can assume that there is an effective reward function, $ \lambda_{eff} $, for every $ \gamma $ so that it becomes the averaged rewards obtained on a large number of agents $ m $. $${\left\langle \sum_l \delta (c_i , c_{k_l} ) \delta (c_j , c_{m_l}) {\lambda}^{(t+1)} \right\rangle }_{ m \to \infty} \longrightarrow \quad \lambda_{eff}^{(t+1)} (c_i , c_j , \gamma ) ,$$ thus $$h^{(t+1)} (c_i , c_j ) = h^{(t)} (c_i , c_j ) - \gamma ( h^{(t)} (c_i , c_j ) -1 ) + \lambda_{eff}^{(t+1)} (c_i , c_j , \gamma ) .
\label{effectivee}$$\
A constant reward for every time in Eq.\[eqtrans\], $${\lambda}^{(t+1)} (c_i , c_j ) = \lambda (c_i , c_j ),
\label{conditionddq}$$ leads to $$h_{max} = \lim_{t \to \infty} h^{(t)} (c_i , c_j ) \, = \, \frac{\lambda_{eff} (c_i , c_j , \gamma ) }{\gamma} + 1.$$ Then $$h_{max} =
\begin{cases}
\frac{\lambda_{eff}^{reward}}{\gamma} + 1 & \textit{for a rewarded pair of $ (c_i , c_j) $} \\
1 & \textit{for a non-rewarded pair of $ (c_i , c_j) $}
\end{cases}$$ Due to the fact that $ \lambda_{eff}^{non-reward} = 0 $, we rename $ \lambda_{eff}^{reward} = \lambda ( \gamma ) $ from so on. Given $ N $ different percept-action but 