Consider the system below of 13 equations with unknowns $\alpha, p_1, p_2, p_3, p_4, q_1, q_2, q_3, q_4$, each belonging to $[0,1]$. Could you help me to show that this system has a unique solution? $$ \begin{aligned} &q_1\Big[ (1-\alpha)^2 p_1 + \alpha^2 (1-p_1)\Big]=x_1\\ &q_1\Big[ \alpha(1-\alpha) p_1 +(1-\alpha) \alpha (1-p_1)\Big]=x_2\\ &q_1\Big[ \alpha^2 p_1 +(1-\alpha)^2 (1-p_1)\Big]=x_3\\ &q_2\Big[ (1-\alpha)^2 p_2 + \alpha^2 (1-p_2)\Big]=x_4\\ &q_2\Big[ \alpha(1-\alpha) p_2 +(1-\alpha) \alpha (1-p_2)\Big]=x_5\\ &q_2\Big[ \alpha^2 p_2 +(1-\alpha)^2 (1-p_2)\Big]=x_6\\ &q_3\Big[ (1-\alpha)^2 p_3 + \alpha^2 (1-p_3)\Big]=x_7\\ &q_3\Big[ \alpha(1-\alpha) p_3 +(1-\alpha) \alpha (1-p_3)\Big]=x_8\\ &q_3\Big[ \alpha^2 p_3 +(1-\alpha)^2 (1-p_3)\Big]=x_9\\ &q_4\Big[ (1-\alpha)^2 p_4 + \alpha^2 (1-p_4)\Big]=x_{10}\\ &q_4\Big[ \alpha(1-\alpha) p_4 +(1-\alpha) \alpha (1-p_4)\Big]=x_{11}\\ &q_4\Big[ \alpha^2 p_4 +(1-\alpha)^2 (1-p_4)\Big]=x_{12}\\ &q_1+q_2+q_3+q_4=1 \end{aligned} $$
Also, each $x_j$ is a number in $(0,1)$ and $\sum_{j=1}^{12}x_j=1$
Some considerations:
(1) If needed to have one unique solution, we can restrict all the parameters to $(0,1)$ and $\alpha>1/2$.
(2) The left-hand side of the 12 equations exhibits some patterns. In fact, it can be more compactly written as: $$ \begin{aligned} & q_i\Big[ \alpha^j(1-\alpha)^h p_i +(1-\alpha)^j \alpha^h (1-p_i)\Big]\\ & i\in \{1,2,3,4\}, (j,h)\in \{\{0,1,2\}^2: j+h=2\} \end{aligned} $$