too. For a multi-agent setting, there are two main streams in literature as interactive partially observable [@Gmytrasiewicz1; @Gmytrasiewicz2; @Panella] and also decentralized partially observable [@Bernstein1; @bookchapter:Oliehoek; @Amato1].

In the PS context, remembering efficiency of Eq.\[eqoriggin\] and using a vector notation for $N$ different percept-actions, then $$r^{(t)} = \frac{1}{N} \sum_{i=1}^N \, \langle a_{s_i}^{*} | s_i \rangle.
\label{eqfullyy}$$

Where we omitted the superscript $ t $ for the probabilities. The environments, so far, have been stochastic, fully observable (since the agent observes real percepts) and containing one learning agent. However, people may think of expanding a vector space on a different basis to make some belief states from the world states $ | s_i \rangle $. As a tangible example, considering one another intelligent agent as an interpreter (an **intelligent projector**) as illustrated in Fig. \[interpreter:1\]. We can span our real states (world states) $ | s_i \rangle  $ on the belief states $ | b_i \rangle  $ which are what is learned by an interpreter (I) concerning $ s_i $. One can write a projection operator [@book:Sakurai] $ B_j $ as $$B_j =  |  b_j \rangle \langle b_j |.$$ Then $$| s_i \rangle = \sum_j \, |  b_j \rangle \langle b_j 