Let $q_i \in Q = \mathbb R_+$ denote the quantity produced by firm $i \in \{1,2\}$. Further let $\pi_i(q_1,q_2) = (1-q_1-q_2)q_i$ denote the profits of $i$. A Nash equilibrium $(q_1^*,q_2^*) \in Q^2$ satisfies \begin{align} &\pi_1(q_1^*,q_2^*) \geq \pi_1(q_1,q_2^*) \quad \forall q_1 \in Q\\ &\pi_2(q_1^*,q_2^*) \geq \pi_2(q_1^*,q_2) \quad \forall q_2 \in Q. \end{align} We are considering symmetric equilibria of the form $q^* = q_1^* = q_2^*$ and therefore apply the symmetric opponents form approach. Define $\pi(q,q^*) = \pi_1(q,q^*)$. There exists a unique symmetric root to the first order condition $\pi_q(q^*,q^*) = 0$ given by $q^* = \frac{1}{3}$.
Claim The candidate $q = \frac{1}{3}$ is the unique symmetric maximizer of $\pi(q,q^*)$.
Problem: The candidate might be a minimum or saddle.
The idea: In economic settings equilibrium quantities are basically restricted by individual rationality, i.e. $\pi(q^*,q^*) = (1-2q^*)q^*$ implies $q^* \in [0,\frac{1}{2}]$. Since $\pi(\frac{1}{3},\frac{1}{3}) = \frac{1}{9} > 0$, the claim follows.
Edit I edit the question to further clarify the issue. Suppose I don't have any information about concavity of $\pi(q,q^*)$ w.r.t. $q$.
A general argument: We need to distinguish 4 cases.
- $q^*$ is a saddle and $\pi(q^*,q^*) > 0$ and $\pi(\infty,q^*) = \infty$.
- $q^*$ is a saddle and $\pi(q^*,q^*) < 0$ and $\pi(\infty,q^*) = -\infty$.
- $q^*$ is a minimum and $\pi(q^*,q^*) < 0$ and $\pi(\infty,q^*) = \infty$.
- $q^*$ is a maximum and $\pi(q^*,q^*) > 0$ and $\pi(\infty,q^*) = -\infty$.
Since case 4 is considered here $\frac{1}{3} = \arg\max_q\pi(q,q^*)$.
