Then, some world percepts would be invisible for each agent while another agent can detect them and help its partner to have more efficiency. Assuming a different forgetting factor for every task to be done for a given agent $ i $ in a multi-agent setting containing $ n $ tasks, we could have $$\gamma_{i} = \sum_{k=1}^n \gamma_{ki} , \quad  0 \leq \gamma_{i} \leq 1.
\label{gamasamam}$$ In such a scenario, the parameter of observability $ \alpha $ can also differ for two given agents $ i $ and $ j $, that is $ \alpha_{i} \neq  \alpha_{j}$.

Specific examples of a multi-agent invasion toy problem
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The original formulation of PS has been introduced using a toy problem called invasion game as elaborated in [@Briegel1; @Briegel2]. To add partial observability employing another agent to the original projective simulation, we can assume an interpreter added to the standard invasion problem. In the basic form of the invasion game, an attacker (A) send some precepts $s \in \{ \Leftarrow , \Rightarrow \}$ (in case $ N=2 $) where a defender (D) perceives and learns them by taking action $a\in \{ - , + \}$ on a percept and getting a reward ($\lambda$). 