Shapley calculator

This component provides the functionality to compute the Shapley value of an individual norm $n_i$ within a normative system $N$ (where $n_i \in N$), with respect to a given value $v$. As explained in Alignment calculator, given a normative system $N$ and a value $v$, it is possible to quantify the degree of alignment $\mathsf{Algn}_{N,v}$ that the norms in $N$ have with respect to value $v$. The Shapley value $\phi_i(v)$ of norm $n_i \in N$ quantifies the contribution that the specific norm $n_i$ makes to that alignment. It is defined as:

$$\phi_{i}(v) = \sum\limits_{N'\subseteq N\setminus \{n_i\}} \frac{|N'|!\left(|N|-|N'|-1\right)!}{|N|!} \cdot \left( \mathsf{Algn}_{N'\cup \{n_i\}, v} - \mathsf{Algn}_{N', v} \right)$$

In the previous equation, the summation is taken over all the subsets of $N$ where $n_i$ is absent, $N \setminus \{n_i\}$. In general, given a normative system $N$ and a set of norm $\{n_j, n_k, ...\} \subseteq N$, the removal of $\{n_j, n_k, ...\}$ from $N$ is represented by a normative system $N' = N \setminus \{n_j, n_k, ...\}$ where the values of the normative parameters tied to $\{n_j, n_k, ...\}$ are substituted by their baseline quantities. These baseline normative parameters are drawn from a baseline normative system $N_{bsl}$. This baseline normative system has the same norms as $N$, tied to the same normative parameters, but the values that these baseline normative parameters are selected in such a way that, when it is implemented on the model, it does not cause it to evolve in any way. In other words, the baseline normative system $N_{\text{bsl}}$ causes transitions to not lead to any change in system state, $\mathbf{s} \xrightarrow{N_{\text{bsl}}} \mathbf{s}$.

A Shapley Calculator is initialized similarly to the Alignment Calculator, by providing (i) a representation of the model or system being examined (i.e. the entity upon which norms apply to), and (ii) the semantics function $f_v$ of the value of interest $v$ whose alignment is computed by the component.