\begin{align}\label{eq:lin-sys2}
    \sum_{j \in \mathcal{S}_1} q^1(i,j)y_j^1(s_1,s_2)&=(f_1(i)s_1 +f_2(i)s_2)y_i^1(s_1,s_2)\\
    &=\alpha (\rho_{11} s_1 +\rho_{12} s_2)I(X_1=i)y_i^1(s_1,s_2), \;\;\; \forall i \in \mathcal{C}_1.
\end{align}
