NE equilibrium with lobbying of cournot producers I have had an exam (exam is now past and submitted, but I want to now understand the solution without waiting) with the following questions:

GAME

Consider two firms playing the following two-stage game: 

Firm face the following inverse demand:

$$
P(Q) = \frac{S+s_1 + s_2}{(Q+k)^{\alpha}}, Q=q_1 + q_2.
$$

in the first stage, firms can simultaneously lobby to ease trade restrictions by increasing $s_i$, for each unit of $s_i$ each firm pays $\frac{s_i^3}{9}$.

In the second stage they observe each other lobbying choice and set quantity simultaneously. Production costs are zero.

Let $\alpha = 3$ and $k=1$. Show that there is a NE with $S+s_1 +s_2 &gt; S + s_1^{spne} + s_2^{spne}$ and explain why this is NOT a subgame perfect.

SPNE

SPNE is easy to find by backward induction, we know that in the second stage firms profits are

$$\pi_i = \frac{S'}{(Q+k)^{\alpha}}q_i - \frac{s_i^3}{9},$$

$S'$ is a fixed constant at this stage, so by differentiating we find the best responses and the optimal quantities

$$
q_1 = (q_2 + k)/(\alpha-1)\to  q^*_i = \frac{k}{\alpha-1}.
$$.

Anticipating this, at stage one, firm will want to maximize

$$\pi_i = \frac{S'}{(Q^*+k)^{\alpha}}q^*_i - \frac{s_i^3}{9},$$

deriving in $s_i$ we find the solution to be

$$
s^*_i = \sqrt{\frac{3q^*_i}{(Q^* +k)^{\alpha}}}
$$

Another SPNE?

I have tried finding an NE, but could not. Only thing I can find is what I think is another SPNE in which strategies are:

Both play $(\hat{s}, q_i^*)$ where $\hat{s} &gt; s_i^*$ . If the other player deviates in the first stage, then the other will punish the other in the second stage by producing some

$$ q^p $$

such that $$\pi_1(\hat{s},s_2^* ,q^p,q_2^*) = \pi^{spne}$$ while
$$\pi_2(\hat{s},s_2^* ,q^p,q_2^*) &lt; \pi_2(\hat{s},\hat{s} ,q_1^* ,q_2^*) $$ This should be -- assuming there is such a $q^p$ -- a credible threat since the player can get the same as the previous SPNE payoff and effective since it lowers player 2 profits by lowering the demand it receives.

Is this is an equilibrium at all and is it an SPNE or just a NE?
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.