$$\{ s_1 , s_2 \} = \{ \Leftarrow , \Rightarrow \}, \qquad
\{ - , + \} = \{ a_1 , a_2 \} ,$$ where we can consider $ | s_i \rangle = \{ | \Leftarrow  \rangle , | \Rightarrow \rangle \} $ and $ | a_i \rangle = \{ | -  \rangle , | + \rangle \} $ for $ N=2 $ in an invasion game.

While the theoretical asymptotic efficiency for a small amount of $ \gamma $ has been derived for a fully observable one defender invasion game [@Briegel1; @Briegel2], that derivation cannot be used for an arbitrary choice of $ \gamma $ even in the simple form of one agent toy problem. The asymptotic efficiency relating to a given agent ”$ D $” that is $r_{max}^D = r_{max}^D ( \alpha , \gamma_{D}, \gamma_{I}) $ occurs with respect to $ \lim_{t \to \infty} h^{(t)} (s,a) $ regarding the adaptation rule of Eq. \[eqtrans\]. This adaptation is changed stochastically, however, in a big enough time, $ t \to \infty $, the averaged efficiencies on a large number of agents ($ m $) reaches a certain asymptote for every forgetting factor $ \gamma $ (see Appendix. \[averagefequa\]). Then one 