just one desirable action $ a_{s_i}^{*} $ for each percept $ s_i $, we have $$p + q \, ( N-1 ) = 1,
\label{diwwonehhha}$$ and $$p_{N} ( \gamma ) = \frac{\lambda_{N} (\gamma) + \gamma}{\lambda_{N} (\gamma) + N \gamma} \quad , \quad q_{N} ( \gamma ) = \frac{ \gamma}{\lambda_{N} (\gamma) + N \gamma} , \qquad p(0) = 1, \qquad q(0) = 0. 
\label{incsioo}$$

Equations \[incsioo\] are the evaluations of the asymptotic probabilities in our partially observable multi-agent model of problems for which Eq. \[conditionddq\] is satisfied. Otherwise, we may use the specific properties of a given scenario to have $ p $ and $ q $.

In the following, we consider an invasion with $ N=2 $. Figure \[effrewardx\] shows the effective reward of the rewarded pairs of $(c_i, c_j )$ for a fully observable one agent invasion with $ \lambda^{t+1} (s, a_{s}^{*} ) =1 $. For instance, $$p_2 (\gamma \rightarrow 1) \longrightarrow \frac{1.2}{2.2}, \quad q_2 (\gamma \rightarrow 1) \longrightarrow \frac{1}{2.2}.$$

![image](effective_reward.eps){width="75.00000%"}

**An example of an absolute partially observable environment; Regarding Equation \[blockeff\]**
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Fully observable environments have been considered widely with respect to Eq. \[eqfullyy\]. Here, we start from Eq. \[blockeff\] which describe an environment in which all of 