$$\begin{gathered}
r_{max}^i (\alpha_{i} , \gamma_{1i}, \gamma_{2j} ) =  \alpha_{i} \, p ( \gamma_{1i} ) + (1- \alpha_{i}) \left[ p ( \gamma_{1i} ) . p ( \gamma_{2j} )  + q ( \gamma_{1i} ) . q ( \gamma_{2j} ) \right] ,
\label{eqrmaxx}\end{gathered}$$\
according to Eq. \[eqrmaxxx\]. For example, if $$\begin{cases}
\alpha_{i} = \alpha_{j} = \alpha , \\
\forall k \quad \gamma_{ki} =  \gamma_{kj} = \gamma_{k} , 
\end{cases}
\Longrightarrow \quad r_{max}^i (\alpha , \gamma_{k}) = r_{max}^j (\alpha , \gamma_{k}),$$\
which refers to the symmetry of the problem between two agents. Otherwise, the efficiency of one agent would differ from that of its partner as a function of its own forgetting factor, its partner’s forgetting factor and also the parameter of observability of the environment for one agent. With the same variables of $ \gamma_{1i}, \gamma_{2j} $ as regarded for Eq. \[eqrmaxx\], one would have $$\begin{gathered}
r_{max}^j (\alpha_{j} , \gamma_{1i}, \gamma_{2j} ) =  \alpha_{j} \, p ( \gamma_{j} - \gamma_{2j} ) + (1- \alpha_{j}) [ p ( \gamma_{j} - \gamma_{2j} ) . p ( \gamma_{i} - \gamma_{1i} ) \\ + q ( \gamma_{j} - \gamma_{2j} ) . q ( \gamma_{i} - \gamma_{1i} ) ]. \end{gathered}$$

### classes of coalitions {#superadditiven .unnumbered}

Considering the collective 