Game Theory Meets Embeddings: a Unified Framework for Word Sense Disambiguation

Abstract

Game-theoretic models, thanks to their intrinsic ability to exploit contextual information, have shown to be particularly suited for the Word Sense Disambiguation task. They represent ambiguous words as the players of a non cooperative game and their senses as the strategies that the players can select in order to play the games. The interaction among the players is modeled with a weighted graph and the payoff as an embedding similarity function, that the players try to maximize. The impact of the word and sense embedding representations in the framework has been tested and analyzed extensively: experiments on standard benchmarks show state-of-art performances and different tests hint at the usefulness of using disambiguation to obtain contextualized word representations.

Full text

Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing
and the 9th International Joint Conference on Natural Language Processing, pages 88–99,
Hong Kong, China, November 3–7, 2019. c
2019 Association for Computational Linguistics
88
Game Theory Meets Embeddings:
a Unified Framework for Word Sense Disambiguation
Rocco Tripodi
Ca’ Foscari University of Venice
rocco.tripodi@unive.it
Roberto Navigli
Sapienza University of Rome
navigli@di.uniroma1.it
Abstract
Game-theoretic models, thanks to their intrin-
sic ability to exploit contextual information,
have shown to be particularly suited for the
Word Sense Disambiguation task. They rep-
resent ambiguous words as the players of a
non-cooperative game and their senses as the
strategies that the players can select in order
to play the games. The interaction among the
players is modeled with a weighted graph and
the payoff as an embedding similarity func-
tion, which the players try to maximize. The
impact of the word and sense embedding rep-
resentations in the framework was tested and
analyzed extensively: experiments on standard
benchmarks show state-of-art performances
and different tests hint at the usefulness of
using disambiguation to obtain contextualized
word representations.
1 Introduction
Word Sense Disambiguation (WSD), the task of
linking the appropriate meaning from a sense in-
ventory to words in a text, is an open problem in
Natural Language Processing (NLP). It is particu-
larly challenging because it deals with the seman-
tics of words and, by their very nature, words are
ambiguous and can be used with different mean-
ings in different situations. Among the key tasks
aimed at enabling Natural Language Understand-
ing (Navigli,2018), WSD provides a basic, solid
contribution since it is able to identify the intended
meaning of the words in a sentence (Kim et al.,
2010).
WSD can be seen as a classification task in
which words are the objects to be classified and
senses are the classes into which the objects have
to be classified (Navigli,2009); therefore it is
possible to use supervised learning techniques to
solve the WSD problem. One drawback with
this idea is that it requires large amounts of data
that are difficult to obtain. Furthermore, in the
WSD context, the production of annotated data
is even more complicated and excessively time-
consuming compared to other tasks. This arises
because of the variability in lexical use. Further-
more, the number of different meanings to be con-
sidered in a WSD task is in the order of thousands,
whereas classical classification tasks in machine
learning have considerably fewer classes.
We decided to adopt a semi-supervised ap-
proach to overcome the knowledge acquisition
bottleneck and innovate the strand of research in-
troduced by Tripodi and Pelillo (2017). These
researchers developed a flexible game-theoretic
WSD model that exploits word and sense simi-
larity information. This combination of features
allows the textual coherence to be maintained: in
fact, in this model the disambiguation process is
relational, and the sense assigned to a word must
always be compatible with the senses of the words
in the same text. It can be seen as a constraint
satisfaction model which aims to find the best
configuration of senses for the words in context.
This is possible because the payoff function of
the games is modeled in a way in which, when a
game is played between two players, they are em-
boldened to select the senses that have the highest
compatibility with the senses that the co-player is
choosing. Another appealing feature of this model
is that it offers the possibility to configure many
components of the system: it is possible to use
any word and sense representation; also, one can
model the interactions of the players in different
ways by exploiting word similarity information,
the syntactic structure of the sentence and the im-
portance provided by specific relations. Further-
more, it is possible to use different priors on the
sense distributions and to use different game dy-
namics to find the equilibrium state of the model.
Traditional WSD methods have only some of these

89
properties.
The main difference between our model and the
model proposed by Tripodi and Pelillo (2017) is
that they did not use state-of-the-art models for
word and sense representations. They used word
co-occurrence measures for word similarity and tf-
idf vectors for sense similarity, resulting in sparse
graphs in which nodes can be disjoint or some se-
mantic area is not covered. Instead, we are advo-
cating the use of dense vectors, which provide a
completely different perspective not only in terms
of representation but also in terms of dynamics.
Each player is involved in many more games and
this affects the computation of the payoffs and the
convergence of the system. The interaction among
the players is defined in a different way and the
priors are modeled with a more realistic distribu-
tion to avoid the skewness typical of word sense
distributions. Furthermore, our model is evaluated
on recent standard benchmarks, facilitating com-
parison with other models.
The main contributions of this paper are as fol-
lows:
1. the release of a general framework for WSD;
2. the evaluation of different word and sense
embeddings;
3. state-of-the-art performances on standard
benchmarks (in different cases performing
better than recent supervised models);
4. the use of disambiguated sense vectors to ob-
tain contextualized word representations.
2 Word Sense Disambiguation
WSD approaches can be divided into two main
categories: supervised, which require human
intervention in the creation of sense-annotated
datasets, and the so-called knowledge-based ap-
proaches (Navigli,2009), which require the con-
struction of a task-independent lexical-semantic
knowledge resource, but which, once that work
is available, use models that are completely au-
tonomous.
As regards supervised systems, a popular sys-
tem is It makes sense (Zhong and Ng,2010), a
model which takes advantage of standard WSD
features such as POS-tags, word co-occurrences,
and collocations and creates individual support
vector machine classifiers for each ambiguous
word. Newer supervised models exploit deep
neural networks and especially long short-term
memory (LSTM) networks, a type of recurrent
neural network particularly suitable for handling
arbitrary-length sequences. Yuan et al. (2016)
proposed a deep neural model trained with large
amounts of data obtained in a semi-supervised
fashion. This model was re-implemented by Le
et al. (2018), reaching comparable results with a
smaller training corpus. Raganato et al. (2017)
introduced two approaches for neural WSD us-
ing models developed for machine translation and
substituting translated words with sense-annotated
ones. A recent work that combines labeled data
and knowledge-based information has been pro-
posed by Luo et al. (2018). Uslu et al. (2018)
proposed fastSense, a model inspired by fastText
(Joulin et al.,2017) which – rather than predicting
context words – predicts word senses.
Knowledge-based models, instead, exploit the
structural properties of a lexical-semantic knowl-
edge base, and typically use the relational infor-
mation between concepts in the semantic graph
together with the lexical information contained
therein (Navigli and Lapata,2010). A popular al-
gorithm used to select the sense of each word in
this graph is PageRank (Page et al.,1999) that per-
forms random walks over the network to identify
the most important nodes (Haveliwala,2002;Mi-
halcea et al.,2004;De Cao et al.,2010). An ex-
tension of these models was proposed by Agirre
et al. (2014) in which the Personalized PageRank
algorithm is applied. Another knowledge-based
approach is Babelfy (Moro et al.,2014), which de-
fines a semantic signature for a given context and
compares it with all the candidate senses in order
to perform the disambiguation task. Chaplot and
Salakhutdinov (2018) proposed a method that uses
the whole document as the context for the words to
be disambiguated, exploiting topical information
(Ferret and Grau,2002). It models word senses
using a variant of the Latent Dirichlet Allocation
framework (Blei et al.,2003), in which the topic
distributions of the words are replaced with sense
distributions modeled with a logistic normal distri-
bution according to the frequencies obtained from
WordNet.
3 Word and Sense Embeddings
A good machine-interpretable representation of
lexical features is fundamental for every NLP sys-
tem. A system’s performance, however, depends

90
on the quality of the input representations. Fur-
thermore, the inclusion of semantic features, in ad-
dition to lexical ones, has been proven effective in
many NLP approaches (Li and Jurafsky,2015).
Word embeddings, the current paradigm for lex-
ical representation of words, were popularized
with word2vec (Mikolov et al.,2013). The main
idea is to exploit a neural language model which
learns to predict a word occurrence given its sur-
roundings. Another well-known word embedding
model was presented by Pennington et al. (2014),
which shares the idea of word2vec, but with the
difference that it uses explicit latent representa-
tions obtained from statistical calculation on word
co-occurrences. However, all word embedding
models share a common issue: they cannot cap-
ture polysemy since they conflate the various word
senses into a single vector representation. Sev-
eral efforts have been presented so far to deal with
this problem. SensEmbed (Iacobacci et al.,2015)
uses a knowledge-based disambiguation system to
build a sense-annotated corpus that, in its turn, is
used to train a vector space model for word senses
with word2vec. AutoExtend (Rothe and Sch ¨
utze,
2015), instead, is initialized with a set of pre-
trained word embeddings, and induces sense and
synset vectors in the same semantic space using
an autoencoder. The vectors are induced by con-
straining their representation given the assumption
that synsets are sums of their lexemes. Camacho-
Collados et al. (2015) presented NASARI, an ap-
proach that learns sense vectors by exploiting the
hyperlink structure of the English Wikipedia, link-
ing their representations to the semantic network
of BabelNet (Navigli and Ponzetto,2012). More
recent works, such as ELMo (Peters et al.,2018)
and BERT (Devlin et al.,2019), are based on lan-
guage models learned using complex neural net-
work architectures. The advantage of these mod-
els is that they can produce different representa-
tions of words according to the context in which
they appear.
4 Game Theory and Game Dynamics
In this work we take a different approach to WSD
by employing a model based on game theory (GT).
This discipline was introduced by Neuman and
Morgenstern (1944) in order to develop a math-
ematical framework able to model the essentials
of decision making in interactive situations. In
its normal-form representation (Weibull,1997), it
consists of a finite set of players N= (1, .., n),
a finite set of pure strategies Si={1, ..., mi}for
each player i∈N, and a payoff (utility) function
ui:S→R, that associates a payoff with each
combination of strategies in S=S1×S2×...×Sn.
A fundamental assumption in game theory is that
each player itries to maximize the value of ui.
Furthermore, in non-cooperative games the play-
ers choose their strategies independently, consid-
ering what choices other players can make and try-
ing to find the best response to the strategy of the
co-players.
A player i, in addition to playing single (pure)
strategies from Si, can also use mixed strategies,
that are probability distributions over pure strate-
gies. A mixed strategy over Siis defined as a vec-
tor xi= (x1, . . . , xmi), such that xj≥0and
Pxj= 1. Each mixed strategy corresponds to
a point in the simplex ∆m, whose corners cor-
respond to pure strategies. The intuition is that
player irandomises over strategies according to
the probabilities in xi. Each mixed strategy pro-
file lives in the mixed strategy space of the game,
given by the Cartesian product Θ=∆m1×∆m2×
· · · × ∆mn.
In a two-player game, a strategy profile can be
defined as a pair (xi,xj). The expected payoff for
this strategy profile is computed as:
u(xi,xj) = xT
i·Aijxj
where Aij is the mi×mjpayoff matrix between
players iand j.
In evolutionary game theory (Weibull,1997),
the games are played repeatedly and the players
update their mixed strategy distributions over time
until no player can improve the payoff obtained
with the current mixed strategy. This situation cor-
responds to the equilibrium of the system.
The payoff corresponding to the h-th pure strat-
egy is computed as:
u(xh
i) = xh
i·
ni
X
j=1
(Aijxj)h(1)
It is important to note here that the payoff in Equa-
tion 1is additively separable, in fact, the summa-
tion is over all the niplayers with whom iis play-
ing the games. The average payoff of player iis
calculated as:
u(xi) =
mi
X
h=1
u(xh
i)(2)

91
To find the Nash equilibrium of the game it is com-
mon to use the discrete time version of the repli-
cator dynamics equation (Weibull,1997) for each
player i∈N,
xh
i(t+ 1) = xh
i(t)u(xh
i)
u(xi)∀h∈xi(3)
This equation allows better than average strategies
to grow at each iteration. It can be considered as
an inductive learning process, in which the play-
ers learn from past experiences how to play their
best strategy. We note that each player optimizes
their individual strategy space, but this operation
is done according to what other players simulta-
neously are doing, so the local optimization is the
result of a global process.
Game-theoretic models are appealing because
they are versatile, interpretable and have a solid
mathematical foundation. Furthermore, it is al-
ways possible to find the Nash equilibrium in
non-cooperative games in mixed strategies (Nash,
1951). In fact, starting from an interior point of
Θ, a point xis a Nash equilibrium only if it is the
limit of a trajectory of Equation 3(Weibull,1997).
Figure 1: Generic scheme of the model. ·,×and σ
refer to elementwise multiplication, matrix multiplica-
tion and normalization, respectively.
5 The Model
The model used in this paper, Word Sense Dis-
ambiguation Games (WSDG), was introduced by
Tripodi and Pelillo (2017). It is based on graph-
theoretic principles to model the geometry of the
data and on game theory to model the learning al-
gorithm which disambiguates the words in a text.
It represents the words as the players of a non-
cooperative game and their senses as the strategy
that the players can select in order to play the
games. The players are arranged in a graph whose
edges determine the interactions and carry word
similarity information. The payoff matrix is en-
coded as a sense similarity function. The play-
ers play the games repeatedly and – at each iter-
ation – update their strategy preferences accord-
ing to what strategy has been effective in previ-
ous games. These preferences, as introduced pre-
viously, are encoded as a probability distribution
over strategies (senses).
Formally, for a text Twe select its content
words W= (1, . . . , n)as the players of the game
I= (1, . . . , n). For each word we use a knowl-
edge base to determine its possible senses. Each
sense is represented as a strategy that the player
can select from the set Si={1, ..., mi}, where mi
is the number of senses of word wi. The set of all
different senses in the text, C={1, ..., m}, is the
strategy space of the games. The strategy space is
modeled, for each player, as a probability distri-
bution, xi, of length m. It takes non-zero values
only on the entries corresponding to the elements
of Si. It can be initialized with a normal distribu-
tion in the case of unsupervised learning or with
information obtained from sense-labeled corpora
in the case of semi-supervised learning.
The payoff of a game depends on a payoff ma-
trix Zin which the rows are indexed according to
the strategies of player iand the columns accord-
ing to the strategies of player j. Its entries Zr,t are
the payoff obtained when player iselects strategy
rand player jselects strategy t. It is important to
note here that the payoff of a game does not de-
pend on the single strategy taken individually by
a player, but always by the combination of two si-
multaneous actions. In WSD this means that the
sense selected by a word influences the choices of
the other words in the text and this allows the tex-
tual coherence to be maintained.
The disambiguation games to build a payoff
function require: a word similarity matrix A, a
sense similarity matrix Zand a sense distribution
xifor each player i.Amodels the players’ inter-
actions, so that similar players play together and
the more similar they are the more reciprocal influ-
ence they have. It can be interpreted as an attention
mechanism (Vaswani et al.,2017) since it weights
the payoffs. Zis used to create the payoff matrices
of the games so that the more similar the senses of
the words are the more the corresponding players
are encouraged to select them, since they give a
high payoff. Aand Zare obtained by comput-
ing vector representations of word and sense (see
Section 3) and then calculating their pairwise sim-

92
ilarity.
The strategy space of each player, i, is repre-
sented as a column vector of length m. It is initial-
ized with:
xh
i=(|mi|−1if sense his in Si,
0otherwise.(4)
This initialization is used in the case of unsuper-
vised WSD, since it does not use information from
sense-tagged corpora. If instead this information
is available, |mi|−1in Equation 4is substituted
with the frequency of the corresponding sense and
then xiis normalized in order to sum up to one.
Once these sources of information are com-
puted, the WSDG are run by using the replica-
tor dynamic equation (Taylor and Jonker,1978)
in Equation 3, where the payoff of strategy hfor
player iis calculated as:
u(xh
i) = xh
i·
ni
X
j=1
(AijZxj)h(5)
where niare the neighbours of player ias in the
graph A. The average payoff is calculated as:
u(xi) =
mi
X
h=1
u(xh
i)(6)
The complexity of WSDG scales linearly with
the number of words to be disambiguated. Differ-
ently from other models based on PageRank, it is
possible to disambiguate all the words at the same
time. As an example, WSDG can disambiguate
200 words (1650 senses) in 7 seconds, on a single
CPU core. A generic representation of the model
is proposed in Figure 1.
Implementation details The cosine similarity
was used as similarity measure for both words and
senses. The Amatrix was treated as the adjacency
matrix of an undirected weighted graph and, to re-
duce the complexity of the model, the edges with
weight lower than 0.1were removed. The sym-
metric normalized Laplacian of this graph was cal-
culated as D−1/2AD−1/2, where Dis the degree
matrix of graph A. Since the algorithm operates
on an entire text, local information is added to ma-
trix A. The mean value of the matrix is added
to the dlog(n)ecells on the left of the main di-
agonal. For BERT, this operation was replaced
with its attention layer, adding to matrix Athe
mean attention distribution of all the heads of the
last layer. The choice of the last layer is moti-
vated by the fact that it stores semantic informa-
tion and its attention distributions have high en-
tropy (Clark et al.,2019). The first singular vec-
tor was removed from Ain the case of word vec-
tors whose length exceeded 500. This was done
to reduce the redundancy of the representations in
line with Arora et al. (2017). The distributions
for each xwere computed according to SemCor
(Miller et al.,1993) and normalized using the soft-
max function. The replicator dynamics were run
until a maximum number of iterations was reached
(100) or the difference between two consecutive it-
erations was below a small threshold (10−3), cal-
culated as Pn
i=1 kxi(t−1) −xi(t)k. The code
of the model is available at https://github.
com/roccotrip/wsd_games_emb.
6 Evaluation
The evaluation of our model was conducted using
the framework proposed by Raganato et al. (2017).
This consists of five datasets which were unified to
the same WordNet 3.0 inventory: Senseval-2 (S2),
Senseval-3 (S3), SemEval-2007 (SE7), SemEval-
2013 (SE13) and SemEval-2015 (SE15). These
datasets have in total 26 texts and 10,619 words
to be disambiguated. Our objective was to test our
game-theoretic model with different settings and
to evaluate its performances. To this end we per-
formed experiments comparing 16 different sets of
pretrained word embeddings and 7sets of sense
embeddings.
Word embeddings As word embedding mod-
els we included 4 pre-word2vec models: the hi-
erarchical log-bilinear model (Mnih and Hinton,
2007, HLBL), a probabilistic linear neural model
which aims to predict the embedding of a word
given the concatenation of the previous words;
CW (Collobert and Weston,2008), an embeddings
model with a deep unified architecture for mul-
titask NLP; Distributional Memory (Baroni and
Lenci,2010, DM), a semantically enriched count-
based model; leskBasile (Basile et al.,2014), a
model based on Latent Semantic Analysis reduced
via Singular-Value Decomposition; 3 models ob-
tained with word2vec: GoogleNews, a set of
300-dimensions vectors trained with the Google
News dataset; BNC-*, vectors of different di-
mensions trained on the British National Cor-
pus including POS information during training;
and w2vR, trained with word2vec on the 2014

93
dump of the English Wikipedia, enriched with
retrofitting (Faruqui et al.,2015), a technique to
enhance pre-trained embeddings with semantic in-
formation. The enrichment was performed us-
ing retrofitting’s best configuration, based on the
Paraphrase Database (Ganitkevitch et al.,2013,
PPDB). We also tested GloVe (Pennington et al.,
2014), trained with the concatenation of the 2014
dump of the English Wikipedia and Gigaword 5,
and fastText (Bojanowski et al.,2017) trained on
Wikipedia 2017, UMBC corpus and the statmt.org
news dataset.
Figure 2: Performances of the model on the union of all
datasets. The results are presented as F1 for all combi-
nations of word and sense embeddings. Word vectors
are on the rows and sense vectors on the columns.
Contextualized word embeddings As contex-
tualized embeddings we used ELMo (Peters et al.,
2018) in three different configurations, namely:
ELMo-avg, weighted sum of its three layers;
ELMo-avg emb, weighted sum of its three layers
and the embeddings it produces; and ELMo-emb,
the word embeddings produced by the model1.
We also tested three implementations of BERT
(Devlin et al.,2019): base cased (b-c); large un-
cased (l-u) and large cased (l-c). They offer pre-
trained deep bidirectional representations of words
1TensorFlow models available at https://tfhub.
dev/google/elmo/2.
in context2. We used seven configurations for
each model: one for each of the last four layers
(numbered from 1to 4), the sum of these layers,
their concatenation and the embedding layer. We
fed all these models with the entire texts of the
datasets. Since BERT uses WordPiece tokeniza-
tion, we averaged sub-token embeddings to obtain
token-level representations.
We also included three models which were built
together with the sense embeddings introduced be-
low.
Sense embeddings As sense embeddings, in ad-
dition to the three models introduced in Section 3
(AutoExtend, NASARI and SensEmbed), we in-
cluded four models: Chen et al. (2014), a uni-
fied model which learns sense vectors by train-
ing a sense-annotated corpus disambiguated with a
framework based on semantic similarity of Word-
Net sense definitions; meanBNC, created using a
weighted combination of the words from WordNet
glosses, using, as word vectors, the set of BNC-
200 mentioned earlier; DeConf (Pilehvar and Col-
lier,2016), also linked to WordNet, a model where
sense vectors are inferred in the same semantic
space of pre-trained word embeddings by decom-
posing the given word representation into its con-
stituent senses; and finally SW2V (Mancini et al.,
2017), a model linked to BabelNet which uses a
shallow disambiguation step and which, by ex-
tending the word2vec architecture, learns word
and sense vectors jointly in the same semantic
space as an emerging feature.
Results The results of these models are reported
in Figure 2. One of the most interesting patterns
that emerges from the heat map is that there are
some combinations of word and sense embeddings
that always work better than others. Sense vectors
drive the performance of the system, contributing
in great part to the accumulation of payoffs during
the games. The sense vectors that maintain high
performances are SensEmbed, AutoExtended and
Chen2014. In particular Chen2014 has high per-
formances with all the word embedding combina-
tions. While these models are specific sense em-
bedding techniques, the construction of BNC-200
follows a very simple method, which in view of
these results can be refined using more principled
gloss embedding techniques. The performances of
2PyTorch models available at https://github.
com/huggingface/pytorch-transformers.

94
model S2 S3 SE07 SE13 SE15 All N V A R
semi-sup.
MFS 64.7∗65.4 53.9 62.9 66.6∗64.1 68.1 49.5 74.1 80.6
Babelfy 67.0 63.5∗51.6∗66.4†70.3 65.5∗68.6∗49.9 73.2 79.8
ppr w2w 68.8 66.1 53.068.870.3 67.3- - - -
WSD-TM 69.0 66.9 55.6 65.3∗69.6 66.9 69.7†51.276.0 80.9
WSDGα68.768.3 58.966.4 70.767.7 71.1 51.9†75.480.9
WSDGβ68.9 65.5 54.5 67.072.867.2 70.4 51.3 75.7 80.6
WSDGγ69.366.4 56.0†65.9 70.8 67.2 70.4 51.5 75.1 80.6
sup.
IMS (2010) 70.9†69.3 61.3 65.3 69.5†68.9†70.5 55.8 75.6 82.9
IMSw2v 72.2 70.4 62.6 65.9 71.5 70.1 71.9 56.6 75.9 84.7
YuanLSTM 73.8 71.8 63.5 69.5 72.6 71.5- - - -
RaganatoBLSTM 72.0 69.1 64.8 66.9 71.5 69.9 71.5 57.5 75.0 83.8
GAS 72.2 70.5†-67.2 72.6- - - - -
fastSense 73.5 73.5 62.4 66.2 73.2- - - - -
Table 1: Comparison with state-of-the-art algorithms: unsupervised or knowledge-based (unsup.), and supervised
(sup.). MFS refers to the MFS heuristic computed on SemCor on each dataset. The results are provided as F1 and
the first result of the semi supervised systems with a statistically significant difference from the best of each dataset
is marked with ∗(χ2,p < 0.1). †indicates the same statistics but including also supervised models.
NASARI are lower compared to lexical vectors:
this may be due to our choice to use NASARI-
embed, whose vectors have low dimensionality.
The word vectors that have consistently high
performances in association with the three sense
vectors mentioned above are BERT, Chen2014,
SensEmbed and SW2V. This is not surprising
since they are able to produce contextualised word
representations, performing, in fact, a preliminary
disambiguation of the words. In particular, SW2V
is specifically tailored for WSD. ELMo and fast-
Text perform slightly worse. The vectors con-
structed using syntactic information and trained on
the BNC corpus have similar performances to the
their counterparts trained on larger corpora with-
out the use of syntactic information. If we focus
on BERT, we can see that it is able to maintain
high performances (F1≈67) with all its configu-
rations, except for the embedding layers of all the
models (*-emb). The contribution of the sum and
concatenation operations is not significant.
Comparison We performed a comparison with
3 configurations of our model, one for each of the
three best sense vectors: WSDGα, obtained us-
ing Chen2014 as sense vectors and BERT-l-u-4 as
word vectors; WSDGβ, obtained using SensEm-
bed as sense vectors and BERT-l-c-4 as word vec-
tors; and WSDGγ, obtained using AutoExtended
as sense vectors and BERT-l-u-3 as word vectors.
As comparison systems we included three semi-
supervised approaches mentioned above, Babelfy
(Moro et al.,2014), pprw2w, the best configura-
tion of UKB (Agirre et al.,2018), and WSD-TM,
introduced by Chaplot and Salakhutdinov (2018)
(for this model we did not have the possibility
to verify the results since its code is not avail-
able). In addition, we also report the performances
of relevant supervised models, namely: It Makes
Sense (Zhong and Ng,2010, IMS), Iacobacci et al.
(2016), Yuan et al. (2016), Raganato et al. (2017),
Joulin et al. (2017) and Uslu et al. (2018).
The results of our evaluation are shown in Table
1. As we can see our model achieves state-of-the-
art performances on four datasets and on S13 and
S15 it performs better than many supervised sys-
tems. In general the gap between supervised and
semi-supervised systems is reducing. This encour-
ages new research in this direction. Our model
fares particularly well on the disambiguation of
nouns and verbs. However, the main gap between
our models and supervised systems relies upon the
disambiguation of verbs.
7 Analysis
Polysemy As expected, most of the errors made
by WSDGαare on highly polysemous words. Fig-
ure 3shows that the number of wrong answers in-
creases as the number of senses grows, and that
the number of wrong answers starts to be higher
than that of correct answers when the number of
senses for a target word is in the range of 10-15
senses. The words with the highest number of er-
rors are polysemous verbs such as: say (34), make
(24), find (21), have, (17), take (15), get, (15) and
do (13). These are words that in many NLP ap-
plications (especially those based on distributional
models) are treated as stopwords.

95
Sense rank Mancini et al. (2017) show that
senses which are not the most frequent ones are
particularly challenging and most sense-based ap-
proaches fail to represent them properly. In Fig-
ure 4we report the results of WSDGαdivided per
sense rank, where it is possible to see how the per-
formances of the system deteriorate as the rank
of the correct sense increases. It is interesting to
see that, in the first part of the plot, the perfor-
mances follow a regular pattern that resembles a
power-law distribution. This requires further anal-
ysis beyond the scope of this work, along the lines
of Ferrer-i Cancho and Vitevitch (2018).
Figure 3: Correct and wrong answers given by
WSDGαgrouped by number of senses
Figure 4: Correct and wrong answers given by
WSDGαper sense rank.
Priors Corroborating the findings of Pilehvar
and Navigli (2014), Postma et al. (2016) con-
ducted a series of experiments to study the ef-
fect that the variation of sense distributions in
the training set has on the performances of It
makes sense (Zhong and Ng,2010). Specifi-
cally, they increased the volume of training ex-
amples (V) by enriching SemCor with senses in-
ferred from BabelNet; increased the number of
least frequent senses (LFS) (V+LFS); and over-
fitted the model constructing a training set pro-
portional to the correct sense distribution of the
test set (GOLD,GOLD+LFS). We used the same
training sets to compute the priors for our system.
The results of this analysis are reported in Table 2.
These experiments show that increasing the num-
System V V+LFS GOLD GOLD+LFS
IMS 68.9 62.0 86.8 85.4
WSDGα66.4 57.5 88.4 90.8
Table 2: Comparison using different priors.
ber of training examples has a small beneficial ef-
fect. Increasing the number of LFS examples leads
to worse results because this is a deviation from a
real sense distribution. Further, to work with bet-
ter semantic representations, this operation should
also be accompanied by a similar selection on the
training set of sense and word embeddings, other-
wise LFS remain underrepresented. Finally, mim-
icking the distribution of the test set is more bene-
ficial for WSDGαthan for IMS, especially when
LFS examples are added, suggesting that semi-
supervised systems can better adapt to specific do-
mains than supervised systems.
8 Exploratory study
We now present three WSD applications in as
many tasks: selection of context-sensitive embed-
dings; sentence similarity; paraphrases detection.
Context-sensitive embeddings We used the
Word in Context (WiC) dataset (Pilehvar and
Camacho-Collados,2019) for this task. It contains
7466 sentence pairs in which a target word appears
in two different contexts. The task consisted of
predicting if a target word has the same sense in
the two sentences or not. The aim of this experi-
ment was twofold: we wanted to show the useful-
ness of contextualized word embeddings obtained
from WSD systems and to demonstrate that the
model was able to maintain the textual coherence.
The experiments on this dataset were conducted
on the development set (1400 sentence pairs). The
comparison was conducted against state-of-the-
art models for contextualized word embeddings
and sense embeddings: Context2Vec (Melamud
et al.,2016) based on a bidirectional LSTM lan-
guage model; ELMo1, the first LSTM hidden
state; ELMo3, the weighted sum of the 3 LSTM
layers; BERTbase; BERTlarge. The results of these
systems were taken from Pilehvar and Camacho-
Collados (2019). We note here that all these mod-
els, including WSDGα, do not use training data.
They are based on a simple threshold-based clas-
sifier, tuned on the development set (638 sentence
pairs). WSDGαdisambiguates all the words in
each pair of sentences separately and, if the cosine

96
C2V ELMo1ELMo3BERTbase BERTlarge WSDGα
59.7 57.1 56.3 63.6 63.866.2
Table 3: Performance on the WiC dataset.
Pearson Spearman MSE
sense 46.5 43.9 7.9
word 39.8 39.9 8.6
Table 4: WSDGαresults on the SICK dataset.
similarity among the senses assigned to the target
words is below a threshold (0.9), it classifies the
pair as different senses, and as the same sense oth-
erwise. As shown in Table 3the disambiguation
step has a big impact on the results.
Sentence similarity We used the SICK dataset
(Marelli et al.,2014) for this task. It consists of
9841 sentence pairs that had been annotated with
relatedness scores on a 5-point rating scale. We
used the test split of this dataset that contains 4906
sentence pairs. The aim of this experiment was to
test if disambiguated sense vectors can provide a
better representation of sentences than word vec-
tors. We used a simple method to test the two
representations: it consisted of representing a sen-
tence as the sum of the disambiguated sense vec-
tors in one case and as the sum of word vectors
in the other case. Once the sentence representa-
tions had been obtained for both methods the co-
sine similarity was used to measure their related-
ness. The results of this experiment are reported
in Table 4as Pearson and Spearman correlation
and Mean Squared Error (MSE). We used the α
configuration of our model with Chen2014 to rep-
resent senses and BERT-l-u-4 to represent words.
As we can see the simplicity of the method leads
to low performances for both representations, but
sense vectors correlate better than word vectors.
Paraphrase detection We used the test set of
the Microsoft Research Paraphrase Corpus (Dolan
et al.,2004, MRPC) for this task. The corpus con-
tains 1621 sentence pairs that have been annotated
with a binary label: 1if the two sentences consti-
tute a paraphrase and 0otherwise. In this task we
also used the sum of word vectors and the sum of
disambiguated sense vectors to represent the sen-
tences, and used part of the training set (10%) in
order to tune the threshold parameter below which
the sentences are not considered paraphrase. The
classification accuracy for the word vectors used
by WSDGαwas 58.1whereas the disambiguated
sense vectors obtained 66.9.
9 Conclusion
In this work we have presented WSDG, a flexi-
ble game-theoretic model for WSD. It combines
game dynamics with most successful word and
sense embeddings, therefore showing the poten-
tial of an effective combination of the two areas of
game theory and word sense disambiguation.
Our approach achieves state-of-the-art perfor-
mances on four datasets performing particularly
well on the disambiguation of nouns and verbs.
Beyond the numerical results, in this paper we
have presented a model able to construct and eval-
uate word and sense representations. This is par-
ticularly useful since it can serve as a test bed for
new word and sense embeddings. In particular,
it will be interesting to test new sense embedding
models based on contextual embeddings.
Thanks to the flexibility and scalability of our
model, as future work we plan to explore in depth
its use in different tasks, such as the creation of
sentence (document) embeddings and lexical sub-
stitution. In fact, we believe that using disam-
biguated sense vectors, as shown in the context-
sensitive embeddings and paraphrase detection
studies, can offer a more accurate representation
and improve the quality of downstream applica-
tions such as sentiment analysis and text classifi-
cation (see, e.g., (Pilehvar et al.,2017)), machine
translation and topic modelling. Encouraged by
the good results achieved in our exploratory stud-
ies, we plan to develop a new model for con-
textualised word embeddings based on a game-
theoretic framework.
Acknowledgments
The authors gratefully acknowl-
edge the support of the ODYC-
CEUS project No. 732942 (first au-
thor) and of the ERC Consolida-
tor Grant MOUSSE No. 726487
(second author) under the European
Union’s Horizon 2020 research and
innovation programme.
The experiments have been run on the SCSCF
cluster of Ca’ Foscari University. The authors
thank Ignacio Iacobacci for preliminary work on
this paper.

97
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