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MATHSPORT INTERNATIONAL 2019
CONFERENCE
— Proceedings —
Crowne Plaza Hotel
Athens, 13 July 2019
Organized by AUEB Sports Analytics Group
Complex 1 in Male Volleyball as a Markov Chain.
Sotirios Drikos, PhD.
National & Kapodistrian University of Athens, School of Physical Education and Sport
Science
sodrikos@phed.uoa.gr
Abstract
In Volleyball, complex 1 consists of the serve’s pass (reception)  setting  attack skills in
this specified order. This sequence is a stable pattern to win a point. Furthermore, it is
important for the teams’ success. Taking into account that this pattern is a firstorder Markov
chain, the creation of a probability transition matrix is feasible. Assuming multinomial
likelihood with a Dirichlet prior on the transition probabilities a Markovian transition matrix
can be constructed and the calculation of conditional success probabilities is, thus,
achievable. Data from the performance analysis of the winning team from recent world
championships in three age categories (U19, U21, Men) of male Volleyball is used. The
findings lead to redefining target pass area and to shrinking the evaluation scale at least for
the teams under study. Moreover, pass accuracy is necessary because it must give at least two
options for attack, but not sufficient condition for the success of attack in all age categories
for male Volleyball. In the U19 age category, there is a lack of stabilization in the complex 1
sequence after pass against jump spin serve.
1 Introduction
Volleyball consists of 3 stable patterns to win a point: passsettingattack after pass outcome serve
outcome and blockdig setting attack after dig or counterattackoutcome (Florence, Fellingham, Vehrs, &
Mortensen, 2008). For each pattern three are the possible outcomes: win a point, continuation of the action
and a point for the opponent. In rally point system the pattern passsettingattack after the pass is the
necessary condition to claim the victory because in terms of probability winning a point when receiving is
easier than winning a point when serving in male volleyball(Calhoun, DargahiNoubary, & Shi, 2002;
Ferrante & Fonseca, 2014).
Winning teams were significantly better in attack after pass than losing teams (Hayrinen, Hoivala, &
Blomqvist, 2004) and attack after pass emerged as a decisive factor for team’s success (Patsiaouras,
Charitonidis, Moustakidis, & Kokaridas, 2009). It is crucial for a team to organise a tactically well
structured and highly synchronised offensive game after receiving opponents serve. It is the hierarchical
order of skills in Volleyball that makes the performance in one skill depends on the performance in the
previous one. The precise pass is a powerful aggressive tool for highlevel teams and is a good predictor for
winning (Zetou, Moustakidis, Tsigilis, & Komninakidou, 2007). For many coaches receiving well is a
guarantee for a winning attack. The connection between the quality of pass and achievement in attack is
undoubted for men age category in many types of research. Α partial rejection of this belief is suggested by
Lobietti, Michele, & Merni (2006) who proposed that passing accuracy does not appear so fundamental but
it is important avoiding passing errors.
The assumption that passsettingattack after pass pattern is a firstorder Markov chain allows the
recording of these sequences in a transition probabilities matrix where data of the matrix represent the
probability to move from one state to another and, finally, to an outcome. With the use of the Bayesian
analysis, the past team’s performance or the coaches’ opinions about passing effects in the attack can be
taken into consideration as a prior distribution in order to create with actual data the posterior distribution
and, consequently, the conditional success probability.
Thus, the aim of this study is to determine the influence of each level of a pass to the success of attack
in 3 different age categories (U19, U21, Men) for male highlevel Volleyball.
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2Method
All recorded data refer to the performance in pass and attack after the pass of the winning team of the
world championship for national teams in three age categories for male volleyball. All data record the
performance on selected matches of the World national team champions (Poland in Men, 2014; Russia in
U21 and in U19 for 2013). Thus the initial sample (N=) was 815 for Men, 525 for U21 and 407 for U19
passes respectively. For the evaluation of pass, a 6level ordinal scale was used with the 1st level being a
passing error and the 6th level to be a pass performed in an optimal way. In Table 1 the performance ratings
and a brief description of each passing level are presented. Attack was evaluated with three possible
outcomes: point for the team under observation, rally continuation and point for the opponent.
Table 1.Performance ratings for a pass (vs Jump Spin & Jump Float Serve)
Level code
(Symbol)
Level brief description
6(#)
The ball was passed accurately with suitable height, speed and parabolic
trajectory
in the target area (3m4m from the right sideline and about 30
50
cm from the net or over 30

50 cm over the net if setter has the ability to
jump setting). The setter could have all the options (location & type) for a
set from the sidelines and the central lane without any adjustments in his
approach to the ball.
5(+)
The ball was passed either away (1m. behind or 2m. in front of the target
area), or travelled higher, or lower (setter’s shoulder level). The setter could
have all the options for attack (location & type) from the sidelines and the
central
lane with adjustments in his approach to the ball.
4(!)
The ball was passed with either 3m away from the net or near the sidelines
or to the top of the net. The setter could have two options for attack only
from the sidelines.
3()
The ball was passed with very poor parabolic trajector y or near the sidelines,
end line or outside of the court. The setter could have just one mandatory
option for attack or the setter could not approach the ball and another player
sets the ball mandatory.
2(/)
The ball was passed directly to the serving team court. No option for attack
for the receiving team.
1(=)
The ball hit the floor directly or after touched by a receiver. The rally was
ended after 1st or 2nd contact.
The observer was a volleyball coach, expert in evaluation and recording of volleyball performance
data and excellent user of the software. The interobserver reliability of the data collection and recording
was checked by a testretest procedure, with a oneweek interval, from a random sample of 100 actions of
stable pattern passset attack after passoutcome for each one of the teams under observation. As the
acceptable value of Adjusted Cohen’s Kappa was set 0.80 (Altman, 1991). The interobserver reliability in
evaluation and recording of data was well established because of acceptable Adjusted Cohen’s Kappa
values calculated after the testretest procedure. The values were 0.91 and 0.90 for a pass against jump spin
and jump float serve respectively.
Every time the opponent serves the ball on the side of the observed team a sequence of events takes
place that follows a speci fic scheme: pass–set–attack after pass–outcome. An assumption that this scheme
is a firstorder Markov chain is stated. This sequence was recorded in a transition probability matrix where
data of the matrix represent the probability to move from one state to another and finally to reach an
outcome. In this way three (one for each team) transition probabilities matrices were created.
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A simple Bayesian model
1
()
tkti
PY S Y S
to estimate the transition probabilities, and through
them, the success probabilities were made. A multinomial likelihood for each row (i.e. level of the pass)
1 , ,1 ,2 1 , ,1 , 2
( ,..., , ,  ,..., , , )
ik
i
y
i in in in i in in in ik
k
fy y y y
SSSS S
v
M
with
3
1
1
i
n
ik ik
kk
SS
¦¦
M
for each i; where
i
M
is the set of indexes corresponding to possible following
skill Si, was assumed. Given that the interest was in what the data suggest on the relationship between the
different states of the sequence, a minimally informative prior distribution is assumed. A conjugate
Dirichlet prior distribution was used where each row of the prior parameters were all assumed to be equal
to one (except those that were constrained to be zero). All conditional probabilities scores were calculated
using a simple Monte Carlo scheme of 10,000 iterations to acquire a random sample from the posterior
distribution. For a detailed description of the model see Drikos, Ntzoufras, & Apostolidis (2019).
3Results
The posterior means of conditional probabilities for each no terminal level of the evaluation scale for
jump spin and jump float serve are presented in Table 2. Level 1 of pass scale is a terminal level and its
probability to move to another state or to reach a positive outcome is zero. For level 2 of the pass, there is a
noticeable finding. After overpass against jump spin serve the receiving team keeps a sufficiently higher
probability (0.45) to win a point than to keep the ball in its court and have a mandatory attack (level 3). As
expected, the pass in level 4, 5, and 6 of the scales have higher conditional probabilities. An important
increase of probability to win a point is obvious when the pass is evaluated as level 4 (two options from
sidelines) contrary to evaluation as level 3 (one mandatory option for the setter). This increase is 0.21, 0.16,
0.28 against jump spin serve and 0.19, 0.19, 0.16 against jump float serve for Men, U21, and U19
respectively. For U19 against jump serve the probability to win a point with pass level 4 is higher than with
more precise passes (levels 5&6). Comparing success probabilities between levels 5 & 6 it is clear that
more precise pass (level 6) does not mean higher success probabilities. Taking into consideration the
standard deviation of each posterior mean, it is clear that success probabilities of a pass in levels 5 & 6 are
almost equal for each age category. Also in Table 2, the tail posterior probability level of differences across
age categories for each level of pass evaluation scale is presented. It is remarkable that the U19 team has a
significantly higher probability of taking a point after a pass level 4 against both types of serve (offensive
options only from sidelines) than U21 and Men team. Also, the U19 team has a higher probability to gain a
point after an overpass against jump spin serve than both U21 and Men. Meanwhile, the U19 team has a
higher probability of winning a point compared to U21 when the pass from a float serve is accurate on the
net (level 6).
Table 2. Posterior means (±sd) of conditional probabilities and summary of posterior differences
across age categories for each no terminal level of pass evaluation scale
Skills (Si
)
Skills
(sub)
Men U21 U19
Posterior
differences
*
Pass in Jump
2(/)
0.274 (±0.058)
0.266 (±0.053)
0.454 (±0.124)
Men,U21<U19
3()
0.308 (±0.038)
0.337 (±0.055)
0.307 (±0.090)
4(!)
0.548 (±0.022)
0.515 (±0.033)
0.631 (±0.045)
Men<U19, U21<<U19
5(+)
0.593 (±0.022)
0.548 (±0.029)
0.605 (±0.045)
6(#)
0.589 (±0.0212)
0.545 (±0.032)
0.565 (±0.048)
Pass in Jump
2(/)
0.256 (±0.046)
0.188 (±0.069)
0.281 (±0.049)
3()
0.325 (±0.039)
0.304 (±0.052)
0.412 (±0.079)
4(!)
0.539 (±0.024)
0.513 (±0.031)
0.603 (±0.035)
Men<U19, U21<<U19
5(+)
0.581 (±0.022)
0.563 (±0.027)
0.616 (±0.031)
6(#)
0.569 (±0.022)
0.558 (±0.026)
0.629 (±0.030)
Men<U19,U21<<U19
*Inequalities indicate important differences between age categories: Age category A has lower success
rates than age category B with posterior probability less than 0.01 ("A<<<B"), between 0.01 and 0.05
("A<<B"), between 0.05 and 0.10("A<B").
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A detailed preview of success conditional probabilities are provided in Figures 1&2.
Figure 1 and 2.Boxplots (with outliers) of success conditional probability of each no terminal level of
evaluation scale for a pass in Jump spin serve and for a pass in Jump float serve.
4 Discussion
The target for the receiver is an area close to the net or sometimes over it (3m4m from the right
sideline and about 3050 cm from the net or over 3050 cm over the net if the setter has the ability for jump
setting). The pass that is directed to the court of the serving team (2nd level, that is to say, overpass) and the
pass with the oneoption setting (3rd level of the evaluation scale) have the same characteristics at all ages,
with an exception of U19 only for a pass against jump spin serve. The penalty for the overpass is higher
compared to this for a 3rd level pass. Also, the pass level 6 on the net or too close to the net does not present
a higher probability compared to the 5th level. Silva, Lacerda, & Joao (2014) have mentioned the possible
difficulty of the setter to handle a ball on the net. These findings follow the conclusions of Miskin et al.
(2010) that, at least for the teams under consideration, the target area of a pass on the net must be more
conservative.
In all age categories, the probability of winning a point in the stable pattern passsetattack after the
pass is above 0.5 when the pass is evaluated on levels 4, 5, 6 of the evaluation scale. Thus, the first priority
for a team should be to keep the ball in its court giving the setter the opportunity to choose at least between
two attackers from the sidelines (outside hitter and opposite). The coaches’ belief that a good pass is a
guarantee for an effective attack can be more specified by pointing out that a pass which secures at least
two attacking options increase the probability of a successful attack for all age categories in male
Volleyball. This is in partial agreement with many studies about the relationship between pass and attack.
The lack of discrimination between the 5th and 6
th level of evaluation scale according to success
probabilities ensures the finding of Lobietti et al. (2006) that passing with high accuracy is not a necessary
condition for a successful attack. Also, this means that, at least for the teams under examination, the
passing rating system has to be changed. A possible junction of 5th and 6th level should be examined.
The large discrepancy of success probability from 1st, 2nd and 3rd in relation to 4th, 5th, 6th level of the
pass evaluation scale is a clear message that the probability of success is not increasable as the evaluation
grade gets higher. This phenomenon is observed in all age categories. There is no a fixed interval between
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levels of the scale, thus the assumption of treating ordinal data as numerical data and the use of descriptive
statistics, such as mean and standard deviation, for the evaluation of teams’ or players’ performance may be
groundless. The same has been concluded by Florence et al. (2008) after examination of a college women’s
volleyball team.
It is difficult to explain the finding that the U19 team has higher probabilities after an overpass against
jump spin serve instead of keeping the ball in its court with only one option for attack. It is clear that this
analysis is applicable only to these teams, their level and their opponents and generalisations may be not
applicable to other teams. In the model, only the next two touches of the team under observation were
recorded, so it is highly likely that a point after an overpass is due to opponents’ error. But even with this
assumption, it is important to mention that the jump spin serve has a higher speed than jump float serve and
the reaction time for receivers is reduced in <0.5s (Pena, Busca, Galceran, & Bauza, 2013).
Consequently, the reaction time is also limited to the serving team too, especially if they are not well
prepared to play an opponents’ overpass as a free ball.
Team U19 after pass level 4 against jump spin serve is more effective than Men & U21 teams. Also, it is
noteworthy that there is not increased the probability to win a point when passing performance rises above
level 4, contrary to Men and U21 teams. Performance of U19 team in passsetattack after pass pattern
confirms the findings of Costa G. C. et al. (2011) that subsequent actions do not have high functional
dependence in relation to the precedent ones in the age category of U19 due to the fact that because of lack
of players’ maturity the game is not well integrated.
To sum up, the present study is validating the sixlevel scale for evaluation of pass, it is developing a
Bayesian model including prior distribution and is applying this model to performance data of world
champion teams in three age categories. The conclusion reached is that for all ages the quality of pass is
important to ensure at least two offensive options for the setter. Furthermore, the discrepancy of success
probabilities among the levels of the scale makes it clear that for this ordinal scale it is unrealistic to use
descriptive statistics, like a mean and standard deviation. Finally, the target area of the pass must be more
conservative and the evaluation scale must be shrunk, at least for teams under observation.
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