NE equilibrium with lobbying of cournot producers
Since the choice of $q_i$ can be conditioned on $(s_i,s_j)$, strategies in this game are of the form $(\hat s_i, \hat q_i(s_i,s_j))$. For the given values $\alpha=3$ and $k=1$, the SPNE can be calculated as the profile where $s^*_i=1/3$ and $q^*_i\equiv 1$. Indeed, production levels $q_i^*=1$ are the unique NE in all subgames, independently of the chosen $s_i$-levels in the first stage. Therefore your suggested "other SPNE" is not an SPNE.

But consider the following strategy of player 1: $\hat s_1=1/3$ and $\hat q_1(s_1,s_2)=\left\{
\begin{array}\ 1 &amp; \ldots &amp; s_2 = 1/3+\epsilon \\ 2 &amp; \ldots &amp; s_2 \ne 1/3+\epsilon \end{array}
\right\}$, where $\epsilon&gt;0$. Let player 2's strategy be $\hat s_2=1/3+\epsilon$ and $\hat q_2(s_1,s_2)\equiv 1$. Then $(\hat s_1,\hat q_1)$ is a best response to $(\hat s_2,\hat q_2)$, and provided $\epsilon$ is small enough that deviating in the first stage doesn't pay for player 2, $(\hat s_2,\hat q_2)$ is also a best response to $(\hat s_1,\hat q_1)$. Therefore the corresponding strategy profile is a NE with higher total lobbying than in the SPNE (but with the same production levels), benefiting player 1.

However, it is not subgame perfect, since if player 2 deviated in the first stage, player 1 would not carry out his threat to double production in the second stage. Technically speaking, the NE induces production levels $\hat q_1=2$ and $\hat q_2=1$ in all subgames following a deviation of player 2 in the first stage, but these production levels are not in equilibrium in these subgames.