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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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