efficiency of two gaents as $ r_{col} = r_{max}^i + r_{max}^j $, one may ask about the classes of coalitions (see [@book:MultiagentShoham] p. 386 for the definitions) in this game comparing two cases: $ r_{col-FO} (\gamma_{i}, \gamma_{j} ) $ in a fully observale environment versus $ r_{col-PO} (\alpha_{i},\alpha_{j} , \gamma_{1i}, \gamma_{2j}) $ in a partially observable environment. The former is straightforward as $$r_{col-FO} (\gamma_{i}, \gamma_{j} )  = p ( \gamma_i ) + p ( \gamma_j)$$ on one hand. However, regarding the latter, one the other hand, we deal with a four variable function. At first, we can see that every *selfish-selfish* coalition $ \gamma_i = (0 , \gamma_i ) \,\, \& \,\, \gamma_j = ( 0 , \gamma_j ) $ will be superadditive, $$\begin{aligned}
r_{max}^i = \alpha_{i} + ( 1 - \alpha_i ) p ( \gamma_j ) = p ( \gamma_j ) +  \alpha_{i} q ( \gamma_j ) , \notag \\
r_{max}^j = \alpha_{j} + ( 1 - \alpha_j ) p ( \gamma_i ) = p ( \gamma_i ) +  \alpha_{j} q ( \gamma_i ) . \notag \end{aligned}$$ Then $$r_{col-PO} (\alpha_i , \alpha_j , \gamma_{1i} = 0 , \gamma_{2j} = \gamma_{j} ) \geq r_{col-FO} (\gamma_{i}, \gamma_{j} ) .
\label{supsupadlll}$$\
Equation 