# Expected output Player norms:[1.77552809 1.69189243 1.68448805] ```
### Experimental Analysis
The merits introduced by GNN models exceed basic skill-based comparisons. Our approach aims to form balanced teams with a minimization of the variance in skill levels between the teams. In addition, the utility of the proposed model within the TFP domain is characterized by its extension capabilities. One of the possible extensions that might arise is to ensure the diversity of the teams, in terms of their demographics, personalities, or working styles. This goal can already be achieved by the proposed pipeline (GCN + MILP Solver). First, the GCN is used to encode participant attributes (e.g., demographic, personality traits, working style) into feature vectors. Second, the GCN can learn embeddings that capture participants' diversity-related attributes based on their connections or features. Third, we can introduce diversity metrics based on the embeddings. For example, if we are looking for demographic diversity, we can measure the distribution of demographic attributes (e.g., gender, age, ethnicity) within each team. Then, to incorporate diversity into the MILP model, we can either add diversity constraints to the MILP model or add a diversity term to its objective function. For the completeness of this discussion, we will go with one option--adding constraints--to check whether this model can meet a diversity requirement (i.e., demographic balance). First, we need to ensure that each team has a balanced representation of different demographic groups. Then, we introduce a binary indicator \(d_{ig}\), where \(d_{ig}=1\) if player \(i\) belongs to demographic group \(g\), and 0 otherwise. Second, we add constraints to ensure that each team \(j\) has at least \(I_{g}\) and at most \(u_{g}\) players from group \(g\):

\[I_{g}\leq\sum_{i=1}^{n}d_{ij}x_{ij}\leq u_{g},\forall j=1,...,m,\forall g.\]

Furthermore, we can argue in favor of team formation under the condition of unbalanced skill distributions. This is achievable if we neutralize the effect of the term \(\alpha\sum_{j=1}^{m}|\sum_{i=1}^{n}s_{i}x_{ij}-\frac{1}{m}\sum_{k=1}^{m}\sum_{ i=1}^{n}s_{i}x_{ik}|\). This could be achieved by setting \(\alpha=0\). The effect of removing the balance objective would be a highly skewed distribution. Moreover, possible extensions of the model to handle more sophisticated constraints to create more controllable team formations are feasible and can be achieved through several approaches. The first approach is a shift from skill balance to providing a certain upper and lower bound for the skill level per team. This is achieved by adding the following two constraints:

\[\sum_{i=1}^{n}s_{i}\times x_{ij}>=MinSkillLevel,\forall j=1,...,m\]

\[\sum_{i=1}^{n}s_{i}\times x_{ij}\leq MaxSkillLevel,\forall j=1,...,m.\]

The second approach avoids placing two players in the same team due to aversion or a lack of chemistry. For example, player \(i_{1}\) and player \(i_{2}\) cannot be on the same team. As such, a new constraint is added, which is as follows:

\[x_{i1,j}+x_{i2,j}\leq 1.\]

In addition, other metrics may be investigated, for example, the within-cluster sum of squares (WCSS), Silhouette Coefficient (SC), Davies-Bouldin Index, etc. However, these metrics need to be adapted to fit the context of our specific team formation model. ## 8 Conclusions and Future Work

Graph neural networks have shown flexibility in understanding the relationships between entities, incorporating preferences and learning complex interactions between domain-specific nodes. Using graph convolution networks (GCNs) to address the team formation problem offers the advantages of flexibility and adaptability in terms of their ability to reflect network dynamics and relationships. In this work, we propose a hybrid GCN and MILP model to solve the team formation problem. Based on experimental work, the proposed model is capable of obtaining the optimal solution if it exists.