I'm trying to find the solution ($x^*_1, y^*_1, x^*_2, y^*_2$) to the following system of equations: $$ gx_1=\lambda\left(\log \frac{x_2}{1-x_2}-\log \frac{x_2 + y_2}{2-x_2-y_2}\right)\\ by_1=\lambda\left(\log \frac{y_2}{1-y_2}-\log \frac{x_2 + y_2}{2-x_2-y_2}\right)\\ bx_2=\lambda\left(\log \frac{x_1}{1-x_1}-\log \frac{x_1 + y_1}{2-x_1-y_1}\right)\\ gy_2=\lambda\left(\log \frac{y_1}{1-y_1}-\log \frac{x_1 + y_1}{2-x_1-y_1}\right)\\ $$
Here, $g>0>b$ and $\lambda>0$. I suspect that this cannot be done analytically, but would love to be corrected. If it can't be done analytically, I'd also like to prove that such a solution must exist. I'd really appreciate any help, or any possible direction in how to attempt this.