# The payoff matrix

We encoded the payoff matrix of the games as a sense similarity matrix among all the senses in the strategy spaces of the game. In this way, the higher the similarity between the senses of two words, the higher the incentive for a player to select that sense, and play the strategy associated with it.

The *c x c* sense similarity matrix *A* is defined in the following equation:

*
*

This matrix can be computed using the information derived from the same knowledge base used to construct the strategy space of the game. It is used to extract the partial payoff matrix *A** _{ij}* for all the single games played between two players

*i*and

*j.*This operation is performed by extracting from

*A*the entries relative to the indices of the senses in the sense inventories

*S*and

_{i}*S*

_{j}. It produces an

*m*x

_{i}*m*payoff matrix, where

_{j}*m*and

_{i}*m _{j}* are the numbers of senses in

*S*and

_{i}*S*

_{j}, respectively.

The semantic measure that we used in this work is the Gloss Vector measure [PAT 06], since it has been demonstrated to have stable performances in different datasets [TRI 17]. It is based on the computation of the similarity between the definitions of two concepts in a lexical database. They are used to construct a co-occurrence vector *v _{i} =* (v

^{1},

*v*

^{2},...,

*v*

^{n}) for each concept

*i*, with a

bag-of-words approach, where *v ^{h}* represents the number of times word v occurs in the gloss and

*n*is the total number of different words in the corpus. From this representation, it is possible to compute the similarity between two vectors using the cosine distance:

*
*

The vectors are constructed using the concept of super-gloss introduced by [PAT 06]. It is the concatenation of the gloss of the synset and the glosses of the synsets connected to it with any relation in the knowledge base.

# System dynamics

At each iteration of the system, each player plays a game with its neighbors *N _{i}* according to the graph

*W.*The payoff of the

*h*-th strategy is calculated as:

*
*

and the playerâ€™s payoff as: *
*

where N represents the neighbors of player *i.* We assume that the payoff of word *i* depends on the similarity that it has with word *j*, *w*_{ij}, the similarities among its senses and those of word *j*, *A*_{ij}, and the sense preference of word *j*, (*x*_{j}), that can be unambiguous if *j* is a labeled player.

We use the replicator dynamics equation (see section 6.3.2) to find the Nash equilibria of the games. During each phase of the dynamics, a process of selection allows strategies with a higher payoff to emerge and at the end of the process each player chooses its sense according to these constraints.