# Stability tests in time of OAM multiplexing schemes in highly disturbed
environments.

F. Tamburini1, B. Thidé2, P. Weibel1, V. Boaga3, F. Carraro3, and M. Del Pup3.

Corresponding author F. Tamburini. Email<EMAIL_ADDRESS>1 ZKM –
Zentrum für Kunst und Medientechnologie, Lorentzstr. 19, D-76135, Karlsruhe,
Germany, EU  2 Swedish Institute of Space Physics, Ångström Laboratory, P. O.
Box 537, SE-75121 Uppsala, Sweden, EU  3 ARI, Sezione Venezia, Santa Croce
1776/B, IT-30123 Venice, Italy, EU

###### Abstract

We report the results of tests of data transmission and signal stability in
time of two different wide-band multiplexing (MUX) schemes, each in a point-
to-point configuration, based on electromagnetic waves carrying Orbital
Angular Momentum (OAM) in noisy real-world settings. Each radio link
transmitted two high definition wide–band analog TV channels in the same
frequency band with FM-carrier centered at $2.414$ GHz and $27$ MHz bandwidth,
encoded with different OAM modes in the same polarization state,
uninterruptedly for $5$ months during the world exhibition “Globale–Digitale”
at ZKM in Karlsruhe and in other $2$ months time slots taken in the following
$4$ years, $24$ hours per day. We show the practical feasibility of the use of
stable OAM radio/TV links in the real world for a long time, paving the way
for for secure and efficient communication schemes also under electromagnetic
jamming conditions.

###### Index Terms:

Orbital Angular Momentum; Electromagnetic fields; Microwave antennas; Antenna
radiation patterns; Electromagnetic propagation; Microwave communication;
Radio link; Multiplexing

##  I Introduction

CUrrent radio science and communication implementations based on the
electromagnetic (EM) linear momentum $\mathbf{P}$ physical layer, associated
with the Poynting vector and the translational dynamics and force action of
the system, are approaching their limits in terms of radio frequency spectrum
availability and occupancy even with the use of multiple-input multiple-output
(MIMO) techniques [1] and the additional use of wave polarization that can at
most double the channel capacity within a given oscillation frequency
bandwidth. This calls for the introduction of new radio paradigms to transfer
more information, namely, to use not only the energy $\mathbf{E}$ and the
linear momentum $\mathbf{P}$ conserved quantities, but also the other existing
partial and total conserved quantities of the EM field [2] still underutilized
in radio science and technology. For this purpose, it has been proposed that
the EM angular momentum conserved quantity $\mathbf{J}$, associated with
rotational dynamics and torque action of the system, as described in the
standard literature [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], could
be fully exploited in radio communications to improve the existing
multiplexing systems [17, 18, 19, 20, 21, 23, thidepreprint2014].

The total EM angular momentum conserved quantity $\mathbf{J}$ can be split in
two sub-components: the spin angular momentum (SAM) $\mathbf{S}$, associated
with the polarization of the EM wave and the spin of a photon, and the orbital
angular momentum (OAM), $\mathbf{L}$ [14, 16]. SAM and OAM cannot in general
be considered as separate quantities that evolve independently during the
evolution of the system. This can happen only in the paraxial approximation
when propagating in free space or in a homogeneous isotropic medium.

As follows from Maxwell’s equations, OAM is a pseudovector, associated with a
variation of the phase of the EM field as a function of the azimuthal angle,
$\varphi$, taken in the plane perpendicular to that of the wave’s propagation
and centered on the axis of symmetry of the beam. OAM is related to the
rotations in space around a given axis i.e. with recurrence modulo $2\pi$.
Because of this, OAM takes discrete values. It is expressed as a discretized
quantity even in a classical framework where a macroscopic EM field is
present. Any EM field, carrying an arbitrary amount of angular momentum, can
be expanded in a superposition of a denumerably infinite set of discrete OAM
modes. An example are Laguerre-Gaussian (LG) modes that present a cyclic phase
factor $\exp{({\mathrm{i}}m\varphi)}$ modulo the integer $m=0,\,\pm 1,\pm
2,\ldots$, also known as topological charge [11, 24, 14]. In these beams each
photon carries a well-defined quantity $m\hbar$ of OAM. A plane wave does not
carry orbital angular momentum and has $m=0$ OAM. OAM modes with different
values of $m$ are mutually orthogonal and are therefore independent even when
they have the same $m$ value with opposite signs. It is interesting to point
out that OAM is a pseudovector and, because of this, any reflection will
change the sign of the OAM and the reflected beam will result orthogonal to
the original one.

OAM, together with SAM, represents a new physical layer for radio science and
technology. In the paraxial approximation, when propagating in free space, SAM
and OAM can be used to set up efficient communication schemes that can be in
certain cases equivalent or even complementary to the already existing MIMO
systems [25]. Different signals, encoded in different OAM states independently
generated in the same frequency band, do not interfere with each other [26]
and can therefore be used to encode in a more efficient way a larger amount of
information down to the single photon level [27]. This has been confirmed by
numerical simulations, experiments [17, 19] in controlled anechoic chamber
[20] and outdoor in a real-world setting in the radio [21, 23, 28] and optical
bands [29, 30, 31] that gave the start to the vast literature present
nowadays.

One of the most important properties of OAM modes is that they are physical
states of the EM field that preserve their topological properties of phase and
intensity during their propagation in free space and represent a viable
solution for a stable multiplexing (MUX) realized with standard linear
momentum–based techniques [32] and the physical properties of the OAM field
[33, 34, 22]. The robustness of the OAM beams during their propagation is
confirmed by the results of a recent work where the topological stability of
radio vortices at $233$GHz permitted to measure the rotation of the
supermassive black hole in the galaxy M87, after having traveled a distance of
$56$ millions of light years in the outer space [19, 35, 36].

In this paper, we report the results of the tests for the stability in time of
two OAM multiplexing systems for wide-band radio/TV transmission indoor and
the robustness with respect to the presence of multiple reflections and
disturbing signals. The two experiments ran uninterruptedly $24$ hours per
day. The longer period was $5-$months long in 2015 during the
Globale–Digitale. Additional runs were made every year for 2 months including
additional tests with digital channels. This represents a common real-world
situation that can be found in MUX TV links indoor.

##  II Experimental results

Channel stability in time: The two OAM links – experiment 1 and experiment 2 –
were part of scientific and artistic installations present in the 2015
Globale-Digitale world exhibition in Karlsruhe, “The New Art Event in the
Digital Age” took place over 300 days in Karlsruhe, where ZKM presented
spectacular installations, innovative works of art at the interface with
natural science, science with scientific experiments as new form of science in
terms of a new expression of art, urban area performances accompanied by
concerts, lectures and conferences [37]. As part of Globale, the “Infosphere”
exhibition presented an overview of art in the era of the digital revolution,
with science and its social consequences, giving a new perspective to the new
data and science worlds [38].

Both the links ran uninterruptedly for $5$ months $24$ hours per day with more
than $20.000$ people assisting at the conferences and exhibitions interacted
with the experiments and $2$ months per year in the following years until
$2019$ in similar conditions. This replicates exactly the conditions in which
OAM links would experience when applied to real world settings where random
disturbing radio signals were generated by the local wifi, bluetooth systems
and mobile phones of the visitors. Additional static interferences were caused
by the reflections from the walls and the metal structures there present. The
installations were mounted in large accessible closed spaces under safe
conditions for people.

In both the two experiments, two different and independent links were
simultaneously transmitting two superimposed high quality (HD standard) audio
and video analog channels on the same frequency band. One channel was twisted,
carrying $m=1$ orbital angular momentum, whilst the other one was untwisted
($m=0$). Each dual link transmitted two FM wide-band TV channels in the same
frequency band centered at $2.414$ GHz and $27$ MHz bandwidth, one twisted
with $m=1$ OAM and the other untwisted ($m=0$).

We used horizontal polarization for both the links: in experiment 1 to reduce
the effects of reflections from the vertical metal structures present in the
Foyer of ZKM, a large closed environment. In the second link (experiment 2),
set in the “Infosphere” zone of the exhibition, the effect of multiple
reflections and high electromagnetic pollution due to the presence of other
artistic/scientific installations reduced more the channel separation than in
experiment 1. The fading caused by reflections is, in fact, one of the most
common complications in wireless communications that can decrease the quality
of the transmissions, including those based on multi-path protocols like MIMO
(Multiple-Input-Multiple-Output) [1]. The main requirement for a good quality
HD video transmission was to obtain a channel separation with carrier-to-noise
ratios of at least $20$ dB in order to have a stable and good quality in both
video streamings (twisted and untwisted). We periodically measured the channel
separation, bit transfer rate and errors with a digital transmission on the
twisted channel. No macroscopic effects due to interferences have been
reported during the long runs and no evident signal degradation was observed
below the lower limit of $20$ dB required. Down to that value, the presence of
continuous or sporadic interferences would have affected the audio-video
streaming with clearly evident effects such as the rising of patterning on the
video, with lines and/or “snow” until the complete blocking of the signal with
a frozen image displayed on the screen by the receiver. The interferences from
the Wi-Fi, instead, can cause horizontal noise bars on the screen, with
additional noise in the audio channel.

Experiment 1: in the foyer of ZKM in Karlsruhe, always crowded with people, we
set up the links with maximum emitting power of $0.1$W for each channel. The
distance between the source and the horizontal bar where the two
interferometers were mounted was $17.8$ metres, equivalent to $\sim
145~{}\lambda$. The two channels transmitted two high quality movies with
artistic and scientific contents that were displayed in two separate screens
(more information can be found in Ref. [39]).

The twisted beam was generated by the helicoidally-deformed parabolic antenna
used for the $442$-metres link in the Venice experiment [21], with $80$ cm
($6.4\lambda$) diameter, whilst the untwisted beam was generated by a dipole
with a $\lambda/4$ retroreflector backfire configuration. The two transmitting
antennas were mounted on a vertical configuration to minimize the advantages
of a standard horizontal MIMO scheme. Because of the different geometric
spatial shapes of the phase fronts of the two beams, the field vectors of the
$m=1$ twisted beam will result in anti-phase at two points on diametrically
opposite sides of the OAM phase singularity at the centre of the two aligned
beams and orthogonal to the propagation axis, whereas they will result in
phase for an $m=0$ beam [17, 20]. Therefore two simple standard linear-
momentum interferometers, in anti-phase with respect to each others were used
to discern between the twisted and untwisted beams [21]. The antennas of the
two interferometers were mounted at the same height from the floor and
collinear on the same horizontal line.

To minimize the possible cross-talking between the antennas of the two
interferometers in each arm and antenna resonances, we used two couples of
different types of antennas with the same gain. The first interferometer was
realized with two identical $26$ cm ($2.1\lambda$) diameter $16$ dBi backfire
antennas, connected together through a signal splitter/combiner and separated
horizontally by a quantity $\sim 25\lambda$. The second interferometer was
instead realized with a couple of $16$ dBi commercial off the shelf Yagi-Uda
antennas, also connected together trough another signal splitter/combiner with
a separation of $\sim 18\lambda$ mounted on the same horizontal bar. As shown
in the scheme of Fig. 1, in both receivers a phase tuner, in the form of a
silver slit waveguide coupled to a Selenia signal circulator, was inserted
into the path of the interferometers’ arms. In the first receiver we set the
cursor of the slit waveguide so to retard the signal received by the first
antenna of half the wavelength antennas with respect to the second one and
tune the receiver on the twisted channel. In the second receiving line,
instead, the slit waveguide was set so to have both the receiving antennas of
the interferometer B in phase, canceling with disruptive interference the
$m=1$ channel and revealing the untwisted $m=0$ one. Once regulated the
positions of the cursor in the waveguides and optimized the phase tuning, they
remained fixed during the whole run. No additional electronic devices were
used to adjust/correct the phase setting in the interferometers to compensate
the phase errors occurring during the runs. In this way we received
simultaneously two channels in the same frequency band. The averaged measured
channel separation during Globale-Digitale with the digital channel was $32.3$
dB and standard deviation (std) of $0.8$. See Tab. I for more details.

\-

Figure 1: Schematic description of the setup of experiment 1 in the Foyer of
ZKM. Two HD audiovideo channels were transmitted simultaneously in the same
$2.414$ GHz frequency band, $27$ MHz bandwith, with the same horizontal
polarization state. The twisted channel ($m=1$) was produced by a helicoidally
deformed parabolic antenna and the untwisted with dipole a standard antenna.
At the receiving end, at a distance of $\sim 145\lambda$ from the transmitting
antennas the instantaneous EM fields of the $m=0$ and $m=1$ modes resulted in
phase on one side of the central axis (azimuthal angle $\varphi=0$) and in
anti-phase at the opposite side ($\varphi=\pi$). Therefore two independent
standard linear-momentum interferometers one in phase with the $m=0$ beam and
the other with the $m=1$ beam were used to receive simultaneously and
uninterruptedly two transmissions. The boxes “X” indicates the relative phase
adjuster present in each of the two interferometers.

Experiment 2: the second experiment was part of a Science/Art installation
located in the “Infosphere” the of Globale–Digitale exhibition. The maximum
output power was $0.1$ Watts per channel, same frequency and bandwidth as in
experiment 1. In this link the reception of the two channels was performed by
a single receiver at the end of an interferometer made with two $16$ dBi Yagi-
Uda antennas. The selection between the two channels - twisted and untwisted -
was obtained by dephasing the interferometer at the receiving end through the
sliding one of the two receiving antennas, as in Ref. [21]. The averaged
channel separation measured with the digital channel was $22.4$ dB with $1.1$
std during the exhibition in $2015$. As for experiment 1, experiment 2 then
moved in another room with similar conditions for the periodic tests in the
following years. The results are reported in Tab I.

\-

Figure 2: Schematic description of the experimental setup 2 in the Infosphere of Globale at ZKM. The two high definition video channels were transmitted simultaneously in the same $2.414$ GHz frequency band with the same horizontal polarization state. At the receiving end, at a distance of $\sim 64\lambda$ from the transmitting antennas the instantaneous EM fields of the $m=0$ and $m=1$ modes resulted in phase on one side of the central axis (azimuthal angle $\varphi=0$) and in anti-phase at the opposite side ($\varphi=\pi$). Therefore, the standard linear-momentum interferometer received the $m=1$ beam when there was a delay of $\lambda/2$ in the arms of the interferometer and the $m=0$ channel when the phase delay was null.  TABLE I: Channel separation of experiment 1 and experiment 2 (dB) with their standard deviations and duration (months) $24$ hours per day. Year | exp. 1 | std | exp. 2 | std | duration
---|---|---|---|---|---
| dB | std | dB | std | months
2015 | 32.3 | 0.8 | 22.4 | 1.1 | 5
2016 | 32.4 | 1.2 | 21.9 | 1.2 | 2
2017 | 32.1 | 0.7 | 22.0 | 1.0 | 2
2018 | 31.9 | 0.8 | 21.9 | 1.7 | 2
2019 | 31.7 | 0.7 | 21.8 | 1.1 | 2

The experimental results reported here show that a multiple-channel
transmission using OAM states is compatible and robust with respect to
reflections, interference from a beam with different OAM value and does not
need error correction code protocols like those used in digital channels. More
interestingly, the advantages offered by the use of digital multiplexing
techniques, even those based on phase-coding such as phase shift keying (PSK),
are preserved and can be used to further improve the channel capacity also in
these complicated environmental situations as shown in Ref. [28, 40, 41]. The
channel separation values measured for the experiments 1 and 2 are drawn in
Fig. 3 for the acquisition in $2015$ and in Tab. I are reported the data of
the acquisitions taken in all the five years of tests.

\-

Figure 3: Result of the tests of channel separation with the digital channel
of both links, experiment 1 and experiment 2 of the data acquisition in $2105$
during the Globale-Digitale exhibition. We notice that in the correspondence
of conferences and events in the Foyer and in the Infosphere the channel
separation were reaching the lower limit of the $20$ dB needed to transmit the
two channels in the same frequency band.

###  II-A Testing OAM multiplexing schemes with QPSK digital protocol

Here we report the results of additional tests outdoor, in an urban setting,
where both the twisted and untwisted channels experience controlled
reflections and interferences with other disturbing signals that may overlap
their frequency band. The OAM beam was emitted by the twisted parabolic
antenna, the untwisted mode was emitted by a $16$ dBi Yagi-Uda antenna. At the
receiver’s side we did not use two independent interferometers as in
experiment 1 because of logistic reasons. A single interferometer realized
with the same $16$ dBi backfire antennas of experiment 1 was set up at the
receiver’s side and the tuning between the twisted and untwisted channels was
realized with one of the phase tuners used in experiment 1. We superimposed on
the same $2.414$ GHz frequency carrier the two collinear, horizontally-
polarized radio beams, one twisted ($m=1$) and one untwisted ($m=0$). As OAM
radio links are compatible and robust with respect to digital multiplexing
techniques, also with those based on phase coding such as the phase shift
keying (PSK) coding protocol and QPSK, with very high coding rates [28, 40,
41, 28], we used a quadrature phase shift coding (QPSK) DVB-S digital signal
to measure with high precision the channel transmission capacity and channel
insulation capacity of the twisted channel with respect to the untwisted
channel. QPSK is a digital phase encoding technique used in many
telecommunications applications today, belonging to the class of PSK
protocols. It employs, at any given time, four different phase states
$\\{01,11,10,00\\}$ for the carrier. These four phase states correspond to
$\\{0,90,180,270\\}$ degrees of relative phase shifts, respectively. For each
time period, the phase can change once with constant amplitude. In this way,
two bits of information are transmitted within each time slot. This is the
procedure used in our periodical tests for experiment 1 and experiment 2: the
untwisted beam, transmitting at the same frequency and power as the twisted
one, acts at all effects as a disturbing jamming signal.

Figure 4: (Left: ) Schematic description of the experimental setup for the
channel test with digital protocols. The $m=1$ OAM beam carried a digital QPSK
signal operating at a carrier frequency of $2.414$ GHz. The untwisted ($m=0$)
signal at the same carrier frequency and with the same polarization was
emitted by a standard Yagi-Uda antenna, fed by an FM $27$MHz wide band signal.
An interferometer at the receiver’s end equipped with a phase adjuster device
was used to discriminate and test the two channels and the signal is
downconverted to $1514$ MHz after being mixed with a local oscillator at $900$
MHz. Then the signal is divided and distributed to $3$ different paths: a
digital receiver for QPSK, a channel analyzer and an analog FM receiver. In
the worst conditions, because of the reflections, the modulation error ratio
(MER) of the OAM $m=1$ QPSK channel superimposed to the $m=0$ one was $12$ dB
and $20$ dB when the disturbing signal was switched off. The best value
obtained without reflections was $50$ dB.

As shown and described in Fig. 4, we probed the phases of the fields of the
received signals at such points in a standard phase interferometric manner,
using the two identical backfire antennas of experiment 1. The transmitted
$m=1$ signal was encoded with a QPSK DVB-S protocol. The digital transmitter
used for the twisted ($m=1$) mode, a Microwave Link QPSK DVB-S transmitter for
the $2.4$ GHz band was tuned to the usual frequency of $2.414$ GHz (free-space
wavelength $\lambda=12.42$ cm) and transmitted encoded video images at a rate
of $11.5$ Megasymbols/s. We used a forward error correction of the FEC=$3/4$
type, i.e., after three bits transmitted, a fourth bit is added. The $m=0$
untwisted channel used instead an analogue frequency modulation (FM)
transmission, centered at $2.414$ GHz and $27$ MHz bandwidth, with $200$ mW
maximum power and horizontal polarization. The transmitting antenna was one of
the $16$ dBi Yagi-Uda antennas used in the previous experiments. We equalized
the transmission channels by varying the output power of the two channels by
inserting signal attenuators in the transmission line in order to obtain the
same power at the receiving end for both the twisted and untwisted channels.
At the receiver’s side, distant $80$ m ($644\lambda$) from the transmitters,
the signals were collected by an interferometer realized with the two $16$ dBi
backfire antennas connected together through a signal combiner/splitter with a
baseline of $2.6$ metres ($\sim 21\lambda$). Both the antennas were placed
about $1.5$ m ($\sim 12\lambda$) above the reflecting ground. As shown in Fig.
4, to select one of the two channels (twisted and untwisted), a phase tuner
was placed in one of the interferometer arms. The phase tuner was realized
with a silver slit waveguide coupled to a Selenia signal circulator. The
received signal was routed to a signal analyzer to draw the frequency spectrum
and intensity of the channels with a spectrum analyzer Hameg HMS3010 or to a
3-way power splitter where, after a down-conversion in frequency with a local
oscillator (LO) of $900$ MHz to $1.514$ GHz was then split up into three
different receiver chains where audiovideo channels were received, analyzed,
and displayed in the monitors: the digital DVB-S digital chain with the QPSK
constellation tester $(m=1)$, the scientific chain for measuring and testing
in real time the analog ($m=0$) and digital ($m=1$) channels. Here the test-
signal was down-converted to $48.25$ MHz and fed into one of the branches of
the scientific channel to measure the received power level. The analogue
frequency modulation (FM) chain of the untwisted channel. In the latter chain
we inserted a DC block that prevented the flow of audio and direct current
(DC) frequencies and minimized the interference with the RF signals $(m=0)$.
The analyzer used was based on a Nokia-Hirschmann Receiver Analog Type HIT CSR
$501$ satellite receiver. In this chain, the spectrum analyzer Hameg HMS3010
was inserted to measure the separation of the two channels, the signal levels,
and the channel bandwidths.

The modulation error ratio (MER) of the QPSK channel alone was larger than
$20$ dB, with a bit error rate (BER) of $10^{-8}$ and a carrier-to-noise ratio
C/N $>15$ dB, including the effect of the reflections mainly from the ground.
The average background noise power in a $200$ MHz bandwidth was $-93$ dBm,
peaking at $-85.6$ dBm at the centre frequency. The power of the received FM
signal was measured by inserting the spectrum analyzer in the reception line
and was found to be $-68.95$ dBm in a $27$ MHz wide transmission band. When
the FM analogue transmitter signal with the same carrier frequency was
switched on, the ensuing interference on the digital twisted signal was quite
small and caused the MER to vary from $9$ to $11$ dB, the BER from $10^{-3}$
to $10^{-5}$, and the C/N from $9$ to $12$ dB indicating that OAM states
improve with their stable intrinsic phase properties future multiport schemes
also in the presence of strong reflections. In the absence of reflections we
measured a channel separation of $50$ dB.

##  III Conclusions

Radio transmissions with twisted waves carrying OAM are a robust tool for
multiplexing schemes. Twisted waves are natural eigenmodes of the
electromagnetic field that preserve in time their phase spatial profile when
propagating in free space, as shown with the detection of twisted waves from
the M87* black hole, located $56$ million light years far away from Earth
[35]. We set up two indoor experiments working for long periods in time, the
first run in $2015$ lasted for $5$ months, $24$ hours per day, and $2$ months
per year, $24$ hours per day, in the following years until $2019$ to simulate
real working conditions for future OAM links indoor where multiple reflections
are present. We periodically measured the separation between the two channels
with digital channel techniques: experiment 1 had an averaged channel
separation value of $32.1$ dB and experiment 2 $22.0$ dB. Additional outdoor
tests with controlled reflections were performed with a $4$-QAM digital
transmission in the twisted channel. We obtained a maximum channel separation
value of $50$ dB when the conditions of phase orthogonality were fulfilled and
no reflections were present. The results obtained clearly show that OAM can be
used to improve the transmission capacity of common-use devices, and re-use
efficiently the same frequency band, ensuring a good quality and stable
channel separation for a long time.

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Focus Entertainment<EMAIL_ADDRESS>Focus Entertainment,
<EMAIL_ADDRESS>Nicolas Grelier, Stéphane Kaufmann
<ccs2012> <concept> <concept_id>10010405.10010476.10010477</concept_id>
<concept_desc>Applied computing Publishing</concept_desc>
<concept_significance>500</concept_significance> </concept> </ccs2012>
[500]Applied computing Publishing John Q. Open and Joan R. Access 2 42nd
Conference on Very Important Topics (CVIT 2016) CVIT 2016 CVIT 2016 December
24–27, 2016 Little Whinging, United Kingdom 42 23

# Automated clustering of video games into groups with distinctive names

Nicolas Grelier  Stéphane Kaufmann

###### Abstract

When doing a study on a large number of video games, it may be difficult to
cluster them into coherent groups to better study them. In this paper, we
introduce a novel algorithm, that takes as input any set of games $S$ that are
released on Steam and an integer $k$, and cluster $S$ into $k$ groups. Each
group is then assigned a distinctive name in the form of a Steam tag. We
believe our tool to be valuable for gaining deeper insights into the video
game market. We show that our algorithm maximises an objective function that
we introduce, the naming score, which assesses the quality of a clustering and
how distinctive its name is.

###### keywords:

Steam tags, naming score, Cohen’s h

###### category:

##  1 Introduction

In this paper, we are interested in understanding better the video game
market. Let us imagine a scenario where Alice, who could be a researcher, a
game developer, or working in a publishing company, faces a large number of
games. She might be looking for relevant games for an academic study, or
trying to find games similar to the one she is currently developing, to serve
as a benchmark. The set of games could be all video games, or could be more
specific: for instance all _Racing_ games, or all _Puzzle_ games released
since $2018$ that have at least $1000$ reviews on the video game platform
Steam.

Obviously, even as an expert, Alice only played a few of the games in the
large set. She might have never heard of most of the games, especially those
that did not sell well. Still, she wants to better understand this set. Maybe
there are some games that are actually not relevant for her purposes, that she
could then dismiss. Maybe she can learn that these games, which she thought
were all similar, can actually be split into several coherent groups of games.
This information may help Alice to do market segmentation [6, 15], since
knowing the types of games that exist within a segment helps in understanding
better the types of players. In this paper, we provide an algorithm for
automatically splitting the games into groups with distinctive name, whatever
the set of games with which Alice started.

To the best of our knowledge, there do not exist clustering algorithms for
this specific task. One issue with video games clustering is that there do not
seem to be clear ways of splitting games into groups, such that each game
would clearly belong to a unique group, even when using genres [1, 10]. This
is what one hopes to do with classical clustering, but we believe that for
video games, one has to relax this desired property that a game should be as
different as possible from the games in the other clusters. Thus, we introduce
a new clustering objective. Our objective is to find an algorithm that could
split a set of games into groups, and assign a name to each group. All games
within a group should relate to its name. Moreover, the name should be as
distinctive as possible, so that it gives information about the group. When
facing different options, meaning that the names of several clusters relate to
one game, we are fine with any arbitrary choice: We do not require that a game
$G$ put into cluster $C$ would have nothing to do with the name of cluster
$C^{\prime}$. We only require that the name of cluster $C$ applies to $G$. In
other words, we aim at high intra-cluster similarity, but lowering the inter-
cluster similarity matters significantly less to us (we still want the name of
a cluster to be distinctive enough from the rest of the games). Note that we
do not require cluster names to be genres.

We want an algorithm that could partition any set of games into groups with
distinctive names, using an explainable method, and that would reflect how
players think. In this paper, we rely on data from the video game platform
Steam, obtained through SteamSpy’s API. On Steam, everyone can assign tags to
games. As of November 2023, there are $448$ tags assigned to the $66908$ games
available on Steam. Among the tags, one can find genre tags like _Adventure_
or _RTS_ , and many other tags like _2D_ , _Funny_ , _Cats_ or _Medieval_.
SteamSpy provides a database with for each game the twenty most assigned tags
to it, along with how many players assigned those tags. We use those values
for the clustering algorithms we test in this paper, and our cluster names are
tags.

###  1.1 Related work

To the best of our knowledge, this is the first study about the problem of
clustering games into groups with distinctive names, where the name of a
cluster should apply to all the games it contains, but allowing that a game
might have been placed in another cluster. Previous works about clustering
games include clustering by genres through survey [10], by game traits through
survey [7], by players behaviours [3], based on how algorithms play games [2,
4], and based on characteristics for video streaming [17]. Note that a major
difference between these works and ours is that we want a program that could
then be applied to cluster any set of games, whereas previous works aim at
clustering all video games for good.

###  1.2 Our contributions

We introduce in Section 2 a new function to assess the quality of a clustering
of video games. To maximise this function, the clustering must first have high
intra-cluster similarity, and with less importance have low inter-cluster
similarity, such that it is possible to give a name to each cluster of games
that is distinctive from the other games. We call this function the _naming
score_ of a clustering.

We provide in Section 3 a clustering algorithm that has significantly larger
naming scores on several testing sets than simply applying a K-means
algorithm, with and without Principal Component Analysis (PCA). Section 4
presents the results of our clustering algorithm applied to the testing sets:
We provide the names of each cluster along with the titles of the five most
reviewed games of that cluster.

##  2 The naming score of a clustering

We are given a set of games $S$ that we want to partition into $k$ clusters.
Each cluster should have a distinctive name. Our idea is that we should be
able to assign a tag $t$ to each cluster $C$, such that $t$ is assigned to
most games in $C$ and to few games in $S\setminus C$.

It is important to note a distinction between our clustering objective, and
what is generally desired when doing clustering of some data points. Usually,
it is wanted that a point should belong well to its cluster, and should be
different from the points in the other clusters. This leads for instance to
the well-known silhouette method to determine the number of clusters [14].
However, in our case, it is not an issue that a game might have been put in
another cluster. We just want that the name of its cluster does strongly
relate to the game. This is why we introduce a new objective function.

###  2.1 Intuitive definition

Let us consider one cluster $C$ of a clustering of a set of games $S$. We have
access to the tags that are assigned by players to these games. Our idea is to
find the most over-represented tag $t$ among those, and to name $C$ as the
cluster of games that have the tag $t$. However, it is a priori not clear how
one can say whether a tag is over-represented.

For this purpose, we use Cohen’s h [5]: a measure of difference between two
proportions, in this case the proportion of games in cluster $C$ that have the
tag $t$ compared to the proportion of games in the set $S$ that have the tag
$t$. The exact formula for computing Cohen’s h is given below. The reason why
we use Cohen’s h instead of, for instance, a simple ratio between the two
proportions is as follows: Imagine there is a tag $t$ that is assigned to
$0.001\%$ of the games in $S$, and to $0.1\%$ of the games in $C$. The
increase in proportion is a multiplicative factor of $100$, but still the
proportion of games in $C$ that have $t$ is only $0.1\%$. Thus, it would be
inappropriate to name $C$ as “the group of games that have the tag $t$”.
Cohen’s h solves this issue by only giving a large value when the new
proportion is both larger than the previous one, and the new proportion is in
itself large.

With Cohen’s h, we have a method of assessing whether a tag is over-
represented in a group of games $C$. Therefore, we name $C$ as the “the group
of games that have the tag $t$”, where $t$ is the tag with highest Cohen’s h
value. We define the naming score of $C$ as the Cohen’s h value of this tag
$t$. Finally, we define the naming score of a clustering of $S$ as the
weighted average naming score over all clusters, where the weight of a cluster
$C$ is $|C|/|S|$. We do weighted average, for otherwise one could obtain a
clustering with an extremely high naming score by putting nearly all games in
one cluster (which would have a naming score close to $0$) and only one game
in each other cluster (with a naming score close to $\pi$, i.e. the highest
value Cohen’s h may take). However, we want clusters to be roughly of the same
size. By taking a weighted average, our naming score function may be large
only if there is no large cluster. We give below a formal definition of this
naming score.

###  2.2 Formal definition

Let us consider a set $S$ of video games and an integer $k$. The problem
consists in partitioning $S$ into $k$ subsets, such that for each subset $C$
in $S$, there exists a tag $t$ that is distinctive to $C$. By “distinctive”,
we mean that $t$ is assigned to almost all games in $C$, and to few games in
$S\setminus C$.

Let $C$ be a subset of video games in $S$ (potentially $S$ itself). We denote
by $T_{C}$ the set of tags that are assigned to at least one game in $C$. For
a tag $t$ in $T_{C}$, we denote by $p(t,C)$ the proportion of games in $C$ to
which $t$ is assigned. We denote by $h(t,C)$ Cohen’s h applied to the
proportions $p(t,C)$ and $p(t,S)$, i.e. $h(t,C)$ is a measure of difference
between the proportion of games in $C$ that have the tag $t$, and the
proportion of games in $S$ that have the tag $t$. It is defined as
$h(t,C):=\phi(p(t,C))-\phi(p(t,S))$, where $\phi(x)$ equals
$2\arcsin{\sqrt{x}}$ [5]. Observe that the maximal value of Cohen’s h is
$\pi$. A greater Cohen’s h value indicates a greater over-representation of
the tag $t$ among the games in $C$ compared to the games in $S$.

We define the naming score $n(C)$ of a subset $C$ of $S$ as $n(C):=\max_{t\in
T_{C}}h(t,C)$. Let $\mathcal{C}=\\{C_{i}\\}_{1\leq i\leq k}$ be a partition of
$S$. We make an abuse of notation and denote by $n(\mathcal{C})$ the naming
score of $\mathcal{C}$, which we define as
$n(\mathcal{C}):=\frac{1}{|S|}\sum_{1\leq i\leq k}n(C_{i})\cdot|C_{i}|$.

##  3 Finding the best clustering algorithm

In this paper, we use the classical K-means algorithm form the scikit-learn
Python library to do the clusterings. We show that a modification of the Steam
database (selecting only a few tags, weighting tags differently) significantly
improves the naming score. We show step by step how we were able to improve
the naming score by doing one modification after another. To speed up the
running time of the K-means program, we only consider the $4065$ games that
have at least $2000$ reviews on Steam.

###  3.1 Priority helps

In [8], we introduced the notion of priority of a steam tag. For a game $G$,
where the most assigned tag $t_{\max}$ was assigned by $n_{\max}$ players, the
priority of a tag $t$ that was assigned to $G$ by $n$ players is defined as
$n/n_{\max}$. If $t$ is not assigned to $G$, then its priority is $0$. We
argued that the priority is a measure of essentiality of tags to describe
games. For a game $G$, tags with high priority are essential to describe $G$,
and tags with low priority give information of less importance.

In this subsection, we show that priority helps to cluster games into groups
with distinctive names. We compare two databases. In both, there is one entry
per game, and as many columns as there are Steam tags. In the first database
$\mathcal{D}_{1}$, there is a $1$ in the column of the tag $t$ for the game
$G$ if the tag $t$ is assigned to $G$, and a $0$ otherwise. In the second
database $\mathcal{D}_{2}$, the value at column $t$ for entry $G$ is the
priority of the tag $t$ with respect to the game $G$.

Figure 1: Comparison of the naming scores of the clusterings depending on
whether the priority is taken into account.

We compare the results of the K-means algorithm from the scikit-learn Python
library applied to the two databases $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.
Surprisingly, doing a Principal Component Analysis (PCA) [13] only lessened
the naming scores. Thus, we apply the K-means algorithm directly to the
databases for the rest of this paper. Figure 1 indicates that the naming
scores of the clusterings are always larger using $\mathcal{D}_{2}$ than when
using $\mathcal{D}_{1}$. Therefore, throughout the rest of the paper, we keep
using the database $\mathcal{D}_{2}$.

###  3.2 Restricting the set of tags

We have the intuition that the information contained in the Steam tags can be
summarised using only a few tags. Our working hypothesis is as follows:

######  Working Hypothesis .

Having too many tags for the clustering brings noise. There exists a small set
of tags that summarise all the important information contained in the Steam
tags, which leads to better clusterings.

This study was started in [9], where we presented a set of seven tags, the
capital tags, that encompass all others. Those tags are _Singleplayer_ ,
_Multiplayer_ , _Action_ , _Casual_ , _Adventure_ , _Strategy_ and _Anime_.
Those tags where obtained by considering the tags that were correlated with
many other tags, and that are assigned the most frequently. In [9], we say
that a group of tags $\mathcal{T}$ _covers_ a set of games $S$ if for each
game $G$ in $S$ there is at least one tag in $\mathcal{T}$ assigned to $G$. It
was observed that the capital tags cover $94\%$ of all the games on Steam.

We iterate this process, by first removing the capital tags from the database.
We refer now to those capital tags as being of rank $1$. To obtain the capital
tags of rank $2$, we did an exhaustive search to find how to cover as many
games as possible while using only a few tags. The number of games was chosen
empirically, when we deemed that adding a new tag would not increase
significantly the number of games covered. After one iteration of this
process, we obtained the capital tags of rank $2$: _RPG_ , _2D_ , _3D_ ,
_Atmospheric_ , _Simulation_ , _Colorful_ and _Puzzle_. After a second
iteration, we obtained the capital tags of rank $3$: _Pixel Graphics_ ,
_Funny_ , _Story Rich_ , _Fantasy_ , _Arcade_ , _Relaxing_ , _Shooter_ ,
_Management_ , _Horror_ , _Sci-fi_ , _Platformer_ , _Co-op_ , _Third Person_ ,
_Open World_ , _Rogue-like_ , _Exploration_ and _Sports_. We observe that out
of the $4065$ games with at least $2000$ reviews, $98\%$ have a capital tag of
rank $1$, $96\%$ have a capital tag of rank $2$, and $97\%$ have a capital tag
of rank $3$. Moreover, $99.7\%$ have at least one capital tag (of any rank).

Figure 2: Comparison of the naming scores of the clusterings depending on
whether they are computed on all tags or only on the capital tags.

Figure 2 shows the naming scores when the K-means algorithm is done only on
the capital tags of rank $1$, $2$ and $3$, compared to the previous results.
Note that when we compute the naming score of a clustering, we do it on all
Steam tags, not only on the capital tags. There are $14$ games out of the
$4065$ that do not have any capital tags, and thus were not put into any
cluster. Since we are missing only about $0.3\%$ of the considered games, this
difference is not significant. We observe that the naming scores are quite
close, although they are generally better when using only the capital tags.

###  3.3 Using the ranks of the capital tags

We have seen in Figure 2 that, for clustering the games into groups with
distinctive names, using only the capital tags does not deteriorate the
results. However, in Working Hypothesis 3.2, we stated the belief that capital
tags do not only preserve the quality of the naming, but even improve it. We
had the following intuition:

######  Working Hypothesis .

To obtain the best naming scores, capital tags should be treated differently
according to their ranks.

Let us consider a game $G$. As said in Subsection 3.2, $G$ is very likely to
have at least one capital tag of each rank. Let us consider only the capital
tags of rank $i$ assigned to $G$. We tried several functions to apply to the
priorities of the tags, we present here the one that gave the best naming
scores. If $G$ has no capital tag of rank $i$, we do nothing. Otherwise, we
can associate $G$ to a vector in $[0,1]^{d_{i}}$, where $d_{i}$ is the number
of capital tags of rank $i$. The coordinate along axis $j$ is the priority of
the capital tag $j$ of rank $i$ assigned to $G$. First, we normalise each
vector by its $\ell_{2}$ norm. Secondly, we multiply it by a scaling factor
denoted as $\lambda_{i}$, where $\lambda_{1}=0.25$, $\lambda_{2}=0.7$ and
$\lambda_{3}=1$. Those values are the ones that maximised the naming scores in
our experimentation.

Figure 3: Comparison of the naming scores of the clusterings depending on
whether they are computed on all tags, only on the capital tags or only on the
normalised and weighted capital tags.

Figure 3 shows a significant improvement of this new solution compared to the
previous ones. This algorithm, using the database with priority and only
keeping the normalised and weighted capital tags, is the algorithm that
maximises the most the naming score among all the methods we tried for this
set of games.

###  3.4 Testing on other sets

Figure 4: Comparisons of the three methods on the $2791$ games that have the
tag _Racing_. Figure 5: Comparisons of the three methods on the $12185$ games
that have the tag _Puzzle_. Figure 6: Comparisons of the three methods on the
$2961$ games that have the tag _Rogue-like_.

We tested the three methods of using the K-means algorithm, that is i) using
all Steam tags, ii) using only the capital tags and iii) using the normalised
and weighted capital tags, on other sets of games. Indeed, we want a program
that would work well for all sets. We observe that Option iii) maximises the
naming scores. Figures 4, 5 and 6 depict the naming scores for the three
methods on the sets of _Racing_ games, _Puzzle_ games and _Rogue-like_ games.
We chose these sets of games as they are sufficiently large so that one would
need a program and could not do a clustering by hand, not too large since we
do not want the running times of the programs to be too long, and finally
well-known because we provide in Section 4 the results of our best algorithm
applied to these sets. As we want the reader to be able to compare our
clusterings with their intuition, we needed set of games that are sufficiently
well-known.

##  4 Some applications

In this section, we apply our best method, i.e. using the normalised and
weighted capital tags, to some concrete examples, and present the cluster
names. We also give for each cluster the titles of the five most reviewed
games in that cluster (so that it is likely that the reader is familiar with
them). As we have presented in Section 3 the curves of the naming scores with
respect to the number of clusters, we can use the elbow method to choose an
adequate number of clusters [11, 12, 16]. The elbow method is not perfectly
reliable, but it is sufficient for our purposes, since we just want to show
that the cluster games and names are coherent.

For the clustering of the $4051$ games that have at least $2000$ reviews and
at least one capital tag, we choose $8$ clusters. We give the list of clusters
with their distinctive tag, along with their naming scores and the five most
reviewed games in that clusters.

  * •

_Sports_ ($2.70$): Rocket League, EA SPORTS FIFA 23, FIFA 22, Assetto Corsa,
TEKKEN 7.

  * •

_Anime_ ($2.36$): Wallpaper Engine, NARAKA: BLADEPOINT, VRChat, Doki Doki
Literature Club!, Mirror 2: Project X.

  * •

_Open World_ ($1.98$): Grand Theft Auto V, Rust, ELDEN RING, Cyberpunk 2077,
ARK: Survival Evolved.

  * •

_Simulation_ ($1.67$): Euro Truck Simulator 2, War Thunder, Sid Meier’s
Civilization VI, Cities: Skylines, RimWorld.

  * •

_RPG_ ($1.40$): The Witcher 3: Wild Hunt, Warframe, Dying Light, Baldur’s Gate
3, The Elder Scrolls V: Skyrim.

  * •

_2D_ ($1.14$): Terraria, Stardew Valley, Hollow Knight, Geometry Dash, The
Binding of Isaac: Rebirth.

  * •

_Atmospheric_ ($1.11$): Dead by Daylight, Portal 2, Sekiro: Shadows Die Twice,
Deep Rock Galactic, Tomb Raider.

  * •

_Multiplayer_ ($1.01$): Counter-Strike: Global Offensive, PUBG: BATTLEGROUNDS,
Dota 2, Tom Clancy’s Rainbow Six Siege, Garry’s Mod.

There are no real surprises in those lists, except maybe for Wallpaper Engine
and VRChat being in the _Anime_ cluster. This is not a mistake from the
program, since _Anime_ has indeed a high priority in both of these games.
Thus, this result truly corresponds to how the Steam community perceives these
games. We use $6$ clusters for the $2091$ games with the tag _Racing_ that
have at least one capital tag:

  * •

_Open World_ ($2.39$): Grand Theft Auto V, BeamNG.drive, Forza Horizon 4,
Forza Horizon 5, Need for Speed Heat.

  * •

_Third Person_ ($2.25$): Sonic Adventure 2, MX Bikes, Sonic Adventure DX, Road
Redemption, Mon Bazou.

  * •

_Arcade_ ($1.87$): TrackMania Nations Forever, FlatOut 2, Retrowave, Garfield
Kart - Furious Racing, AudioSurf.

  * •

_Sports_ ($1.85$): Rocket League, F1 2021, SpeedRunners, Need For Speed: Hot
Pursuit, Sonic & All-Stars Racing Transformed Collection.

  * •

_Simulation_ ($1.73$): Assetto Corsa, CarX Drift Racing Online, F1 2020, DiRT
Rally, DiRT Rally 2.0.

  * •

_Indie_ ($0.23$): Stumble Guys, Crossout, Transformice, Notes of Soul, Car
Mechanic Simulator 2015.

We have $8$ clusters for the $12185$ games with the tag _Puzzle_ (that
necessarily have at least one capital tag since _Puzzle_ is a capital tag of
rank $2$):

  * •

_Platformer_ ($2.42$): Ori and the Will of the Wisps, BattleBlock Theater,
GRIS, Ori and the Blind Forest: Definitive Edition, Ori and the Blind Forest.

  * •

_Pixel Graphics_ ($2.36$): Papers, Please, The Escapists 2, CARRION, There Is
No Game: Wrong Dimension, Baba Is You.

  * •

_Horror_ ($2.34$): Resident Evil 2, Inscryption, Little Nightmares, Resident
Evil 7 Biohazard, DEVOUR.

  * •

_Exploration_ ($2.19$): Stray, Outer Wilds, Inside the Backrooms, SOMA, Atomic
Heart.

  * •

_Story Rich_ ($2.17$): Portal 2, Tomb Raider, Half-Life 2, Portal, Batman:
Arkham Knight.

  * •

_Relaxing_ ($2.14$): PC Building Simulator, Dorfromantik, Unpacking, Mini
Motorways, ISLANDERS.

  * •

_Adventure_ ($0.52$): Hogwarts Legacy, STAR WARS Jedi: Fallen Order, Rise of
the Tomb Raider, The Stanley Parable, Mirror’s Edge.

  * •

_Casual_ ($0.19$): Human Fall Flat, It Takes Two, Helltaker, Plants vs.
Zombies GOTY Edition, Mirror.

We choose $6$ clusters for the $2961$ games with the tag _Rogue-like_ (that
necessarily have at least one capital tag since _Rogue-like_ is a capital tag
of rank $3$). We observe that there are two clusters that share the same tag
with highest Cohen’s h: _Pixel Graphics_. This is not a contradiction to our
objective, since we are fine with having games relating to more than one
cluster. However, if one wants to better understand the differences between
the two clusters, it is possible to consider for each cluster the two tags
with highest Cohen’s h, instead of only the highest one. In this case, one
cluster would be named _Pixel Graphics_ -_2D_ and the other _Pixel Graphics_
-_RPG_. We discuss this further in Section 5.

  * •

_Arcade_ ($2.30$): Paint the Town Red, Crab Champions, Soulstone Survivors,
Rogue Legacy 2, Nova Drift.

  * •

_Pixel Graphics_ -_RPG_ ($1.83$): Noita, Realm of the Mad God Exalt, Hero
Siege, Loop Hero, Stoneshard.

  * •

_Pixel Graphics_ -_2D_ ($1.82$): The Binding of Isaac: Rebirth, Vampire
Survivors, Dead Cells, Enter the Gungeon, Skul: The Hero Slayer.

  * •

_RPG_ ($1.66$): Hades, Slay the Spire, Darkest Dungeon, The Scroll Of Taiwu,
The Binding of Isaac.

  * •

_2D_ ($1.52$): FTL: Faster Than Light, Rogue Legacy, Spelunky 2, Spelunky,
Have a Nice Death.

  * •

_3D_ ($0.60$): Risk of Rain 2, Don’t Starve, Inscryption, Gunfire Reborn, Lobotomy Corporation | Monster Management Simulation.

There are issues with the last cluster, _3D_ , which contains games that do
not use 3D game graphics. This can be partly understood by the fact that its
naming score is quite low, $0.60$. Therefore, our clustering algorithm failed
in this case to have all clusters with a distinctive name. However, it is
unclear whether it is truly possible to do better when clustering the _Rogue-
like_ games into $6$ clusters.

##  5 Conclusion

We have introduced a new optimisation function for clustering games into
groups with distinctive names. We showed that using the priority notion on
normalised and weighted capital tags improves the naming score of the
clusterings when using the K-means algorithm. We also provided a few example
of clusterings that are coherent. We noticed that two clusters may share the
same tag with highest Cohen’s h. It could be interesting not only to consider
the tag with highest Cohen’s h, but for instance the three best tags. The name
of a cluster would then not consist in one tag but in the three tags with
highest Cohen’s h. We conjecture that our algorithm would still provide better
naming scores than the other algorithms we tested in this paper, but we leave
this question to future work. Our clustering algorithm works only on games
that have at least one capital tag. As $99.7\%$ of the games with at least
$2000$ reviews have at least one capital tag, this is not too problematic.
Still, it would be interesting to try to guess for the remaining games what
their capital tags could have been. We believe this could be done using
traditional machine learning methods.

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##  Appendix A Tables corresponding to the figures of the paper

number of clusters | without priority | all tags | capital tags | normalised and weighted capital tags
---|---|---|---|---
1 | 0 | 0 | 0 | 0
2 | 0.59 | 0.61 | 0.54 | 0.66
3 | 0.59 | 0.8 | 0.86 | 0.93
4 | 0.95 | 1.02 | 0.95 | 1.09
5 | 0.98 | 0.98 | 1.15 | 1.3
6 | 1.02 | 1.06 | 1.07 | 1.35
7 | 1.11 | 1.25 | 1.36 | 1.37
8 | 1.09 | 1.3 | 1.35 | 1.48
9 | 1.24 | 1.28 | 1.47 | 1.51
10 | 1.33 | 1.34 | 1.47 | 1.62
11 | 1.36 | 1.46 | 1.48 | 1.7
12 | 1.37 | 1.51 | 1.55 | 1.78
13 | 1.41 | 1.49 | 1.62 | 1.84
14 | 1.38 | 1.66 | 1.61 | 1.87
15 | 1.48 | 1.56 | 1.65 | 1.93
16 | 1.54 | 1.65 | 1.71 | 1.96
17 | 1.56 | 1.67 | 1.73 | 1.93
18 | 1.5 | 1.69 | 1.71 | 2.03
19 | 1.52 | 1.74 | 1.71 | 2.11
20 | 1.6 | 1.76 | 1.72 | 2.1

Table 1: Naming scores of the algorithms presented in Section 3 applied to the $4065$ games with at least $2000$ reviews. number of clusters | all tags | capital tags | normalised and weighted capital tags
---|---|---|---
1 | 0 | 0 | 0
2 | 0.41 | 0.69 | 0.69
3 | 0.68 | 0.93 | 1.13
4 | 0.77 | 1.12 | 1.24
5 | 0.89 | 1.11 | 1.25
6 | 0.86 | 1.17 | 1.43
7 | 0.89 | 1.33 | 1.44
8 | 1.02 | 1.4 | 1.42
9 | 1.13 | 1.42 | 1.57
10 | 1.15 | 1.51 | 1.59
11 | 1.25 | 1.49 | 1.58
12 | 1.28 | 1.47 | 1.64
13 | 1.29 | 1.48 | 1.73
14 | 1.32 | 1.54 | 1.73
15 | 1.33 | 1.52 | 1.73
16 | 1.37 | 1.53 | 1.74
17 | 1.44 | 1.55 | 1.74
18 | 1.36 | 1.56 | 1.77
19 | 1.44 | 1.58 | 1.78
20 | 1.42 | 1.59 | 1.8
Table 2: Naming scores of the algorithms presented in Section 3 applied to the $2791$ games with the tag _Racing_. number of clusters | all tags | capital tags | normalised and weighted capital tags
---|---|---|---
1 | 0 | 0 | 0
2 | 0.67 | 0.71 | 0.53
3 | 0.6 | 0.71 | 0.77
4 | 0.7 | 1.08 | 1.02
5 | 0.93 | 1.12 | 1.18
6 | 1.02 | 1.34 | 1.34
7 | 1.16 | 1.34 | 1.44
8 | 1.21 | 1.41 | 1.55
9 | 1.22 | 1.45 | 1.6
10 | 1.35 | 1.44 | 1.68
11 | 1.39 | 1.49 | 1.66
12 | 1.36 | 1.51 | 1.66
13 | 1.48 | 1.53 | 1.72
14 | 1.51 | 1.56 | 1.78
15 | 1.57 | 1.64 | 1.83
16 | 1.58 | 1.65 | 1.88
17 | 1.56 | 1.65 | 1.88
18 | 1.53 | 1.7 | 1.88
19 | 1.57 | 1.71 | 1.88
20 | 1.58 | 1.71 | 1.88
Table 3: Naming scores of the algorithms presented in Section 3 applied to the $12185$ games with the tag _Puzzle_. number of clusters | all tags | capital tags | normalised and weighted capital tags
---|---|---|---
1 | 0 | 0 | 0
2 | 0.85 | 0.93 | 0.57
3 | 0.84 | 0.88 | 1.18
4 | 0.86 | 1.29 | 1.36
5 | 0.92 | 1.4 | 1.42
6 | 1.05 | 1.41 | 1.54
7 | 1.15 | 1.39 | 1.38
8 | 1.21 | 1.57 | 1.6
9 | 1.2 | 1.62 | 1.52
10 | 1.27 | 1.64 | 1.65
11 | 1.3 | 1.59 | 1.7
12 | 1.33 | 1.63 | 1.69
13 | 1.4 | 1.66 | 1.84
14 | 1.39 | 1.66 | 1.84
15 | 1.47 | 1.64 | 1.88
16 | 1.53 | 1.65 | 1.93
17 | 1.52 | 1.65 | 1.94
18 | 1.51 | 1.68 | 1.97
19 | 1.5 | 1.7 | 1.99
20 | 1.51 | 1.68 | 2
Table 4: Naming scores of the algorithms presented in Section 3 applied to the
$2961$ games with the tag _Rogue-like_.

Table 1 depicts the naming scores of the algorithms presented in Section 3
applied to the $4065$ games with at least $2000$ reviews. Likewise, Table 2
does the same for the $2791$ games with the tag _Racing_ , Table 3 for the
$12158$ games with the tag _Puzzle_ and Table 4 for the $2961$ games with the
tag _Rogue-like_.