
is not a concave function in the parameters $\boldsymbol{\Theta}$ and thus
finding a global maximum it not easy. Specifically, the problem stems from the
terms $\text{Pr}(H_{t}^{i}\mid H_{t-1}^{i},\mathbf{O}_{t-1})$ which contain
products of variables. Thus, we opted for a round-robin heuristic algorithm.
This heuristic iteratively optimizes one parameter from
$\boldsymbol{\Theta}=\\{\mathcal{M},\mathcal{N},\mathcal{R}^{i},\mathcal{E}\mid
i\in[n]\\}$ per iteration, while keeping the other parameters fixed; in all
cases fixing all but one of the parameters results in a concave function.
Hence, we can use standard convex-optimization tools to optimize for each
parameter separately. In practice, we found that 1-2 iterations of the round
robin heuristic are adequate.

###  4.2. Likelihood calculation

Given the model parameters $\boldsymbol{\Theta}$, we can calculate the
likelihood of the data $\Pr(\mathbf{O})$, based on the results of the previous
section:

(6) $\displaystyle\log\Pr(\mathbf{O})$
$\displaystyle=\sum_{t=1}^{T}\sum_{i=1}^{n}\log\Pr(O_{t}^{i}\mid\mathbf{O}_{t-1})$
(7)
$\displaystyle=\sum_{t=1}^{T}\sum_{i=1}^{n}\log\left[\sum_{h\in\mathcal{H}}\Pr(O_{t}^{i}\mid
H_{t}^{i}=h)\Pr(H_{t}^{i}=h\mid\mathbf{O}_{1:t-1})\right].$

###  4.3. Decoding the hidden states

In decoding , we find the most likely hidden variables path
$\mathbf{H}_{1},\dots,\mathbf{H}_{T}$, given the model parameters
$\boldsymbol{\Theta}$ (evaluated by EM) and the observed variables
$\mathbf{O}=\mathbf{O}_{1:T}$. Formally, this problem can be expressed as

(8)
$\displaystyle\overline{\mathbf{h}}=\operatorname*{arg\,max}_{\mathbf{h}}\text{Pr}(\mathbf{H}=\mathbf{h}\mid\mathbf{O}).$

We solve the problem expressed in Eq. (8) via a dynamic-programming algorithm,
which is very similar to the _Viterbi_ algorithm used for standard HMMs
(Bishop, 2006). To this end, first we decouple the chains

$\displaystyle\text{Pr}(\mathbf{H}=\mathbf{h}\mid\mathbf{O})=\prod_{i=1}^{n}\text{Pr}(\mathbf{H}^{i}=\mathbf{h}^{i}\mid\mathbf{O}),$

using the independence of $\mathbf{H}^{i}$ and $\mathbf{H}^{j}$ given
$\mathbf{O}$. For each entity $i$, we define $\delta_{i}(t,h)$ to be the
probability of the maximum-probability path ending at $H_{t}^{i}=h$; formally:

$\displaystyle\delta_{i}(t,h)=\max_{h_{1},\dots,h_{t-1}\in\mathcal{H}^{t-1}}$
$\displaystyle\text{Pr}(H_{1}^{i}=h_{1},\cdots,H_{t-1}^{i}=h_{t-1},H_{t}^{i}=h\mid
O_{1}^{i},\cdots O_{t}^{i}).$

That is, $\delta_{t}(h)$ is the probability of the most probable path ending
at $H_{t}^{i}=h$. Now the values of $\delta_{i}(t,h)$ can be computed using
the following dynamic-programming recursion:

$\displaystyle\delta_{i}(t,h)=\text{Pr}(O_{t}^{i}\mid H_{t}^{i}=h)\times$
$\displaystyle\max_{h^{\prime}\in\mathcal{H}}\left(\delta_{i}(t-1,h^{\prime})\text{Pr}(H_{t}^{i}=h\mid
H_{t-1}^{i}=h^{\prime},\mathbf{O}_{t-1})\right),$

$\forall h\in\mathcal{H}$ and $t\in\\{1,\ldots,T\\}$.

The base case is

$\displaystyle\delta_{i}(1,h)=\text{Pr}(H_{1}^{i}=h)\text{Pr}(O_{1}^{i}\mid
H_{1}^{i}=h),\forall h\in\mathcal{H}.$

Once all $\delta_{i}(t,h)$ values are computed one can trace back the
solutions starting from $\delta_{i}(T,h)$, where
$h=\operatorname*{arg\,max}_{h^{\prime}\in\mathcal{H}}\delta_{i}(T,h^{\prime})$.

###  4.4. Evaluating the learning algorithm on synthetic data

We evaluated our implementation of the EM algorithm on synthetic data as
follows. We created a 1Pillar team profile and generated samples that
correspond to $T=100$ timesteps. We then ran the EM algorithm and found that
it correctly identifies the structure of the team. Figure 3 shows the Negative
Evidence Lower Bound (N-ELBO) as a function of the iterations of EM. The
figure shows the average and the standard deviation of N-ELBO over $10$
different initializations. As expected, N-ELBO decreases with iterations until
it stabilizes to a low value. The negative likelihood of the (best) learned
model is $416$, while the negative likelihood of the true model is $417$
(slightly larger). That is, EM finds a model that has even better likelihood
than the model used to generate the data.

Figure 3. Negative Evidence Lower Bound (N-ELBO) vs iterations of EM; average
and standard deviation over 10 random initializations.

Negative Evidence Lower Bound (N-ELBO) vs iterations of EM

(a) Dallas Mavericks

(b) Cleveland Cavaliers

(c) Boston Celtics

(d) Houston Rockets

Figure 4. Team structure ($\mathcal{R}$) for the Dallas Mavericks, Cleveland
Cavaliers, Boston Celtics and Houston Rockets.

##  5\. Real-world experiments

In this section, we validate the practical utility of our model by using it to
identify the structure of all NBA teams for the season 2021-2022. We also show
how our model can be used to analyze and obtain significant insights for
specific games.

###  5.1. Data

For each NBA team of the 2021-2022 season we selected their starting lineup
players. That is, the players that played the most minutes in each position
(Point Guard, Shooting Guard, Small Forward, Power Forward, Center)222The
starting lineups are obtained from the basketball reference website. For
example, for Boston Celtics the starting lineup is here:
https://www.basketball-reference.com/teams/BOS/2022_depth.html.

For our experiment,s we used a Play-by-Play dataset (PBP) that contains a
detailed record of events for all NBA games of the season. Using PBP, for each
player, we computed his average PIE metric333See
https://www.nba.com/stats/help/faq across all quarters (each game has four
12min quarters) of all games. At a high level, the PIE metric of a player for
a given time window, captures the percentage of events (points, assists,
rebounds, etc) that the player achieved in the given window. Finally, we
labeled the performance of every player for every quarter he played as average
($O_{t}^{i}=0$), if his PIE metric was within $\pm\epsilon$ of his average
PIE, where $\epsilon=0.05$. If his performance was above this region, we
labeled his performance as over-performance $O_{t}^{i}=1$. Accordingly, if his
performance was below the average region we labels his performance as under-
performance $O_{t}^{i}=-1$. If the player didn’t play in a quarter we used his
latest PIE measurement in the game, or if there was no previous measurement we
assumed average performance $O_{t}^{i}=1$. We name the resulting dataset
NBAseason.

For section 5.4, we created the NBAgames dataset. This dataset contains a few
selected games of interest. We divided each of these games into 3 minute time
windows so as to have more fine-grained measurements. As before, for each
player we computed his average PIE metric in each 3 minute window and labeled
his performance.

###  5.2. Implementation details

In order to apply EM in real data, we created a collection of approximately
$10000$ team profiles, i.e., instantiations of our model. These profiles were,
to a large extent, informed by our findings in Section 4.4. Then, we found the
profile from the collection that best matches each team by calculating the
likelihood of the profile. Finally, for each team we ran the EM algorithm
initialized with the aforementioned profile.

The implementation of our algorithms, along with the datasets we used in our
experiments are publicly available at https://github.com/jasonNikolaou/Team-
collapse.git.

###  5.3. Team structure in NBA teams

Using dataset NBAseason, we applied the EM algorithm to find the maximum-
likelihood estimates of the parameters of our model as described in Section
5.2.

(a) Nets vs Knicks game (16 Feb 2022)

(b) Celtics vs Knicks game (6 Jan 2022)

(c) Lakers vs Thunder (27 Oct 2021)

Figure 5. Team (hidden) state and points difference (PTS +/-) per 3 minute
window for different games.

Thus, for every one of the 30 NBA teams we obtained an estimate of its _team
structure_ $\mathcal{R}$. While of course each team has a different structure,
there are two main team-structure types that emerge. In Figure 4 we present
four representative teams of these team structures (two for each), while we
present the rest of the team structures in Appendix E.1.

The team structures correspond to the matrix $\mathcal{R}$ of the team profile
and therefore, they are presented as a $5\times 6$ heatmap. The first column
in the heatmap corresponds to the dependency of each row player $i$ to its
hidden state (i.e., $r_{i}$), and the rest $5\times 5$ matrix represents the
dependency between the hidden state of the row player $i$ and the observed
state of the column player $j$.

The first two team structures, shown in Figures 4(a) and 4(b) correspond to
the structures of Dallas Mavericks and Cleveland Cavaliers respectively. For
the Mavericks, we see that the state of every player is mainly dependent on
their own hidden state (first column of the matrix), while they do not depend
on their own observations. Simply put, these teams consist of players that
typically can showcase “short memory” with respect to their observed
performance/state and their state only depends on their own hidden (mental)
state. The Cavaliers’ team structure is very similar.

The other two team structures shown in Figures 4(c) and 4(d) correspond to
Boston Celtics and Houston Rockets respectively. For Celtics, we see a smaller
dependency of their players’ on their own hidden state and a larger dependency
for some of them on their observed state. However, there is also a dependency
on another player’s observed state (last column in Figure 4(c)). In this
particular case , this player is Al Horford, who is the most experienced
player on the team and his teammates can be thought of as “feeding off” his
observed state/performance. The Rockets team structure (Figure 4(d)) is very
similar.

Interestingly, while both the Dallas Mavericks and Boston Celtics have very
different structures, they both reached the conference finals and finals
respectively. On the other hand, the Houston Rockets, which have structure
similar to Boston Celtics, had a record of 20 wins and 62 loses and finished
last in the Western conference. This demonstrates that there are many ways to
succeed (or fail) in a team sport. The team structure is just one
characteristic of a team, among others that create different vulnerabilities
and strengths.

###  5.4. Diving deeper into specific games

For the results we report here, we use the NBAgames dataset described in
section 5.1. Using the parameters learned in the previous section, we decode
the most probable hidden states of players by applying the decoding algorithm
described in Section 4.3.

We now analyze three games in detail. In all these games, one of the teams
managed to comeback and win the game despite trailing behind in score for the
most part of the game. More analyzed games can be found in the Appendix E.2.

Figure 5 visualizes the hidden team state we obtained from our model over the
3 minute windows, and overlays the actual point differential for the game
during these states. In these games, a team was performing well to start the
game (Knicks, Celtics and Lakers respectively in the figure order), but later
in the game there was a “transition” and they surrendered the lead and
ultimately lost the game (i.e., collapsed). As we can see the inferences on
the team state made by our model follow closely the point differential, and
hence the status, of the game. These results indicate that our model captures
the underlying hidden states of the players to a great extent and can explain
what one observes at the game.

##  6\. Conclusions

In this paper, we proposed a probabilistic graphical model that allowed us to
formalize the dependencies between team players’ hidden/mental states and the
observed performance of themselves and their teammates. Then, we used this
model to identify, analyze and explain team collapses, i.e., events where a
team severely under-performed. We also showed that we can learn the parameters
of this model using an EM algorithm. Furthermore, we used our model to analyze
real-world team data from NBA games in 2021-22 season. As a result, we found
some interesting team structures and identified games where according to our
model a team collapsed.

One shortcoming of our model is the assumption that players in the team do not
change over time. Considering a dynamic version of the model, where players
can get substituted is an interesting direction of future work. In our
experiments with real-world data, we relied on the use of the PIE metric to
measure the observed performance of players. In the future, we plan to explore
other metrics as well as make use of machine learning methods in order to
classify the observed performance. Finally, we can potentially use our model
to calculate collapse probabilities, as predictors of future collapses. In
such a scenario, our model can guide coaching decisions, e.g., substitutions,
time-out, etc.

Reproducibility. We make the code and the data we used in our experiments
publicly available at https://github.com/jasonNikolaou/Team-collapse.git.

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##  Appendix A Appendix for Section 2

###  A.1. Proof of Lemma 1

###### Proof.

$\displaystyle\sum_{h\in\mathcal{H}}(H_{t}^{i}=h\mid\mathbf{O}_{t-1},H_{t-1}^{i})$
$\displaystyle=\sum_{h\in\mathcal{H}}\left(\sum_{j=1}^{n}R_{ij}\text{Infl}(H_{t}^{i}=h\mid
O_{t-1}^{j})+R_{i}\text{Infl}(H_{t}^{i}=h\mid H_{t-1}^{i})\right)$
$\displaystyle=\sum_{j=1}^{n}R_{ij}\sum_{h\in\mathcal{H}}\text{Infl}(H_{t}^{i}=h\mid
O_{t-1}^{j})+R_{i}\sum_{h\in\mathcal{H}}\text{Infl}(H_{t}^{i}=h\mid
H_{t-1}^{i})$ $\displaystyle=\sum_{j=1}^{n}R_{ij}+R_{i}$ $\displaystyle=1.$

∎

###  A.2. On the influence matrices

In this section, we answer the following question: why isn’t $\mathcal{M}$ the
same as the probability distributions $\text{Pr}(H_{t}^{i}\mid O_{t-1}^{i})$.
Consider the following proof by contradiction. Let $n=1$ and $r_{1}=0$,
$R_{11}=1$. We define the following parameters:

(9) $\displaystyle\mathcal{N}=\begin{bmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{bmatrix},\mathcal{N}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},\mathcal{E}=\begin{bmatrix}0.7&0.2&0.1\\\ 0.2&0.6&0.2\\\
0.1&0.2&0.7\end{bmatrix},$

Assume that:

(10) $\displaystyle\text{Pr}(H_{t}^{i}=h\mid O_{t-1}^{i}=o)=\mathcal{M}[h,o]$

We have

(11) $\displaystyle\text{Pr}(H_{t}^{1}=0\mid H_{t-1}^{1}=1,O_{t-1}^{1}=2)$
$\displaystyle=r_{1}\mathcal{N}[1,0]+R_{11}\mathcal{M}[2,0]$ (12)
$\displaystyle=0+1\times 1$ (13) $\displaystyle=1$

and

(14) $\displaystyle\text{Pr}(H_{t}^{1}=0\mid
H_{t-1}^{1}=1)=\mathcal{N}[1,0]=0.$

We also have

(15) $\displaystyle\text{Pr}(H_{t}^{1}=0\mid H_{t-1}^{1}=1)$
$\displaystyle=\sum_{o}\text{Pr}(H_{t}^{1}=0\mid
H_{t-1}^{1}=1,O_{t-1}^{1}=o)\text{Pr}(O_{t-1}^{1}=o\mid H_{t-1}^{1}=1)$ (16)
$\displaystyle\geq\text{Pr}(H_{t}^{1}=0\mid
H_{t-1}^{1}=1,O_{t-1}^{1}=2)\text{Pr}(O_{t-1}^{1}=2\mid H_{t-1}^{1}=1)$ (17)
$\displaystyle\geq 1\times 0.2$ (18) $\displaystyle=0.2,$

which is a contradiction.

In the same way, we can prove that $\mathcal{N}$ isn’t the same as the
probability distributions $\text{Pr}(H_{t}^{i}\mid H_{t-1}^{i})$.

##  Appendix B Appendix for Section 3

###  B.1. Examples of model parameters used in our experiments

$\mathcal{N}$ instance used in the experiments presented in Section 3:

(19) $\displaystyle\mathcal{N}=\begin{bmatrix}0.7&0.2&0.1\\\ 0.2&0.6&0.2\\\
0.1&0.2&0.7\end{bmatrix}$

$\mathcal{E}$ instance used in our experiments:

(20) $\displaystyle\mathcal{E}=\begin{bmatrix}0.8&0.2&0\\\ 0.2&0.8&0.2\\\
0&0.2&0.8\end{bmatrix}$

$\mathcal{M}$ instances used in our experiments in Section 3

(21) $\displaystyle\mathcal{M}=\begin{bmatrix}0.8&0.2&0\\\ 0.2&0.8&0.2\\\
0&0.2&0.8\end{bmatrix},$

Additional $\mathcal{M}$ instances we considered:

(22) $\displaystyle\mathcal{M}=\begin{bmatrix}0.9&0.1&0\\\ 0.2&0.6&0.2\\\
0.2&0.3&0.5\end{bmatrix}$

and

(23) $\displaystyle\mathcal{M}_{\text{anti}}=\begin{bmatrix}0&0.2&0.8\\\
0.1&0.8&0.1\\\ 0.8&0.2&0\end{bmatrix}.$

Additional $\mathcal{R}$ instances we considered In addition to the
$\mathcal{R}$ instances we described in Section 3, we also experimented with a
few more $\mathcal{R}$ instances. We describe the full set of those instances
here:

  * $\bullet$

$\mathcal{R}_{\text{H}}$: All players depend on their previous hidden state.
That is, $r_{i}=1,\forall i\in[n]$ and $R_{ij}=0,\forall i,j\in[n]$. We denote
the profile that corresponds to this $\mathcal{R}$ with H.

  * $\bullet$

$\mathcal{R}_{\text{HO}}$: All players depend only on their previous hidden
and observed states. That is, $r_{i}>R_{ii}>0,\forall i\in[n]$ and
$R_{ij}=0,\forall i,j\in[n]$. For our experiments we use $r_{i}=0.7$ and
$R_{ii}=0.3$, for all $i\in[n]$.

  * $\bullet$

$\mathcal{R}_{k}$: There are $k$ _pillar players_ $P$ that affect all others;
i.e., $r_{i}=R_{ii}=0$ and $R_{ij}=1/|P|$ for all $i\notin P$ and $j\in P$ and
$r_{i}=1$ for $i\in P$.

  * $\bullet$

$\mathcal{R}_{\text{kH}}$: There are $k$ pillar players $P$ that affect all
others, but all non-pillar players are also affected by their hidden states.
That is, $r_{i}=1$ for all $i\in P$, $r_{i}>0$ for all $i\notin P$ and
$R_{ij}=(1-r_{i})1/|P|$ for all $i\notin P$ and $j\in P$. In our experiments
we set $r_{i}=0.5$ for all $i\notin P$.

  * $\bullet$

$\mathcal{R}_{\text{kD}}$: In this case, there are again $k$ pillar players
$P$ (as in $\mathcal{R}_{k}$, but this time they depend on each other. That
is, for every $i\in P$ $0<r_{i}<1$ and $R_{ij}=(1-r_{i})1/|P|$ for every
$i,j\in P$. In our experiments we again set $r_{i}=0.5$.

  * $\bullet$

$\mathcal{R}_{\text{kDH}}$: This is a combination of $\mathcal{R}_{H}$ and
$\mathcal{R}_{kD}$ where all pillar players are set as in $\mathcal{R}_{kD}$
and all non-pillar players are set as in $\mathcal{R}_{\text{kH}}$.

  * $\bullet$

$\mathcal{R}_{\text{uniform}}$: In this case we assume that for every player
$i$, $r_{i}=R_{ij}=1/(n+1)$.

###  B.2. Team profiles

We create the following extended sets of profiles:

H: $\mathcal{M}$ as above and $\mathcal{R}^{i}=\mathcal{R}_{H}$.

HO. $\mathcal{M}$ as above and $\mathcal{R}^{i}=\mathcal{R}_{\text{HO}}$.

1Pillar (1P). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{1}}$.

1Pillar + Hidden (1P + H). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{1H}}$.

2Pillars (2P). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2}}$.

2Pillars + Dependence (2P + D). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2D}}$.

2Pillars + Hidden (2P + H). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2H}}$.

2Pillars + Hidden + Dependence (2P + H + D). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2DH}}$.

3Pillars (3P). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{3}}$.

3Pillars + Dependence (3P + D). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{3D}}$.

3Pillars + Hidden (3P + H). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2H}}$.

3Pillars + Hidden + Dependence (3P + H + D). $\mathcal{M}$ as above and
$\mathcal{R}^{i}=\mathcal{R}_{\text{2DH}}$.

Uniform. Players equally depend on their hidden states and other players, i.e.
$R_{i}=1/6,R_{ij}=1/6,\forall i,j\in[n]$.

1Pillar Bad Teammate (BT1P). Same as the 1Pillar profile, but we change the
$\mathcal{M}$ matrix to the following:

(24) $\displaystyle\mathcal{M}=\begin{bmatrix}0.9&0.1&0\\\ 0.2&0.6&0.2\\\
0.2&0.3&0.5\end{bmatrix}.$

We are trying to capture a pillar who has significantly negative impact when
his/her observed state is $-1$ (i.e. below-average performance), and weakly
positive impact when their observed state is $1$ (i.e. above-average
performance).

1Pillar Bad Teammate + Hidden (BT1P). Same as the 1Pillar + Hidden profile,
but with the $\mathcal{M}$ matrix of the BT1P profile.

1Pillar Great Teammate (GT1S). Same as the 1Pillar profile, but we change the
$\mathbf{M}$ matrix to the following:

(25) $\displaystyle\mathcal{M}=\begin{bmatrix}0.5&0.3&0.2\\\ 0.1&0.8&0.1\\\
0&0.1&0.9\end{bmatrix}.$

This profile captures a pillar who has significantly positive impact when
their observed state is $1$ (i.e. above-average performance), and weakly
negative impact when their observed state is $-1$ (i.e. below-average
performance).

1Pillar Great Teammate + Hidden (GT1P). Same as the 1Pillar + Hidden profile,
but with the $\mathcal{M}$ matrix of the 1-Pillar Great Teammate profile.

Finally, we create two profiles to capture anti-dependence. That is, when one
player’s performance improves, another player’s performance drops. We define
the anti-dependence $\mathcal{M}$ matrix as:

(26) $\displaystyle\mathcal{M}^{\text{antiD}}=\begin{bmatrix}0&0.2&0.8\\\
0.1&0.8&0.1\\\ 0.8&0.2&0\end{bmatrix}.$

1Pillar + anti-Dependence (1Pillar + antiD). Same as the 1Pillar profile, but
we change the $\mathcal{M}$ to $\mathcal{M}^{\text{antiD}}$.

2Pillars + anti-Dependence (2Pillars + antiD). Same as the 2Pillar profile,
but we change the $\mathcal{M}$ matrix of both pillars to
$\mathcal{M}^{\text{antiD}}$. In this way, the two pillars are anti-dependent
on each other. Note that the pillars don’t depend at all on non-pillar
players.

###  B.3. Team state over time

In order to better understand our models we experiment with the 19 profiles
described in Appendix B.2. For each profile we types of teams and then run
$1000$ simulations. Each simulation consists of $100$ samples. We assume that
every player $i\in[n]$ starts from hidden state $H_{1}^{i}=0$. In our
experiments, we use 19 different team profiles, which correspond to different
combinations of $\mathcal{R}$ and $\mathcal{M}$ pairs.

We define the _hidden team state_ (resp.  _observed team state_) at each
timestep as the average of the hidden (resp. observed) states of the
individuals in the team at this timestep; values close to $0$ correspond to
average team performance, values close to $-1$ (resp. $1$) correspond to an
under-performing (resp. over-performing) team. The team (hidden) state of the
different teams over time is shown in Figures 6 \- 9. In these plots the thick
green region corresponds to the 25-th and 75-th percentiles of the team state
across simulations. The light green region corresponds to the min and max
values of the team state across simulations.

Depending on the team profile, the thick green band might vary its width.
Teams with a narrow green band, are more stable, since in $50\%$ of the time
they are close to their average mental state. Furthermore, its worth noticing
that for some profiles the green band is shifted either upwards (e.g. GT1P) or
downwards (e.g BT1P). This shift reveals an inclination of a team towards
higher (or lower) mental states.

(a) 1Pillar

(b) 1Pillar + H

(c) 1Pillar + H + antiD

(d) 2Pillars

(e) 2Pillars + H

(f) 2Pillars + D

Figure 6. Average hidden and observed states for each team profile

(a) 2Pillars + H + D

(b) 2Pillars + antiD

(c) 3Pillars

(d) 3Pillars + H

(e) 3Pillars + D

(f) 3Pillars + H + D

Figure 7. Average hidden and observed states for each team profile

(a) 1Pillar Bad Teammate (BT1P)

(b) 1Pillar Bad Teammate (BT1P) + H

(c) 1Pillar Great Teammate (GT1T)

(d) 1Pillar Great Teammate (GT1T) + H

(e) H

(f) HO

Figure 8. Average hidden and observed states for each team profile (a)
Balanced Figure 9. Average hidden and observed team states for each team
profile; average is taken over different simulations

###  B.4. Team collapse probability vs Team profile

Figure 10. Team collapse probability vs Team profile

team collapse probability vs team profile

###  B.5. Maximum and average time in collapse state

(a) Maximum time in team collapse

(b) Average time in collapse

Figure 11. Maximum and average time team team collapse

In figure 11 (a), we observe that the more pillars a team has, the longer the
maximum window of team collapse is. In general adding self-dependence, i.e.
dependence on the hidden states, doesn’t always make the team more resilient.
However, adding self-dependence in the 3Pillars profile resulted in higher
maximum collapse time. Moreover, we observe that having a Great Teammate
player on whom all other players depend (1Pillar Great Teammate (GT1T)
profile), significantly reduces the length of the maximum time collapse
window. This is because when the pillar player under-performs, he/she only
mildly affects non-pillar players. On the contrary, if the pillar player over-
performs, non-pillar players are greatly affected in a positive way.

In Figure 11 (b), we observe that the average collapse time doesn’t
significantly vary among team profiles. That said, it is clear that when the
players only depend on their hidden state, i.e.H profile, the average duration
of the collapse is longer. This is because, in the rare case that all players
choke, it is very difficult to get out of the choke. For example, this is not
the case for the 1Pillar profile. When the pillar of the team chokes, non-
pillar players are likely to choke. But, once the pillar player gets out of
the choke, non-pillar players are likely to do the same.

##  Appendix C Independence

In graphical models, independence between random variables is defined using
the concept of $d$-separation. For a definition and analysis of $d$-separation
see (Koller and Friedman, 2009). We use $X\perp Y\mid Z$ to denote that $X$ is
independent of $Y$ given $Z$. In our proofs we used the following independence
relationships:

  * •

$H_{t}^{i}\perp H_{t}^{j}\mid\mathbf{O}_{1:t}$

  * •

$O_{t}^{i}\perp\mathbf{O}_{t}^{-i}\mid H_{t}^{i}$

  * •

$H_{t}^{i}\perp\mathbf{O}_{t}^{-i}\mid\mathbf{O}_{1:t-1}$

  * •

$O_{t}^{i}\perp\mathbf{O}_{t}^{1:t-1}\mid H_{t}^{i}$

  * •

$(H_{t}^{i},O_{t}^{i})\perp(H_{t}^{j},O_{t}^{j})\mid\mathbf{O}_{1:t-1}$

  * •

$\mathbf{H}^{i}\perp\mathbf{H}^{j}\mid\mathbf{O}$

Apart from checking the above relationships using the rules of $d$-separation,
we also checked them using independence checker programs that are freely
available online.

##  Appendix D Appendix for Section 4

###  D.1. Expectation step

Calculating the posterior. Using the definition of conditional probability we
have that the posterior can be written as

(27)
$\displaystyle\text{Pr}(\mathbf{H}\mid\mathbf{O})=\frac{\text{Pr}(\mathbf{H},\mathbf{O})}{\text{Pr}(\mathbf{O})}.$

Now, using the structure of the graphical model (see Fig. 1), we can decompose
the joint probability of the variables as a product of probabilities where
each variable is conditioned on its parent nodes (Koller and Friedman, 2009).
Thus, we have

(28)
$\displaystyle\text{Pr}(\mathbf{H},\mathbf{O})=\text{Pr}(\mathbf{H}_{1})\prod_{t=1}^{T}\prod_{i=1}^{n}\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\prod_{t=2}^{T}\prod_{i=2}^{n}\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{t-1})$

Thus, the difficulty of sampling from the posterior stems from the denominator
$\text{Pr}(\mathbf{O})$ which can be written as

(29)
$\displaystyle\text{Pr}(\mathbf{O})=\text{Pr}(\mathbf{O}_{1})\prod\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1}).$

In general, calculating $\text{Pr}(\mathbf{O})$ is difficult, however in our
model it can be done efficiently by calculating
$\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$. We do this as follows.
First, we define the forward parameters

(30)
$\displaystyle\text{Pr}({\mathbf{H}_{t}\mid\mathbf{O}_{1:t}})=\prod_{i=1}^{n}\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t}),$

where we used the independence of $H_{t}^{i}$ and $H_{t}^{j}$ given
$\mathbf{O}_{1:t}$. We now calculate
$\text{Pr}(H_{t}^{i}|\mathbf{O}_{1:t-1})$.

(31) $\displaystyle\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})$
$\displaystyle=\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1},\mathbf{O}_{t})$ (32)
$\displaystyle=\frac{\text{Pr}(\mathbf{O}_{t}\mid
H_{t}^{i},\mathbf{O}_{1:t-1})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})}$
(33)
$\displaystyle=\frac{\text{Pr}(\mathbf{O}_{t}^{-i}\mid\mathbf{O}_{1:t-1})\text{Pr}(O_{t}^{i}\mid
H_{t}^{i},\mathbf{O}_{1:t-1})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(O_{t}^{i}\mid\mathbf{O}_{1:t-1})\text{Pr}(\mathbf{O}_{t}^{-i}\mid\mathbf{O}_{1:t-1})}$
(34) $\displaystyle=\frac{\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(O_{t}^{i}\mid\mathbf{O}_{1:t-1})},$

where in the second equation we used Bayes’ rule, in the third equation we
used the independence between $O_{t}^{i}$ and $\mathbf{O}_{t}^{-i}$ given
$\mathbf{O}_{1:t-1}$ and the independence of $\mathbf{O}_{t}^{-i}$ and
$H_{t}^{i}$ given $\mathbf{O}_{1:t-1}$, and in the last equation we used the
independence of $O_{t}^{i}$ and $\mathbf{O}_{1:t-1}$ given $H_{t}^{i}$.
Furthermore, we have

(35)
$\displaystyle\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})=\sum_{h\in\mathcal{H}}\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i}=h,\mathbf{O}_{1:t-1})\text{Pr}(H_{t-1}^{i}\mid\mathbf{O}_{1:t-1}),$

where $\text{Pr}(H_{t-1}^{i}\mid\mathbf{O}_{1:t-1})$ can be calculated
recursively.

We also have

(36)
$\displaystyle\text{Pr}(\mathbf{H}_{t},\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$
$\displaystyle=\prod_{i=1}^{n}\text{Pr}(H_{t}^{i},O_{t}^{i}\mid\mathbf{O}_{1:t-1})$
(37) $\displaystyle=\prod_{i=1}^{n}\text{Pr}(O_{t}^{i}\mid
H_{t}^{i},\mathbf{O}_{1:t-1})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})$ (38)
$\displaystyle=\prod_{i=1}^{n}\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1}),$

where in the first equation we used the independence of
$(H_{t}^{i},O_{t}^{i})$ and $(H_{t}^{j},O_{t}^{j})$ given
$\mathbf{O}_{1:t-1}$, and in the third equation we used the independence of
$O_{t}^{i}$ and $\mathbf{O}_{1:t-1}$ given $H_{t}^{i}$.

Moreover, we have

(39) $\displaystyle\text{Pr}(\mathbf{H}_{t}\mid\mathbf{O}_{1:t})$
$\displaystyle=\frac{\text{Pr}(\mathbf{H}_{t},\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})}$

Having computed $\text{Pr}(\mathbf{H}_{t}\mid\mathbf{O}_{1:t})$ and
$\text{Pr}(\mathbf{H}_{t},\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$ we can
calculate $\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$ using the above
equation. Thus, we can compute $\text{Pr}(\mathbf{O})$ and the posterior
distribution.

The E-step concludes by defining
$Q(\boldsymbol{\Theta}\mid\boldsymbol{\Theta}^{\text{old}})$ as the expected
value of the log likelihood function of $\boldsymbol{\Theta}$, with respect to
the current conditional distribution of $\mathbf{H}$ given $\mathbf{O}$ and
the current estimates of the parameters:

(40) $\displaystyle
Q(\boldsymbol{\Theta}\mid\boldsymbol{\Theta}^{\text{old}})=\mathbb{E}_{\mathbf{H}\sim\text{Pr}(\mathbf{H}\mid\mathbf{O},\boldsymbol{\Theta}^{\text{old}})}\left[\log\text{Pr}(\mathbf{H},\mathbf{O}\mid\boldsymbol{\Theta})\right]$

In practice, we estimate the expectation by drawing samples from the posterior
distribution
$\text{Pr}(\mathbf{H}\mid\mathbf{O},\boldsymbol{\Theta}^{\text{old}})$, and
using the empirical mean. To this end, we next show how to generate samples
from the posterior distribution.

Sampling from the posterior. We are now ready to prove Lemma 1 that allows us
to decompose the posterior distribution into products of simpler
distributions.

###### Proof.

First, based on the previous analysis, we have

(41) $\displaystyle\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$
$\displaystyle=\frac{\text{Pr}(\mathbf{H}_{t},\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(\mathbf{H}_{t}\mid\mathbf{O}_{1:t})}$
(42)
$\displaystyle=\prod_{i=1}^{n}\frac{\text{Pr}(H_{t}^{i},O_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}$
(43) $\displaystyle=\prod_{i=1}^{n}\frac{\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}.$

Hence, the evidence becomes

(44) $\displaystyle\text{Pr}(\mathbf{O})$
$\displaystyle=\prod_{t=1}^{T}\text{Pr}(\mathbf{O}_{t}\mid\mathbf{O}_{1:t-1})$
(45)
$\displaystyle=\prod_{t=1}^{T}\prod_{i=1}^{n}\frac{\text{Pr}(H_{t}^{i},O_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}$
(46)
$\displaystyle=\prod_{t=1}^{T}\prod_{i=1}^{n}\frac{\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}.$

Finally, substituting the above into the definition of the posterior
distribution and combining with the joint distribution, we have

(47) $\displaystyle Pr(\mathbf{H}\mid\mathbf{O})$
$\displaystyle=\frac{\text{Pr}(\mathbf{H},\mathbf{O})}{\text{Pr}(\mathbf{O})}$
(48)
$\displaystyle=\prod_{t=1}^{T}\prod_{i=1}^{n}\frac{\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{t-1})}{\text{Pr}(O_{t}^{i}\mid
H_{t}^{i})\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})/\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})}$
(49)
$\displaystyle=\prod_{t=1}^{T}\prod_{i=1}^{n}\frac{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t})\text{Pr}(H_{t}^{i}\mid
H_{t-1}^{i},\mathbf{O}_{t-1})}{\text{Pr}(H_{t}^{i}\mid\mathbf{O}_{1:t-1})}$

. ∎

Thus, we sample $H_{t}^{i}$ from a propability distribution that is
proportional to

(50)
$\displaystyle\frac{\text{Pr}(H_{t}^{i}=h\mid\mathbf{O}_{1:t})\text{Pr}(H_{t}^{i}=h\mid
H_{t-1}^{i},\mathbf{O}_{t-1})}{\text{Pr}(H_{t}^{i}=h\mid\mathbf{O}_{1:t-1})}.$

##  Appendix E Real-world experiments

###  E.1. Team structures

Figure 12 presents the structure matrix for all 30 NBA teams in our dataset.
As discussed in Section 5 there are two main profiles and each team can be
thought of as falling closer to one of them. For example the Cleveland
Cavaliers, fall closer to the profile we saw for the Dallas Mavericks, but
with the added difference that two of their players also depend slightly to
their own observed state (not only their hidden state). At the same time,
Houston Rockets are closer to the profile we saw for the Boston Celtics, where
the observed state of Martin seems to impact the state of his teammates.

(a) Boston Celtics

(b) Brooklyn Nets

(c) New York Knicks

(d) Philadelphia 76ers

(e) Toronto Raptors

(f) Chicago Bulls

Figure 12. Team structure $\mathcal{R}$

(a) Cleveland Cavaliers

(b) Detroit Pistons

(c) Indiana Pacers

(d) Milwaukee Bucks

(e) Atlanta Hawks

(f) Charlotte Hornets

Figure 13. Team structure $\mathcal{R}$

(a) Miami Heat

(b) Orlando Magic

(c) Washington Wizards

(d) Denver Nuggets

(e) Minnesota Timberwolves

(f) Oklahoma City Thunder

Figure 14. Team structure $\mathcal{R}$

(a) Portland Trail Blazers

(b) Utah Jazz

(c) Golden State Warriors

(d) LA Clippers

(e) Los Angeles Lakers

(f) Phoenix Suns

Figure 15. Team structure $\mathcal{R}$

(a) Sacramento Kings

(b) Dallas Mavericks

(c) Houston Rockets

(d) Memphis Grizzlies

(e) New Orleans Pelicans

(f) San Antonio Spurs

Figure 16. Team structure $\mathcal{R}$

###  E.2. Additional examples of specific games

The following figure provides some additional examples of games with the
inferred from our model team state. As we can see here also, the team states
inferred from our model follows closely in the vast majority of the cases the
observed score differential at the game. All in all, the expressivity of our
model is very good and can explain well the observed output of the game.

(a) Denver Nuggets vs Los Angeles Lakers(11 Jan 2022)

(b) Dallas Mavericks vs Golden State Warriors (20 May 2022)

(c) Miami Heat vs Boston Celtics (23 May 2022)

(d) Milwaukee Bucks vs Boston Celtics (15 May 2022)

Figure 17. Team collapse games


Quant Trefoil ivanviktd December 2021

#  Calculating HOMFLY-PT polynomials on a photonic processor

Ivan Dyakonova,b, Ilya Kondratyeva, Sergey Mironovc,d,e, Andrey Morozovd,f,g
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we discuss an approach to calculate knot polynomials on a
photonic processor. Calculations of knot polynomials is a computationally
difficult problem and therefore it is interesting to use new advanced
calculation methods to find them. Here we present a proof of concept by
calculating the simplest knot polynomial of the trefoil knot in fundamental
representation. This approach, however, can easily be generalized to more
complex knots and representations. Same operators can also be realized on a
quantum computer with the same effect.

ITEP/TH-13/24

IITP/TH-11/24

MIPT/TH-9/24

a Quantum Technology Centre, Faculty of Physics, MSU, Moscow, 119991, Russia
b Russian Quantum Center, Moscow, 121205, Russia
c INR RAS, Moscow 117312, Russia
d NRC “Kurchatov Institute”, Moscow 123182, Russia
e ITMP, MSU, Moscow, 119991, Russia
f IITP RAS, Moscow 127994, Russia
g MIPT, Dolgoprudny, 141701, Russia

##  1 Introduction

Coherent photonic processors have recently attracted attention due to
capability to significantly improve speed and energy efficiency of executing a
particular kind of operations - unitary matrix multiplication [1]. The
principle of operation is simple: a multimode input light field $a^{in}_{i}$
undergoes a transformation $U$ imposed by an interferometric circuit. The
output field is then expressed as $a_{j}^{out}=U_{ij}a^{in}_{i}$. The advance
of integrated photonic technologies enabled fabrication of precisely
reconfigurable universal interferometers [2, 3] that provide full control over
the unitary matrix $U$ via the phase modulation inside interferometer arms.
Reconfigurable photonic processors find wide application in boosting routine
tasks of machine learning algorithms [4] and in experimental simulation of
fundamental physics models [5]. The perspective of higher signal processing
bandwidth and reduced energy consumption provides steady development of
integrated photonics [6].

We want to apply coherent photonic processor to calculations in mathematical
physics. The calculations we are interested in in the context of this paper
are the calculations of the knot polynomials or Wilson-loop averages of the
Chern-Simons theory.

Chern-Simons theory with the action

$S=\frac{k}{4\pi}\int\mathcal{A}\wedge
d\mathcal{A}+\frac{2}{3}\mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A}=\frac{k}{4\pi}\int
d^{3}x\varepsilon^{ijk}\left(\delta_{ab}\mathcal{A}^{a}_{i}\partial_{j}\mathcal{A}^{b}_{k}+\frac{2}{3}if_{abc}\mathcal{A}^{a}_{i}\mathcal{A}^{b}_{j}\mathcal{A}^{c}_{k}\right).$
(1)

is usually considered in the literature as potential model for topological
quantum computer as it describes non-Abelian anyons. Though we do not have
such physical system yet, topological quantum computer would have radically
better fidelity in comparison to many other quantum computer models as the
states are topologically stable [7].

In this paper though, we consider the opposite problem. As Chern-Simons theory
itself when employed in knot theory presents technical interest, we suggest to
use an existing quantum computer to calculate interesting quantities in such a
theory and, consequently, in topological theory of knots. In Chern-Simons
theory the most interesting and complex observables are Wilson-loop averages:

$\left<W\right>_{Q}^{\mathcal{K}}=\text{Tr}_{Q}\text{
Pexp}(\int\limits_{\mathcal{K}}\mathcal{A}dx),$ (2)

here $\mathcal{K}$ is a contour in 3d space along which we calculate Wilson-
loop, which is in fact a knot, $T$ is a representation of the gauge group in
which we calculate the Wilson-loop averages. These observables, as it is known
[8], are equal to the knot polynomials from the mathematical knot theory, with
the knot being equivalent to the loop $\mathcal{K}$. These knot polynomials
are interesting from the point of view of many theories because of numerous
dualities with other theories, for example conformal field theory [8]-[19],
topological string theory [20], integrable theories [21] and matrix models
[22]-[25] and others. In the context of topological quantum computers, knot
polynomials are potential observables. Due to the fact that the knot
polynomials on one hand are constructed from a product of unitary matrices
and, on the other hand, their calculations prove to be a difficult task for
complex knots, calculations of knot polynomials is a natural problem for the
advanced computing machines, such as quantum computer and photonic processor.

We study Chern-Simons theory with the gauge group $sl_{N}$ and calculate the
resulting knot polynomials which are called HOMFLY-PT polynomials [26, 27] on
a photonic processor. HOMFLY-PT are polynomials of two variables $q$ and $A$
which are connected to the parameters of the theory

$q=exp(\frac{4\pi i}{k+N}),\ \ \ A=q^{N}.$ (3)

##  2 Two-bridge knot calculus

We adopt the Reshetikhin-Turaev approach to knot calculus [29]-[33] and use a
simple construction of the two-bridge knot. Such knots are constructed as a
four strand braid (see Fig.1), closed on each side.

Figure 1: Trefoil knot as a two-bridge knot.

The polynomials are constructed as matrix elements of the product of operators
that correspond to each crossing in the knot [34]-[37]. Moreover, crossing-
operators between different strands are connected via Racah matrices.
Important property of all these operators is that they are represented by the
unitary matrices for $q$ from (3) and large enough $k$. Such matrices can also
be used as universal quantum gates in topological quantum computer [38, 39,
40].

Knot polynomials and operators depend on the representation of the group
$sl_{N}$. Chosen representation define the size of matrices and thus change
the size of quantum computer required to calculate corresponding knot
polynomial. For the simplest fundamental representation that we consider in
this paper the size of matrices is $2$ by $2$ and thus one-qubit quantum
computer is needed. For the fundamental representation required operators are
equal to

$\begin{array}[]{lcrlcclcrl}&&\begin{array}[]{cc}[2]&[1,1]\end{array}&&&&&\begin{array}[]{cc}[2]&[1,1]\end{array}\\\
\\\ \\\
S_{[1]}&=&\frac{1}{\sqrt{(q+q^{-1})(A-A^{-1})}}\left(\begin{array}[]{cc}\sqrt{\frac{A}{q}-\frac{q}{A}}&\sqrt{Aq-\frac{1}{Aq}}\\\
\sqrt{Aq-\frac{1}{Aq}}&-\sqrt{\frac{A}{q}-\frac{q}{A}}\end{array}\right)&\begin{array}[]{l}\,\emptyset\\\
\,\text{adj}\end{array}&&T_{[1]}&=&\left(\begin{array}[]{cc}\frac{q}{A}&0\\\
0&-\frac{1}{qA}\end{array}\right)&\begin{array}[]{l}\,[2]\\\
\,[1,1]\end{array}\\\ \\\ \\\ \\\
&&\begin{array}[]{lc}\emptyset&\text{adj}\end{array}&&&&&\begin{array}[]{lc}\emptyset&\text{adj}\end{array}\\\
\\\
\bar{S}_{[1]}&=&\left(\begin{array}[]{cc}\frac{q-q^{-1}}{A-A^{-1}}&\frac{\sqrt{(Aq-\frac{1}{Aq})(\frac{A}{q}-\frac{q}{A})}}{A-A^{-1}}\\\
\frac{\sqrt{(Aq-\frac{1}{Aq})(\frac{A}{q}-\frac{q}{A})}}{A-A^{-1}}&-\frac{q-q^{-1}}{A-A^{-1}}\end{array}\right)&\begin{array}[]{l}\,\emptyset\\\
\,\text{adj}\end{array}&&\bar{T}_{[1]}&=&\left(\begin{array}[]{cc}1&0\\\
0&-A\end{array}\right)&\begin{array}[]{l}\,\emptyset\\\
\,\text{adj}\end{array}\\\ \\\ \end{array}$ (4)

The knot polynomial is then given by the matrix element of the product of
aforementioned operators. For the simplest trefoil knot the answer is given by
the following formula (see also Fig.1):

$\mathcal{H}_{[1]}^{3_{1}}=\frac{A-A^{-1}}{q-q^{-1}}\left(ST^{3}S^{\dagger}\right)_{\emptyset\emptyset}=A^{2}-A^{4}(q^{2}+q^{-2})$
(5)

Actually the simplest way to calculate this polynomial is to use non-diagonal
operators:

$T_{nd}=STS^{\dagger}=\left(\begin{array}[]{cc}-A^{2}\cfrac{q-q^{-1}}{A-A^{-1}}&A\cfrac{\sqrt{(Aq-A^{-1}q^{-1})(A/q-q/A)}}{A-A^{-1}}\\\
\\\
A\cfrac{\sqrt{(Aq-A^{-1}q^{-1})(A/q-q/A)}}{A-A^{-1}}&\cfrac{q-q^{-1}}{A-A^{-1}}\end{array}\right)$
(6)

Then the trefoil polynomial is a matrix element of $T_{nd}^{3}$. In general
powers of $T_{nd}$ will give different two-strand knots and links.

##  3 Experiment and results

Coherent photonic processors have recently drawn attention due to ability to
perform computation of specific mathematical functions effectively in terms of
time and energy. The most vivid example is matrix multiplication implemented
simply by transmitting light through an optical interferometer and detecting
output power. In this work we devised a method to map the problem of computing
the knot polynomial to a coherent photonic platform.

Figure 2: a) Schematic of the photonic processor. The emission of a diode
laser is coupled into one of the input ports of the integrated Mach-Zehnder
interferometer. The relative phase $\theta$ between the arms is controlled by
a thermooptical phaseshifter, driven by a programmable current source connect
to a personal computer. The output power distribution is registered by a pair
of detectors $D_{1}$ and $D_{2}$. The measured values are normalized to a sum
of output power in both ports. b) The result of the $T^{3}_{nd,11}$
measurement. The value of the matrix element is estimated as the normalized
output power, measured by the detector $D_{1}$. The dots represent the
experimental results, the solid violet curve represents theory.

Our photonic processor implements multiplication of $2\times 2$ unitary matrix
and a two-dimensional complex field amplitude vector. The schematic of the
processor can be found in Fig. 2. We use the femtosecond laser writing method
to fabricate a reconfigurable Mach-Zehnder interferometer. The photonic
waveguides are inscribed by a tightly focused beam of the ytterbium-fiber
femtosecond laser (Avesta Antaus) inside the volume of a fused silica sample
mounted on the 3D air-bearing stage (Aerotech FiberGlide 3D). The phase
modulation of one the interferometer arms is implemented using a thermo-optic
phaseshifter. The phaseshifer is engraved on top of the photonic circuit after
the NiCr metal film is sputtered onto the top surface of the chip. The
electrical connection is established via spring-loaded electric contacts
soldered onto the interface PCB board. We drive the circuit with constant
current set by a programmable current source.

The input light field is generated by a diode laser (Toptica CTL 950) and
injected into port X of the circuit through a single-mode fiber array. The
output of the chip is coupled to another single-mode fiber array which is
connected to a pair of PIN-diode-based photodetectors which measure the output
power.

Using photonic processor, we have implemented (6). Then we measured the
absolute value of the corresponding matrix elements for different values of
$q$ (different $k$). As a result we get graph on Fig. 2. On this graph we
present experimental measurements, depending on the values of $k$. As we can
see it coincides with theoretical curve, which shows that we can indeed
measure knot polynomials, using this approach.

##  4 Conclusion

This paper should be considered as a proof-of-concept for calculating knot
polynomials on a quantum computer or photonic processor. We used the simplest
case of fundamental representation for the 2-bridge knots which can be
calculated using a one qubit quantum computer or similar photonic processor.
We show that corresponding matrix element calculated on a photonic processor
indeed reproduces the analytical answer. This approach should be also expanded
to higher representations and more involved knots which would require higher
dimensional photonic processors or multiqubit quantum computers.

##  5 Acknowledgements

We are grateful to S. Straupe for very useful discussions. This work was
supported by Russian Science Foundation grant No 23-71-10058.

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