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J. Quant. Anal. Sports 2020; aop
Ivan Gonzalez-Cabrera, Diego Dario Herrera and Diego Luis González*
Generalized model for scores in volleyball
matches
https://doi.org/10.1515/jqas-2019-0060
Abstract: We propose a Markovian model to calculate the
winning probability of a set in a volleyball match. Tradi-
tional models take into account that the scoring probabil-
ity in a rally (SP) depends on whether the team starts the
rally serving or receiving. The proposed model takes into
account that the different rotations of a team have differ-
ent SPs. The model also takes into consideration that the
SP of a given rotation complex 1 (K1) depends on the play-
ers directly involved in that complex. Our results help to
design general game strategies and, potentially, more effi-
cient training routines. In particular, we used the model
to study several game properties, such as the importance
of having serve receivers with homogeneous performance,
the effect of the players’ initial positions on score evo-
lution, etc. Finally, the proposed model is used to diag-
nose the performance of the female Colombian U23 team
(U23 CT).
Keywords: Markov chains; volleyball; winning
probability
1Introduction
Mathematical models have proved to be useful in the study
of technical properties in several sports, allowing for the
generation of new game and training strategies. In par-
ticular, Markovian models have been used to describe
the statistical properties of several sports, including vol-
leyball, tennis and racquetball, Carrari, Ferrante, and
Fonseca (2017); Newton and Aslam (2009); Strauss and
Arnold (1987); Simmons (1989); González (2013); Walker
and Wooders (2011); Hoffmeister and Rambau (2017);
Hoffmeister (2019). A common feature of these sports is
*Corresponding author: Diego Luis González, Departamento
de Física, Universidad del Valle, A.A. 25360, Cali, Colombia,
e-mail: diego.luis.gonzalez@correounivalle.edu.co
Ivan Gonzalez-Cabrera: Konrad Lorenz Institute for
Evolution and Cognition Research, Klosterneuburg, Austria,
e-mail: ivan.gonzalez-cabrera@kli.ac.at
Diego Dario Herrera: Federación Colombiana de Voleibol, Bogotá,
Colombia, e-mail: dherrerafarfan@gmail.com
that a match is divided into a certain number of sets and
each set is divided into several rallies. A rally is a sequence
of playing actions which start from the serve until the ball
falls and a team/player scores a point. The winner of the
match is the first team/player that achieves the victory on
a certain number of sets.
As a first approximation, the probability for a team/
player to win a set can be calculated by considering a
sequence of Bernoulli trials between contenders. In this
approach each trial corresponds to a rally with an asso-
ciated probability of success, which is assumed to be a
constant for all the set. The probability that team Awins
at least ntrials and be ahead of their opponent by at least
ktrials at the end of the set was calculated in Siegrist
(1989). The case k=1 and arbitrary ncorresponds to the
Banach match problem. But for k=nthe gambler’s ruin
problem is obtained. The models based on Bernoulli tri-
als assume that the probability of each team/player to
win a trial does not change throughout the game. This is,
of course, a strong assumption in the case of modeling
sports, where the scoring probability usually depends on
several factors that could change throughout the game. For
instance, psychological factors, such as stress and anxiety,
could affect the performance of a team/player especially
under extreme pressure. These factors could lead to non-
Bernoulli models or/and non-Markovian models. Another
scenario which leads to non-Bernoulli processes is given
by the properties of the game itself. In the particular case
of volleyball, it is well known that the scoring probability
in a single rally depends on whether the team starts that
rally serving or receiving, Lee and Chin (2004). Bernoulli
models are particular cases of a broader kind of games,
usually called binary Markov games. General results about
binary games are found in Walker and Wooders (2011).
A major result of this work is that, in sports like tennis,
if a simple monotonicity condition is satisfied, then play
does not depend upon the score, or upon the history of
points won or actions taken. Another general Markovian
model for non-Bernoulli trials for games in which there is
a form of Markovian dependence between the outcomes of
successive trials can be found in Haigh (1996).
This paper focuses on the particular case of indoor
volleyball matches under the playing rules of the Fédéra-
tion Internationale de Volleyball (FIVB). In FIVB’s compe-
titions, a complete volleyball match is divided into sets.
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2|I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
The winning team is the one that wins three sets out of a
maximum of five. Each set is composed of a certain num-
ber of rallies, the team that wins the rally scores a point. If
the serving team loses a rally, the players of the opposing
team rotate one position clockwise and serve at the start of
the next rally: the player in position 2 rotates to position 1
to serve, the player in position 1 rotates to position 6, and
so on, see Figure 1. Conversely, when the serving team wins
the rally, it scores a point and continues serving. All sets
(except the deciding fifth set) end when a team scores 25
points with a difference of at least two points with respect
to the other team’s score. If a team reaches 25 points with a
difference of less than two points, the set is extended until
one of the teams reaches a difference of two points. The
fifth set is played to 15 points instead of 25, also ending
with a minimum difference of two points.
As mentioned above, the simplest approximation for
the score evolution in a set corresponds to Bernoulli tri-
als where, for instance, the probability that team Awill
score a point in a rally is p. Consequently, the probability
team Bwill score a point is simply q=1−p. However, a
more sophisticated model based in binary Markov games
can be considered. In these models, the probability of win-
ning a trial depends on the current score. This score must
be interpreted as a state variable rather than the score as it
is usually understood. This score can include factors such
as whether the team is serving or receiving, whether it is
too early or too late in the match to distinguish the first
rallies of the set from the last ones, etc. Thus, in volley-
ball, the model can be improved easily if it is taken into
account that the scoring probability in a rally (SP) depends
on the game situation of the team, Ferrante and Fonseca
(2014); Kovacs (2009); Fellingham, Collings, and McGown
(1994). The SP is psif team Ais serving. If the team is
receiving, the SP is pr. Both psand pr, are conditional prob-
abilities, i.e. they are the SP of team Agiven that it starts
the rally serving or receiving, respectively. In these Marko-
vian models, the key quantities used as inputs are SPs,
which can be used as a measure of team performance.
Because of this, many authors have studied skill perfor-
mance and its effect on the final score of the game, Miskin,
Fellingham, and Florence (2010); Zetou et al. (2007). Most
of the existing models in the literature are based on the
assumption that the SPs do not change throughout the
game, even though intuition suggests that psand prcould
change depending on the evolution of the match, and thus
lead to non-Markovian models. However, there are some
studies that suggest that the independence assumption
is compatible with the actual game experience, Schilling
(1994); Gilovich, Vallone, and Tversky (1985), Avugos et
al. (2013). Markovian models for volleyball matches have
been successfully used to determine the average duration
of a game, the effect and efficiency of the score system,
the strategies to serve or receive, etc. Ferrante and Fonseca
(2014); Kovacs (2009); Fellingham et al. (1994); Pfeifer and
Deutsch (1981). A quite general and detailed model for
beach volleyball can be found in Hoffmeister and Rambau
(2017); Hoffmeister (2019). This model takes into account
the influence of individual players on scoring probabili-
ties. Furthermore, it considers different SPs for the each
type of attacking actions.
Nonetheless, in order to describe more precisely the
score evolution, it is necessary to take into considera-
tion additional details of the game. In modern volleyball
schemes, the rallies are interpreted by using the concept of
a complex. There are four basic complexes that are defined
by the game actions they involve, Conejero et al. (2017).
As mentioned before, each rally starts with a team serv-
ing. This game situation is usually called complex 0 (K0).
The main objective of the serve is to achieve a direct point
(ace), or at least reduce the attacking options of the oppos-
ing team. The team which is not serving is in a defensive
formation called complex 1 (K1), or attack phase. Typically,
K1includes the actions of receiving, setting, attacking,
and covering the attack. The objective of the K1is to neu-
tralize the opposing serve, organize an attack to win the
rally, and score a point. Complex 2 (K2), or defense phase,
includes the actions of blocking, defending, setting, coun-
terattacking, and covering the counterattack. The main
goal of this complex is to neutralize the opponent’s attack
coming from its K1and to organize a counterattack in order
to obtain the point. Finally, Complex 3 (K3), or counterat-
tack phase, includes the actions of blocking, defending,
setting, counterattacking, and covering the counterattack.
Its main objective is to neutralize the counterattack from
the K2of the opposing team, and to build a new counter-
attack. It is worth noting that complex K2and K3usually
involve more playing actions than K0and K1. Therefore,
K2and K3are intrinsically more difficult to model than K0
and K1.
A team is in complex K1when it has lost the previous
rally. Thus, the main function of this complex is recover
the serve and prevent the rival team from scoring an addi-
tional point. If a team succeeds in K1, the difference in the
score increases/decreases by one point. A team is in com-
plex K2when it has won the previous rally. In order to score
two consecutive points, a team must succeed in at least
in one K2complex. Therefore, this complex is fundamen-
tal for reaching the two-point difference needed to win the
set. In this sense, while complex K1is important to avoid
losing the game, complex K2is needed to win the match. A
team performs complex K3when it was not able to score in
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |3
complexes K2and/or K1. Usually, when a team is in com-
plex K3, the SP is smaller than that of complex K2due to
several factors such as mental stress.
The SP of a team not only depends on whether the
team starts that rally serving or receiving but also on
another factors, as follows. However, in volleyball, each
time that a team recovers the serve, the players of that team
have to move in a clockwise direction through the different
positions on the court. The positions are labeled from 1 to
6, as shown in Figure 1. If a team is serving, the players in
the same row can switch positions once the serve is per-
formed. However, if the team is receiving, the players can-
not change positions until K1is performed. This game rule
ensures that the players eventually move from the front
row to the back row, and vice versa. Consequently, the SP
depends also on the rotation of the team. There are six dif-
ferent rotations, Ri={P1,. . . ,P6}, with i=1, . . ., 6 and
Pjrepresenting the player in the position 1≤j≤6. Each
rotation can be in one of the following states: serving or
receiving. The initial rotation at the beginning of the set
is denoted by R1. A complete cycle is achieved when the
initial rotation returns to the initial state. In Figure 1, the
setter, the opposite, and the libero are represented by S,O,
and L, respectively. Ois usually the most efficient spiker
of the team. Consequently, Ois covered when the team is
in K1to prevent him from receiving a serve, increasing the
O’s chances of attacking. The setter takes the ball from the
receiver and sets it to one of the spikers. There are two
middle-blockers and two right-side spikers, represented
by MBiand RSi, with i=1, 2, respectively. The middle-
blockers are usually the fastest blockers of the team, cover-
ing position 3 and supporting the block at positions 4 and
2. They are also used in attack as spikers. In modern vol-
leyball schemes, once a MB loses the serve, it is replaced
by one of the liberos, which are usually the players with
the best reception performance. This substitution allows
Figure 1: The dierent positions of the players in the court are rep-
resented by numbers from 1 to 6. Each time that a team gains the
serve, the players must rotate in clockwise direction moving from
one rotation to the next one. For this particular case, team Amoves
from rotation {S,RS1,MB1,O,RS2,L}to {RS1,M B1,O,RS2,L,S}.
the physical recovery of the MBs, which often perform
many defensive and offensive actions in the front row. The
right-side spikers are used in attack and reception. In fact,
the serves from the opposing team are usually received
by them and L. The spikers perform offensive actions try-
ing to force the rival team to lose the control of the ball.
In Figure 1, the rotation R1={S,RS1,MB1,O,RS2,L}
is shown. For simplicity, we focus on the 5-1 formation
where the teams use a single setter. This is perhaps the
most used formation in modern volleyball but different
configurations can also be used in high-level volleyball.
In this paper we propose a generalized Markovian
model to calculate the winning probability in a set, 𝒫A
S.
Two different scenarios are considered. The first scenario
takes into account the differences of the SP among the
different rotations of a team. The second one goes fur-
ther, considering that the SP depends not only on the rota-
tion but also on the performance of the players directly
involved in K1. The proposed model allows us to reproduce
previous results and explore the effect of performance
differences between the different rotations of a team.
2Model description
In order to improve existing models, it is necessary to
include the differences among the SPs of different rota-
tions. For team A, the SPs at rotation Riare denoted by
pr,iand ps,i, depending on whether the team is receiv-
ing or serving, respectively. However, the SP of a rota-
tion in a given rally also depends on the subset of play-
ers who are directly involved in the rally. For the case of
K1, we have three basic actions: receiving, setting, and
attacking. For each rotation, K1can be performed by dif-
ferent sets of players. However, in most of the rallies,
the setting is performed by S, the attack by RSi,MBi, or
O, while the reception is carried out by RSior L. Con-
sequently, there are nine different basic configurations
for performing K1per rotation. For the particular rota-
tion shown in Figure 1, we have the following config-
urations: (L,S,O),(L,S,RS1),(L,S,MB1),(RS1,S,O),
(RS1,S,RS2),(RS1,S,MB1),(RS2,S,O),(RS2,S,RS1),
and (RS2,S,MB1). The first player inside the parenthe-
ses performs the reception of the serve, the second player
performs the setting, and the last one performs the attack.
Of course, in real game situations, other combinations are
possible. For example, when the reception of the serve is
not completely successful, the setter is usually unable to
perform the setting, so another player has to set the ball to
one of the spikers. Another example is when Sattacks with
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4|I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
the second touch, which might be an efficient strategy if
the opponent does not expect it. Therefore, it can only
be used a few times during a match. However, in most
cases, K1is performed with one of the nine configurations
previously mentioned. Hereafter, the configuration jwith
j=1, . . ., 9 of the rotation Ri, with i=1, . . ., 6, is denoted
by 𝒞i,j.¹
For the sake of simplicity, we propose a model that
involves the following assumptions:
–Salways performs the setting. As stated before, this is
true for most of the rallies in a match.
– The attacking actions in K1are performed by one of the
following players: O,RS1/RS2, or MB. Only Ois used to
attack from the back row. In this way, in each rotation,
Salways has three available spikers to choose from.
– The reception of the serve is always performed by one of
the following players L,RS1, or RS2.
– All rallies are defined by complexes K0,K1, and K2. The
K3complex is not taken into account, i.e. we neglect the
counterattack of the counterattack.
– Only complex K1is modeled explicitly. It is assumed
that K1is always composed by the actions of receiving,
setting, and attacking.
– The probability that a given configuration 𝒞i,jof the
rotation Riparticipates in the K1is qi,j. The SP of
configuration 𝒞i,jis pr,i𝒞i,j. In this way, the SP in
K1depends on the performance of the three players
directly involved in the rally. Therefore, the SP for a
given rotation i,pr,i, can be written as
pr,i=
9
j=1
qi,jpr,i𝒞i,j.(1)
– When team Ais in complex K2, all of the team players
perform a role. Thus, we assume that, in that case, the
SP depends on the performance of all players of team A.
In our model, the probability to win a set depends on the
starting rotation and whether the team starts serving or
receiving. The probability that team Twins a given set can
be written as
𝒫T
ℳ(𝒮,ω)=P𝒩
ℳ(𝒮,ω)+P>𝒩
ℳ(𝒮,ω),(2)
1A more detailed model could include an additional configura-
tion that takes into account all the game situations neglected in the
present model. However, these situations have been intentionally
excluded in our model because we do not have enough experimen-
tal data to make a quantitative estimate of the associated probability
that goes beyond the reasonable assumption that it is much smaller
than that of the above considered configurations.
where 𝒮represents the set of parameters that defines the
model ℳ, i.e. it includes the SPs and the initial rotation,
while ωgives the initial state (serving or receiving). 𝒩is
the minimum number of points required to win a set. For
the first four sets, 𝒩=25, and for the fifth set, 𝒩=15.
The first term on the right side in Eq. (2) corresponds to
the probability of finishing the set with a score (𝒩,n)with
n≤ 𝒩 − 2. The second term gives the probability that the
set ends with more than 𝒩points for the winning team.
If team Tstarts serving, the probability P𝒩
ℳ(𝒮,ω)can be
written as
P𝒩
ℳ(𝒮,ω)=
2(𝒩 −2)
k=1
{n}
k
l=1
𝒪lδ(𝜐)Θ(ϑ),(3)
where
𝜐=𝒩 −
k
′′
j=1
ns,j−
k
′
j=1
1and
ϑ=𝒩 − 2−
k
′
j=1
nr,j−
k
′
j=1
1.(4)
As usual, δ(x) is the Kronecker delta and θ(x) is the
unitary step function. The sums ′
jand ′′
jare per-
formed over the odd and even integers, respectively. The
factor 𝒪lis given by
𝒪l=pns,l
s,l,(5)
for odd lwhile for even l
𝒪l=1−ps,l−11−pr,lnr,lpr,l.(6)
In Eq. (3) the sum over kconsiders all the possible rota-
tions and states at the end of the set. For k=1 to 12 the set
ends at the first cycle; for k=13 to 24, the set ends at the
second cycle; and so on. For odd values of k, the winning
team scores the last point at serve, while for even values,
it scores the last point at reception. Lets assume that the
initial rotation is R1. Thus, for k=1, 2, the winning team
wins with R1; for k=3, 4, it wins with R2; etc. Due to the
periodic nature of the rotations, for k=13, 14, the winning
team wins with R1but only at the second cycle. Higher val-
ues of kcan be interpreted similarly. Note that ns,jis the
number of points scored by the winning team while serv-
ing at the rotation defined by the odd integer j. Similarly,
nr,jis the number of points scored against the winning
team while it is receiving at the rotation defined by the
even integer j. Therefore, for a given k, the sum over the
set {n}represents the sum over all the possible values ns,j
and nr,j. For instance, if the winning team starts serving at
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |5
R1, the first four terms of Eq. (3) are given by
P𝒩
ℳ(𝒮,s)=p𝒩
s,1+p𝒩 −1
s,1(1 −ps,1)1−(1 −pr,2)𝒩 −2
+p𝒩 −1
s,1(1 −ps,1)pr,2×
×
𝒩 −1
ns,3=1
𝒩 −3
nr,2=0ps,4
ps,1ns,4
(1−pr,2)nr,2
+p𝒩 −2
s,1(1 −ps,1)pr,2(1 −ps,3)pr,4×
×
𝒩 −1
ns,3=0
𝒩 −4
nr,2=0
𝒩 −4−nr,2
nr,4=0ps,3
ps,1ns,3
×
×(1−pr,2)nr,2(1−pr,4)nr,4+· · · (7)
The first term of (7) corresponds to the probability of
winning the set scoring 25 consecutive points at R1. The
second term corresponds to the probability of winning
the set without performing any rotation, scoring 24 points
while R1is serving and the last point while it is receiv-
ing. For the third and fourth terms, a single rotation is
performed and the set is won by R2serving and receiv-
ing, respectively. Higher terms can be interpreted similarly.
For large/small values of the SPs the probability to finish
the set after several rotations is negligible. Thus, for those
cases, only the first few terms in Eq. (3) must be evalu-
ated. Note that the higher the value of k, the more sums
are involved in Eq. (3). In fact, for a given k, the set {n}is
defined by k−1 variables. Therefore, the number of sums
is the same.
In general, the probability P>𝒩
ℳ(𝒮,ω)depends on the
rotation that reaches the score (𝒩 − 1,𝒩 − 1) required to
extend the set. In our model, P>𝒩
ℳ(𝒮,ω)is given by
P>𝒩
ℳ(𝒮,ω)
=
6
j=1QA,s
jGs
j+QA,r
jGs
j+1+QB,s
jGr
j+QB,r
jGr
j,
(8)
where QT,ω
jis the probability that the winning team
reaches the score (𝒩 −1,𝒩 −1) with the rotation jand the
team T=Aor Bscoring the last point of the sequence. The
index ωrepresents the state of the team that scores the last
point in the rally where the tie is reached. Gω
jcorresponds
to the probability that, given the score (𝒩 − 1,𝒩 − 1), the
rotation j, and the state ωthe team obtains the victory in
the set. If the winning team starts serving, the probability
QT,ω
jcan be written following the same procedure used for
the first term of Eq. (2)
QA,ω
j=
κj,ω
{n}
k
l=1
𝒪(l)δ(𝜐)δ(ϑ),(9)
with
𝜐=𝒩 − 1−
k
′′
j=1
ns,j−
k
′
j=1
1
and
ϑ=𝒩 − 1−
k
′
j=1
nr,j−
k
′
j=1
1.(10)
where κj,ωrepresents the set of values of kfor which Rj
reaches the score (𝒩 − 1,𝒩 − 1) at state ω. For instance,
{κ1,s}is given by 13, 25, and 37; {κ1,r}is given by 2,
14, 26, and 38; etc. The probability QB,ω
jcan be written
similarly.
As shown in next section, it is only possible to find a
simple expression for the probability Gω
jfor few particu-
lar cases. As shown, in its most general form, our model
leads to complex expressions which have to be evaluated
numerically. The implicit sums in Eq. (2) can be evaluated
analytically only in few particular cases, as it is shown
in the next section. Because of this, for the most general
cases discussed in this work, our approach is to perform
a Monte Carlo simulation to calculate the score evolution
as follows, instead of explicitly evaluate Eq. (2). If team A
starts a rally serving in rotation Ri, the SP is given by ps,i.
This probability implicitly takes into account not only the
serve performance, but also the probability of scoring in
K2. If, on the contrary, team Astarts the rally in K1, the con-
figuration 𝒞i,jperforming K1is chosen with a probability
qi,jand the SP is given by pr,i(𝒞i,j). Note that the spiker is
often selected by the setter, while the player who receives
the serve is usually chosen by the opposing team’s server.
Then, the probability qi,jimplicitly depends on both the
setter and the server from the opposing team. The proba-
bility of finishing a set with teams Aand Bscoring mand
npoints, respectively, P𝒩(m,n;𝒮), is calculated directly
from the numerical data obtained by simulating more than
106sets. The average number of rallies played in a set, Nr,
can calculated performing the sum
Nr=
∞
m,n=0
(m+n)P𝒩(m,n;𝒮),(11)
directly during the simulation or using the final numer-
ical results for P𝒩(m,n;𝒮). Our model is more general
than any other in the existing literature for indoor vol-
leyball because it considers the individual performance of
each configuration of players in K1and the dependence of
the SP on the different rotations. Additionally, as will be
shown in the next section, we are able to replicate previous
results with the proposed model.
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6|I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
3Recovering previous results
3.1 Model 1
The simplest model assumes that SP is equal for all rota-
tions, whether the team starts the rally serving or receiv-
ing. In order to build this simple model, we set for team A
ps,i=p qi,j=1
9and pr,i𝒞ij =p,(12)
for all iand j. In Model 1, the SP is completely defined by
a single parameter, and the winning probability depends
neither on the starting rotation nor whether the team starts
the set serving or receiving. This model is simple and
allows for the explicit calculation of many quantities. In
this simple case, the sums in Eq. (3) can be reduced to
a single one noticing that there are ℬ𝒩−1,n=(𝒩 − 1+
n)!/(n! (𝒩 −1)!) terms, where team Aand Bhave scored 𝒩
and npoints, respectively. Therefore, the probability that
team T, with T=Aor B, takes the victory in a set scoring
𝒩points is given by
P𝒩
1(α)=
𝒩 −2
n=0
ℬ𝒩 −1,nα𝒩(1 −α)n,(13)
with α=pif T=A, and α=1−pif T=B. The subindex
1 in 𝒫𝒩
1(α)stands for Model 1. The probability of winning
the set with more than 𝒩can be calculated easily because
the probabilities QT,ω
jand Gω
jin Eq. (8) does not depend
on the rotation jor the state ω. Then, Eq. (8) reduces to
P>𝒩
1(α)=Q G, where Q=ℬ𝒩 −1,𝒩 −1α𝒩 −1(1 −α)𝒩 −1is
the probability to reach the score (𝒩 −1,𝒩 − 1) and Gthe
probability to score the two consecutive points required to
win regardless jor ωwhich can be calculated by using a
simple tree diagram González (2013). Thus
P>𝒩
1(α)
=ℬ𝒩 −1,𝒩 −1α𝒩 −1(1 −α)𝒩 −1α2
2α2−2α+1.(14)
The probability that team T, with T=Aor B, takes the
victory in a set, regardless of the score, 𝒫T
1(α), is given by
the sum of Eqs. (13) and (14) which coincides with the one
reported previously by González (2013).
From Eq. (11), the average number of rallies played in
a set, Nr, can be written as González (2013)
Nr=
𝒩
n=0
(n+25)ℬ𝒩 −1,np𝒩qn+pnq𝒩
+2(2𝒩 − 2)!
(𝒩 − 1)!2
q𝒩+1p𝒩 −1(𝒩 − (2𝒩 − 2)qp)
(1 −2qp)2(15)
Figure 2 shows the behavior of 𝒫A
1(p)for 𝒩=15, 25,
and 100. One important result derived from Model 1 is that
volleyball is a selective sport. This means that the team
with the largest SP usually wins. A quantitative measure of
the selectivity is the slope of 𝒫A
1(p)at p=0.5. The larger
the slope, the more selective the score system is. As 𝒩
increases, 𝒫A
1(p)gets closer to a unitary step function, as
can be expected for a perfectly selective game (dotted line
in Figure 2). Thus, sets played to a small 𝒩favor the team
with the smallest SP. Another important result is the aver-
age number of rallies represented by a continuous line in
Figure 3. As expected, Nrreaches its maximum when the
SP of both teams are equal and decreases to 25 for p=1
and p=0.
0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
1.0
Model 1
= 15
= 25
= 100
p
A
(p)
1
Figure 2: Probability of victory for team Aas a function of pfor
dierent values of 𝒩.
0.0 0.2 0.4 0.6 0.8 1.0
20
25
30
35
40
45
50
0.0 0.2 0.4 0.6 0.8 1.0
20
25
30
35
40
45
50
ps = 0.3
ps = 0.5
ps = 0.8
Nr
pr
Model 1
Model 2
ps = 1
Figure 3: Average number of rallies played per set in Model 1 and
Model 2. In Model 2, team A starts the rally serving.
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |7
3.2 Model 2
Model 2 takes into account that the SP actually depends on
whether the team starts the rally serving or receiving. Let
psand prbe the SP for team Aif it starts the rally serving or
receiving, respectively. In elite male volleyball matches, ps
and prare around 0.35 and 0.75, respectively, Ferrante and
Fonseca (2014); Kovacs (2009); Fellingham et al. (1994).
Analogously for team B,qr=1−psand qs=1−pr.
Before the beginning of the first set, the referee tosses a
coin to determine which team starts serving. The winner
of the toss can choose to serve or pick a court side to play
on. From the second to the fourth set, the team which starts
serving is the one which started receiving in the previous
set. In the fifth set, a new coin toss is performed. Thus, we
obtain Model 2 by setting
ps,i=ps,qi,j=1
9and pr,i𝒞ij =pr,(16)
for all iand j. In Model 2, all rotations are equivalent, but
the probability to win the set depends on whether the team
starts the set serving or receiving, i.e. it depends on ω.
Consequently, the sums in Eq. (3) can be reduced to just
two. Note that there are ℬ25−k,kways that Team Ascores
25 points, out of which kof them are performed in rallies
where the team starts receiving. In the same way, there are
ℬl−k,k−1forms that team Bscores lpoints, out of which
l−kare scored in rallies where the team starts serving.
Consequently, we have
P𝒩
2(𝒮,s)=p25
s+
23
l=1l
k=1
Akα25−kγl−kδkβk(17)
with α=1−δ,β=1−γ, and Ak=ℬ25−k,kℬl−k,k−1. On
the other hand, from Eq. (8), for T=A,α=psand β=pr,
while α=qsand β=qrfor T=B.
In Model 2, QT,ω
j=QT,ω
kand Gω
j=Gω
kare satisfied for
all j,k, and ω. Thus, Eq. (8) can be reduced to
P>𝒩
2(𝒮,s)=𝒬A𝒢s+𝒬B𝒢r,(18)
where 𝒬Tis the probability that team Treaches the score
(24, 24) serving next and 𝒢ωis the probability of that the
winning team wins the set given that starts the tie break at
state ω. In this way, we have
P>𝒩
2(𝒮,s)=24
k=1
Ckα24−kγ24−kβkδk×
×α2
α2β+γα+γ2δ+
+24
k=1
Bkγ24−kα25−kβk−1δk×
×(αβ(γ+α))
α2β+γα+γ2δ,(19)
where Bk=ℬ24−k,k−1ℬ25−k,k−1is the number of ways
in which the winning team scores 25 −kpoints at serve
and k−1 at reception while the opposite team scores
24 −kat serve and kat reception, given that the oppo-
site team starts the tie break serving. Similarly, Ck=
ℬ24−k,k−1ℬ24−k,kis the number of ways in which the win-
ning team scores 24 −kat serve and kat reception while
the opposite team scores 24 −kpoints at serve and kat
reception, given that the tie break starts with the winning
team serving. The probability 𝒢ωcan be calculated by con-
sidering a sub Markov chain consisting of only the six
states Ferrante and Fonseca (2014).²
The winning probability, 𝒫T
2(α,β,s), is the sum of
Eqs. (17) and (19). It is important to highlight that
𝒫T
2(α,β,s)depends on three parameters, two SPs and the
initial state of the set (serving or receiving). Thus, the
probability that team Awins the set, given that it starts
receiving, can be written as
𝒫A
2(ps,pr,r)=1− 𝒫B
2(qs,qr,s).(20)
Note that Model 2 reduces to Model 1 by setting ps=
pr=p. The average number of rallies per set as a func-
tion of the SP is shown by dashed lines in Figure 3. It is
not possible to find a simple analytical expression for Nrin
Model 2. However, Nrcan easily be evaluated numerically.
As mentioned before, for Model 1, Nrreaches its maximum
for pr=0.5. In the case of Model 2, four different values of
pswere considered. For the trivial case of ps=1, Nr=25,
regardless the value of pr. In this case, team A starts serv-
ing, scoring 25 consecutive points. For 0<ps<1,Nrhas
a maximum localized at pr=p*
r. As psdecreases, the max-
imum of Nrand p*
rincrease. In the limit case of ps=0, the
maximum is given at p*
r=1and Nr→∞as expected. In
this scenario, team A never scores in K0, but it always does
so in K1. Consequently, the two-point difference required
to win the set cannot be achieved in a finite number of
rallies.
The probability of winning a set, 𝒫A
2(ps,pr,s), for
Model 2 is shown in Figure 4 as a function of prfor three
2At this point, it is important to highlight that for the most general
case, the calculation of Gω
jrequires to consider a sub Markov chain
with 36 states making difficult to find a simple analytical expression
for it.
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8|I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
ps = 0.5
ps = 0.4
A (ps, pr, s)
pr
Model 2
= 25
= 15
ps = 0.3
0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
1.0
p
ps = pr
ps = 0.5
ps = 0.4
ps = 0.3
2
Figure 4: Probability of winning a set, 𝒫A
2(ps,pr,s), for Model 2.
As before, the continuous lines represent 𝒩=25 and the
discontinuous ones represent 𝒩=15.
different values of ps, 0.3, 0.4, and 0.5. The continuous
lines represent 𝒩=25, and the discontinuous ones rep-
resent 𝒩=15. As expected, the smaller the value of ps,
the larger the value of prrequired to have a chance of win-
ning. The inset shows the behavior of 𝒫A
1(¯
p)as a function
of the average SP, ¯
p=(ps+pr)/2, for the same values
of psused in the main panel for 𝒩=25. The lines repre-
senting pr/=psare almost indistinguishable from the one
corresponding to pr=ps. Then, Model 1 gives almost the
same 𝒫A
1(¯
p)for different sets of prand ps, while in Model 2
those parameters have completely different probabilities
of victory.
In Figure 5, the difference between the probability of
team Awinning a set, given that it starts serving, and
0.0 0.2 0.4 0.6 0.8 1.0
–0.04
–0.02
0.00
0.02
0.04
0.0 0.2 0.4 0.6 0.8 1.0
–0.04
–0.02
0.00
0.02
0.04
ps = 0.6
pr
ps = 0.4
ps = 0.5
∆
2
A
Figure 5: Dierence between the probability of winning a set
given that the team starts the set serving and the one given that
it starts receiving. Continuous lines correspond to 𝒩=25, while
discontinuous to 𝒩=15.
the one when it starts receiving, ∆𝒫A
2=𝒫A
2(ps,pr,r)−
𝒫A
2(ps,pr,s), is shown. As expected for the case where
ps<pr,∆𝒫A
2>0, implying that under this condition, the
probability of winning a set is larger if the team starts
the set receiving. The importance of starting the set serv-
ing/receiving becomes more relevant for the fifth set,
where 𝒩=15 (see dashed lines in Figure 5). Then, for
ps<pr, it is better to start the set receiving than serv-
ing. Naturally, the difference becomes more important for
smaller values of ps. In fact, for ps=0.3, we found that
∆𝒫A
2can reach a value close to 0.05, which corresponds to
an increase of about 9% in 𝒫A
2(ps,pr,s).
Additionally, Model 2 can be used to estimate the
importance of players with specific functions such as Land
O. Let’s assume that team Aplays with neither Lnor Oand
its respective SPs are ps=0.3 and pr=0.6. From Figure 4,
the probability of winning a set for that team is about 0.2.
Now suppose that the team includes Land Oin the forma-
tion such that serve-reception and attack increase prfrom
0.6 to 0.7. This increase in the SP means that 𝒫A
2(pr,ps,s)
is now close to 0.5. Thus, under these conditions, L
and Ocan increase the probability of victory by about
2.5 times.
4New results
4.1 Model 3
As mentioned before, we want to quantify the importance
of the differences in the SP among the different rotations.
From now on we assume that team Astarts the set serv-
ing. Consequently, team Aserves in rotation R1to team
B, which is also in R1, but performing K1. Once team A
loses the serve, rotation R2of team Bserves to rotation 1
of team A, and so on. There are six different rotations in
total, R1,. . . ,R6. Each one can be either serving or receiv-
ing. If team Ais serving, the SP associated with rotation
Riis denoted by ps,iwith i=1, . . ., 6. In the same way, pr,i
is the SP for Ri, given that the team starts the rally receiv-
ing. Model 3 reduces to Model 2 if we take ps,i=psand
pr,i=prfor all i. For this model, the probability of winning
a set is denoted by 𝒫A
3(𝒮)to emphasize that it depends on
the set of SP, 𝒮 ≡ {ps,i,pr,i}.
The probability that rotation Riof team Awins ncon-
secutive rallies losing the (n+1)-th given that it starts the
rally serving is given by (1 −ps,i)pn
s,ifor 24 ≥n≥0and
p25
s,ifor n=25. Consequently, the average number of con-
secutive points scored in K0by rotation i,¯
mK0,i, can be
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |9
written as
¯
mK0,i=25 p25
s,i+
24
n=0
n(1 −ps,i)pn
s,i
=ps,i1+ps,i+p2
s,i+p3
s,i+p4
s,i×
×1+p5
s,i+p10
s,i+p15
s,i+p20
s,i.(21)
For the case where the rotation Riof team Astarts the
rally receiving, the average consecutive points scored by
the opposing team before it loses the serve, ¯
nK0,i, can be
calculated from
¯
nK0,i=qs,i1+qs,i+q2
s,i+q3
s,i+q4
s,i×
×1+q5
s,i+q10
s,i+q15
s,i+q20
s,i,(22)
with qs,i=1−pr,i. In order to win a set, a team requires to
have large ¯
mK0,iand small ¯
nK0,i. In many cases, the serve is
neutralized by the opposing team, which is able to perform
K1successfully. Thus, the probability of scoring a point for
the team that starts the rally serving turns out to depend on
its performance in K2. In order to guarantee a large ¯
mK0,i,
it is necessary to have a good performance not only in serv-
ing but also in K2. That is why many coaches consider K2
crucial for winning. In the proposed model, K2is implic-
itly included in ps,i. Additionally, the average number of
scored points per rotation allow coaches to strategize in
order to have their best rotation available to score during
the last points of the set/match or, equivalently, having the
weakest rotation of the opposing team during this stage of
the game.
The average score (¯
m,¯
n)of points scored after the
teams performed a complete cycle can be estimated from
¯
m=6+
6
i=1
¯
mK0,iand ¯
n=6+
6
i=1
¯
nK0,i.(23)
Figure 6 shows the behaviors of Eqs. (21) and (22) as
a function of the respective SP, ps,ifor ¯
mK0,iand pr,ifor
¯
nK0,i. The inset shows the behaviors of both functions for
values of SP close to 0.5. As expected, for small values of
the SP (ps,i<0.5), ¯
mK0,iis close to 0, implying that the
team is not able to retain the serve. This is typical for teams
with poor performance in K0and/or K2. In this regime,
¯
mK0,i<1implies that, on average, the respective rotation
requires more than one full cycle to score one point in K0.
For ps,i>0.8,¯
mK0,iincreases quickly, reaching its limit
value for ps,i=1. The results for ¯
nK0,ican be interpreted
in a similar way. For pr,i>0.5,¯
nK0,iis smaller than one,
meaning that team Aneeds on average just one rally to
regain the serve. In the opposite case, when pr,i<0.5,
team Ais not able to score easily in K1and, consequently,
0.0 0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
0.0 0.2 0.4 0.6 0.8 1.0
5
10
15
20
25
SP
0.4 0.5 0.6
0.5
1.0
1.5
2.0 0.4 0.5 0.6
0.5
1.0
1.5
2.0
SP
mK ,i, nK ,i
0
0
mK ,i
0
nK ,i
0
mK , nK1
0
Figure 6: Behavior of Eqs. (21) and (22) as a function of their respec-
tive SP. The continuous line represents ¯
mK0,i, while the dashed one
represents ¯
nK0,i.
the opposing team scores several points before team Agets
the serve back.
In general, model 3 leads to complicated equations for
the probability of winning a set 𝒫A
3(𝒮). For this reason, we
prefer to illustrate the results obtained from a computer
simulation of Model 3, following the method described in
Sec. II, for some particular cases.
4.1.1 Example 1: outstanding server
Now, consider the case where team Ahas an outstand-
ing player serving. Because of this, the SP in K0for the
rotation where this player serves is 20% higher than the
SP of any other rotation. Thus, for this case, we assume
that the set of SPs of different rotations, 𝒮={ps,i,pr,i},
is given by pr,i=pr(equal for all rotations) and ps,i=
0.35(1 +0.2δi,j), where δi,jis the Kronecker’s delta and
jis the index of the rotation Rjwhere the best server (BS)
serves. From Eq. (21), the average number of consecutive
points scored in K0,¯
mK0,j, is 0.72 for rotation Rj, while it
is 0.54 for the other rotations. The effect of the BS can be
better understood as follows. Let’s define ∆𝒫A
3(𝒮1,𝒮2)=
𝒫A
S(𝒮1)− 𝒫A
S(𝒮2)as the difference between the proba-
bilities of winning a set for two different sets of SPs, 𝒮1
and 𝒮2. The initial position of the BS can be chosen at the
beginning of the set and, consequently, the probability of
winning a set depends on that choice. In order to quan-
tify the effect of the initial position of the BS, we calculate
∆𝒫A
3(𝒮1,𝒮2)as follows: for 𝒮1, the BS serves in the initial
rotation R1(j=1), while for 𝒮2, it starts serving at rotation
Rj(j̸=1).
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10 |I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.002
0.004
0.006
0.008
0.010
0.012
j = 3
j = 6
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
pr
pr
∆ 3
A
3
A (Sj)
Figure 7: Eect of the starting position of the BS. The dashed line
represents j=3 and the continuous line represents j=6.
Figure 7 shows the results for ∆𝒫A
3(𝒮1,𝒮2)with j=3
(dashed line) and j=6 (continuous line). For j=3, the
BS starts the set at position 3, while it starts at position 6
for j=6. The inset shows 𝒫A
3(𝒮1,𝒮2)for j=1, 3, and 6. In
these cases, the different curves are almost indistinguish-
able from each other, indicating small differences between
these values. It is not surprising that the largest differences
are found for j=6 because, in such a case, the BS is the
last player to serve. As expected, ∆𝒫A
3is almost zero for
small and large values of pr. This means that the selection
of the initial position of the BS does not make a difference
if the team has either a poor or an outstanding SP in K1.
However, for more realistic values of pr, the selection of
the initial position of the BS could lead to an increase of
up to one percent in the probability of winning a set. This
increase may seem negligible, but it must be taken into
account that it is only due to the action of a single player
in a specific game action, the serve.
4.1.2 Example 2: outstanding opposite
In this case, we assume that team Ahas an outstand-
ing performance in K1in all the rotations where Ois in
the front row. For simplicity, it is assumed that ps,i=ps
is equal for all rotations. If the rotation istarts the rally
receiving, the SP is given by pr,i=pr(1 +0.2jδi,j)
where the sum is extended over the rotation Rjin which
Ois in the front row. Figure 8 shows the behavior of
∆𝒫A
3(𝒮1,𝒮2)for the following sets of SPs: for 𝒮1,Ostarts
the set in position 4, while for 𝒮2,Ostarts the set in
position 1. As in the previous example, for small or large
prvalues, the effect of the choice of O’s starting position is
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
pr
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
pr
3
A (Si)
∆ 3
A
Figure 8: Eect of the starting position of Oat the beginning of the
set.
negligible. However, for values of prthat are close to 0.6,
the increase in the probability of winning a set is above3%.
4.2 Model 4
Model 4 takes into account that the SP of a given rotation
in K1depends on the players directly involved in that com-
plex. The SP in K1for rotation Ri,pr,i, is given by Eq. (1)
where qi,jis the probability that the configuration jof Ri,
𝒞i,j, participates in the rally and pr,i𝒞i,jis the SP of that
specific configuration.
Model 4 leads to even more complex equations than
Model 3 because it requires more probabilities to be com-
pletely defined. However, as in the case of the previous
section, Model 4 can be easily studied numerically by con-
sidering some particular cases of interest. In all the cases
considered in this section, team Astarts the set serving at
rotation R1.
4.2.1 Example 1: right-side spiker with poor reception
performance
In this case, we assume that the RS1has a poor perfor-
mance in reception. Thus, the SP in K1for the configura-
tions where RS1receives the serve is pr,i𝒞i,j=ρ. In all
other cases, where RS2or Lreceives, the SP is pr,i𝒞i,j=
0.7. For the sake of simplicity, it is assumed that all rota-
tions have the same SP at serve, ps,i=0.35.
In order to quantify the effect of the difference in the
SP given by the poor performance in reception of RS1, we
plot in Figure 9 the probability of winning a set, 𝒫A
S(𝒮)as
a function of the probability that RS1receives, Q=3qi,j.
The factor 3 comes from the fact that, for all rotations, there
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |11
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
ρ = 0.3
nK1
ρ = 0.7
ρ = 0.4
ρ = 0.6
ρ = 0.5
Q
0.0 0.3 0.5 0.8
0.5
1.0
1.5
ρ = 0.7
ρ = 0.3
A
Figure 9: Probability of victory 𝒫A
S(𝒮)as a function of the
probability Q.
are three different configurations where RS1receives. Five
values of ρwere tested, ρ=0.3, 0.4, 0.5, 0.6, and 0.7.
For this set of parameters, the SP, pr,i, given by Eq. (1),
increases linearly with ρaccording to pr,i=0.7+Q(ρ−
0.7). Thus, for ρ=0.7 the team Ahas a reception line with
a uniform performance and, consequently, 𝒫A
4(𝒮)is inde-
pendent of Q. However, for smaller values of ρ, there are
differences in the SP of the different configurations. As
expected, for Q=0, the serve is never received by the RS1
and 𝒫A
4(𝒮)does not depend on ρ. In other words, all curves
converge to the same value for Q=0. For ρ<0.7, 𝒫A
4(𝒮)
decreases as Qgets close to 1. For ρ=0.6,𝒫A
4(𝒮)decreases
from 0.625 at Q=0 to 0.35 at Q=1. The effect is more pro-
nounced for smaller values of ρ. In fact, for ρ=0.3, 𝒫A
4(𝒮)
is almost zero for Q=1. A highly non-homogeneous per-
formance in the reception line puts team Aat a disadvan-
tage because the opposing team can direct the serve to the
weakest serve receiver, RS1, making Qclose to 1 and, conse-
quently, decreasing considerably the probability of victory
𝒫A
4(𝒮).
The inset shows the behavior of ¯
nK1as a function of Q
(Eq. (22)). For Q=0, the average number of points scored
by the opposing team is smaller than one, implying that
team Aregains the serve, on average, during the first K0.
However, ¯
nK1increases algebraically as Qgets closer to 1.
For instance, in the case of ρ=0.3 and Q=1, ¯
nK1≈2.5,
and the opposing team scores on average more than two
points before team Aregains the serve.
4.2.2 Example 2: importance of the libero
In this case, we evaluate the importance of L. In order to
do this, the SP in K1, given that Lreceives the serve, is
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
nK
δ
ps = 0.25
ps = 0.35
ps = 0.45
δ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0
A
Figure 10: Eect of Lon the probability of winning a set.
pr,i𝒞i,j=δ. In all other cases, where RS1or RS2receives,
the SP is pr,i𝒞i,j=0.7. For simplicity, ps,i=psis inde-
pendent of the rotation involved in the rally. For δ>0.7, L
has better performance than the right-side spikers, while
for δ<0.7, the opposite happens. It is also assumed that
all configurations of a given rotation have the same proba-
bility of performing K1,qi,j=1/9. Given this set of param-
eters and using Eq. (1), it is found that pr,i=(1.4+δ)/3.
Figure 10 shows the results for 𝒫A
4(𝒮)as a function of
the probability of δ; three different values of ps,iwere con-
sidered. The dotted line separates two different regimes:
on the left side, there is the region where Lis not the best
player at receiving, while on the right, the opposite is true.
The importance of Lis more relevant for small values
of ps,i. This is so because, for δ>0.7, the slope of 𝒫A
4(𝒮)for
a given δincreases as psdecreases. In the same way, for
ps=0.25, if δincreases from 0.7 to 0.8, 𝒫A
4(𝒮)increases
from 0.31 to 0.41. The inset shows the average number
of points scored to team A before regaining the serve. As
expected, if δincreases, ¯
nkdecreases. For the ideal case
δ=1, the reception of Lguarantees scoring the point, and
the opposing team requires on average almost five com-
plete cycles to score a point in K0. For δclose to zero, team
Bscores more than one point to team A.
5Analysis of the female U23
colombian team
In this section, the model is used to study the particular
case of the female Colombian U23 team (U23 CT). The SPs
were calculated by using the data obtained from the 2018
Panamerican Cup (PC). The SPs calculated from the data
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12 |I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
of the PC, ¯
mK0,iand ¯
nK0,i, for the different rotations, are
included in Table 1. Note that there are important differ-
ences among the SPs of the different rotations, confirming
the need for a model which takes those differences into
account. As usual, the inequality pr,i>ps,iholds for all
rotations. Particularly striking is the SP of R4, which has
a great efficiency scoring in K1, but a poor performance in
K0. It is worth noting that R5has the lowest average SP,
while the highest belongs to R6. The most efficient rotation
in K0is R3and the least efficient is R4. In fact, R4requires
three complete cycles to score, on average, a single point
in K0, while R3needs just one. In the case of K1, the best
performance is found in R4and the worst in R5. The aver-
age number of points scored per rotation by the opposing
team to R4is 0.28 and to R5is 1.08. Thus, on average, the
opposing team requires more than three complete cycles to
score a point to R4in K1, while it only requires about one
to score a point to R5.
A rough way to estimate the average number of points
scored in a complete rotation is given by Eq. (23). By using
the data shown in Table 1, we found that ¯
m≈10.6and ¯
n≈
10.7. Consequently, after two complete cycles, the score is
expected to be close to (21.2, 21.4). The final score depends
on the rotations that participate in the last part of the set,
which in turn depend on the selection of the initial rota-
tion. Unfortunately, the simple approach given by Eq. (23)
is qualitatively but no quantitatively correct, making it
necessary to study Model 3 numerically. Figure 11 shows
the probability of winning a set according to Models 1, 2,
and 3 for the U23 CT. According to Model 1, the probability
of winning a set is close to 0.52. Given that pris larger than
ps, in Model 2 𝒫A
2(ps,pr,r)>𝒫A
1>𝒫A
2(ps,pr,s), with a
difference of about 2%. This means that, in general, for
the U23 CT, it is more convenient to start the set receiving
than serving. However, Model 3 is the one that gives the
most valuable information, see symbols in Figure 11. As
predicted by Eq. (23), the probability of winning depends
on the initial rotation. The higher probability of victory is
found when the rotation 4 starts the set receiving because
this practically ensures starting the set scoring a point.
The opposite occurs when the set starts with R4serving
Table 1: The SPs, ¯
mK0,iand ¯
nK0,ifor the dierent rotations of the
U23 CT obtained from the data of the PC.
SP R1R2R3R4R5R6
ps,i0.45 0.44 0.51 0.25 0.40 0.48
pr,i0.56 0.52 0.55 0.78 0.48 0.61
¯
mK0,i0.82 0.79 1.04 0.33 0.67 0.92
¯
nK0,i0.79 0.92 0.81 0.28 1.08 0.64
¯
mPC
K0,i0.71 0.70 0.92 0.34 0.58 0.94
1 2 3 4 5 6
0.45
0.50
0.55
0.60
1 2 3 4 5 6
0.45
0.50
0.55
0.60
Model 3 U23 CT receiving
Model 3 U23 CT serving
Ri
Probability to win a set
Model 1
Model 2 serving
Model 2 receiving
Model 3 U23 CT receiving
Model 3 U23 CT serving
Figure 11: Probability of winning a set for the U23 CT team. The Ri
axis indicates the rotation that starts the set. The squares corre-
spond to the U23 CT starting the set receiving, while the circles
correspond to the U23 CT starting the set serving.
because, in this case, the opposing team will more likely
score the first point.
As shown, the probability of winning a set is less than
0.5 if the team starts the set serving in rotations R1,R2,R4,
and R5. For all other cases, the probability of winning is
greater than 0.5. This probability is maximal when the U23
CT starts the set receiving in R4.
As mentioned before, one of the objectives of these
models is to analyze the performance of a team and gen-
erate strategies that increase the probability of victory. For
the case of the U23 CT, it is clear that the SP in K0of R4is
far below the SPs of any other rotation of the team. This
could be due to, for instance, the poor serving skills of the
player who performs the serve in that rotation or poor per-
formance in the execution of K2. Let’s assume that the SP
in K0of R4increases from 0.25 to 0.46, which is the aver-
age SP in K0of the other rotations. The winning probability
for this set of parameters is included in Figure 11 by open
squares and circles. Under this assumption, a significant
increase in the probability of victory (up to 8%) for all rota-
tions is observed, regardless of whether the team starts the
set serving or receiving.
In order to check the validity of the model in the last
row of Table 1, we also include the value of the average
number of consecutive points scored in K0by rotation i
calculated directly from the data collected in the PC, ¯
mPC
K0,i.
It seems that Eq. (21) overestimates the value of the aver-
age number of points scored in K0but it preserves the
qualitative behavior found by using the experimental data.
In order to check the method used for parameter esti-
mation, we compare the SPs calculated from the data of
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I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches |13
Table 2: The SPs, ¯
mK0,iand ¯
nK0,ifor the dierent rotations of the
U20 CT obtained from the data of the SAC.
SP R1R2R3R4R5R6
ps,i0.52 0.56 0.47 0.60 0.56 0.48
pr,i0.64 0.65 0.50 0.50 0.59 0.54
the PC with those obtained by the female Colombian U20
team (U20 CT) in the 2018 South American Cup (SAC). The
starting team was the same for both cups. In fact, the PC
was used as preparatory tournament for the SAC. However,
two things have to be taken into account. First, we must
expect changes in the SPs because the team was farther
below the age category in PC than in the SAC. Therefore,
it is reasonable to expect a better team performance in the
SAC than in the PC. Additionally, the opponent teams were
different in both tournaments. Secondly, an important tac-
tical change was implemented in the SAC: the opposite
and one of the right-side spikers exchanged their posi-
tions. This change was implemented to balance the attack
performance of all rotations. The SPs obtained from the
data collected in the SAC is shown in Table 2. There are dif-
ferences between both sets of SPs. The average of ps,iand
pr,ichanges from 0.42 and 0.58 in the PC to 0.53 and 0.57
in the SAC, respectively. The most important difference is
found in R4where ps,iis about 2.5 times larger in the SAC
than in the PC. This is partially due to the tactical change
mentioned before.
It is important to highlight that our model requires
several parameters as input. Nevertheless, as shown in
Tables 1 and 2, the parameter estimation depends on the
data used, the opponent teams, the initial rotation, etc. A
general problem in sports analysis, is that it is usually hard
to collect enough representative data such that it reflects
the relevant opponents in a way that allows to estimate
the SPs with enough precision. Despite this, a major con-
tribution of models like the one presented here is that it
helps us to analyze the performance of a given team, diag-
nose possible issues, and develop strategies to increase
its performance. For instance, the experimental data from
the PC shows a poor performance of R4in K0, which was
remedied in the SAC by exchanging the positions of two
players.
6Conclusions
The model we have presented replicates previous results.
For instance, it shows that volleyball is a selective sport,
i.e. in a match, the team with the larger SP usually wins.
Sets played to a small 𝒩, such as the decisive fifth set,
favor teams with smaller SP, Figure 2. In other words, the
probability that the team with the smallest SP wins a set
increases as 𝒩decreases. For this reason, the team with
the largest SP should avoid reaching the fifth set where the
random factors are more relevant. Sets with a large num-
ber of rallies Nrshould be expected when teams have sim-
ilar SPs, see Figure 3. In these cases of large sets/matches,
the physical condition of the players becomes a crucial fac-
tor because it can be expected that the performance, and
consequently the SPs of the teams, decreases if the players
are not well prepared physically. As shown in Figure 4, the
model highlights the importance of starting the set receiv-
ing. This becomes even more relevant for smaller values
of ps. Additionally, the importance of starting the set serv-
ing becomes significant in the fifth set, when 𝒩=15, as
can be seen in Figure 5. It was found that models based
on fewer parameters, such as Model 1 and Model 2, have
limited practical applicability. For instance, Model 2 shows
that different sets of prand pshave different winning prob-
abilities, 𝒫A
2(pr,ps,s). However, for Model 1 those param-
eters are reduced to a single one, ¯
p=(pr+ps)/2, which
could lead to similar winning probabilities, 𝒫A
1(¯
p,s), see
Figure 4. On the other hand, the statistical analysis of
the U23 CT shows that the SPs of different rotations have
relevant differences which are neglected in Model 1 and
Model 2. Those differences have an important impact on
the probability of winning a set.
The serve plays a crucial role in volleyball games
because it can be used to decrease the chances of the
opposing team in K1and, consequently, increase one’s
team probability of obtaining a point in K2, i.e. increasing
ps,i. As shown by Model 3, the increase of 20% in ps,ifor
a single rotation increases the probability of winning the
set by 1%. The increase of ps,iin all rotations leads to a
substantial increase in 𝒫A
2, as shown in Figure 4.
In particular, our multi-parameter model allows us
to quantify the effect of having a single rotation with a
SP larger than any other rotation. This model supports
the idea that starting a set with the best possible rotation
(i.e. the rotation with the largest ps,iand pr,i) leads to a
significant increase in the probability of winning the set.
This kind of model allows us to generate game strategies
that increase a team’s probability of winning. For instance,
Model 3 and 4 can be used to explore the effect of the initial
formation of the players on score evolution. Depending on
the SP of the initial rotation, it could be more convenient
for a team to either choose serving first or choose their side
of the court at the initial coin toss. The relevance of the ini-
tial position of the player which starts serving is discussed
in the first example of Model 3. The initial position of the O
is also studied in the second example of the same model.
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14 |I. Gonzalez-Cabrera et al.: Generalized model for scores in volleyball matches
In the first and second example, we found an increase of
1% and 3% in the probability of wining, respectively. The
two examples of Model 4 show the importance of having
a homogeneous reception line. The presence of a serve
receiver with poor performance on serve-reception can be
used by the opposing team to increase its probability of
winning.
However, the applicability of the model goes beyond
the formulation of general game strategies. The models
discussed here can also be used to diagnose the perfor-
mance of a particular team and to create more efficient
training strategies for that specific case. In the particular
case of the U23 CT, it is clear that the probability of win-
ning a set can be significantly increased if the performance
in K0of R4is improved. This suggest specialized train-
ing for the player who executes the serve at that rotation
and/or complex-specific training to improve the efficiency
of the team’s K2during this particular rotation. The low SP
when serving in R4is common in modern competitive vol-
leyball. During this rotation, the Sis in charge of blocking
one of the RSs of the opposing team. The opposing team
often tries to attack using this player to take advantage
of the usual height difference between their RS and the
rival S. This puts the serving team at a disadvantage. This
issue can be overcome by several tactical strategies. One of
them is a double substitution of players: the Sin position
4 is substituted by the reserve opposite and the Oin the
court is replaced by the reserve setter. Another strategy is
to increase the performance of the defense in the rotations
where Sis in the front row. The implementation of one or
several of these strategies could lead to a non-negligible
increase in the probability of winning a set for the U23 CT.
A weaknesses of the approach used in our model is
that it does not take into account some important factors.
For example, a more complete model requires to assign a
fixed scoring probability to an attacking action or complex
rather than a complete rally. The duration of a rally can
also influence the scoring probability. In order to account
for these factors, a more detailed model for a single rally
is required. Finally, given the importance of the K2, it is
necessary to propose a model which explicitly takes into
account the performance of a team in this complex taking
into account the factors just mentioned. This issue should
be addressed in future work.
Acknowledgments: The work of D.L.G was supported by
the Vicerrectoría de investigaciones de la Universidad del
Valle C.I. 1164.
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