\\
q ( \gamma_{pa} ) , \quad k \neq l 
\end{cases} \\
,{\langle b_k | s_l \rangle}_{max} = 
\begin{cases}
p ( \gamma_{I} ) , \quad k = l \\
q ( \gamma_{I} ) , \quad k\neq l 
\end{cases}$$ where $ \gamma_{pa} $ and $ \gamma_{I} $ stand for the protagonist agent’s forgetting factor and interpreter’s forgetting factor respectively. Hence, for a simple fully observable problem with $ a_{s_i}^{*} = a_i $ we have $ r_{max} = p ( \gamma_{pa} ) $ according to Eq. \[eqoriggin\] or Eq. \[eqfullyy\]. Yet, considering our partially observable two-agent model with $ N'=N $ ($ |S| = |A| = |B|= N $) one can use Eq. \[jankan\], regarding $ r_{PO} $ for the asymptotic efficiency in a partially observable environment (versus $ r_{FO} $ preserved for a fully observable one) and write

$$\begin{gathered}
r_{PO} \equiv r_{max}^{pa} (\alpha , \gamma_{pa}, \gamma_{I} ) =  \alpha \, p ( \gamma_{pa} ) + (1- \alpha ) \left[ p ( \gamma_{pa} ) . p ( \gamma_{I} )  + q ( \gamma_{pa} ) . q ( \gamma_{I} ) \right].
\label{eqrmaxxx}\end{gathered}$$

\
$$r_{PO} = p ( \gamma_{pa} ) \left\lbrace 1 - (1 - \alpha ) \left[ 1 - p ( \gamma_{I} ) 