I am dealing with a problem in economics, specifically game theory, where $n$ agents have a best response $x_i$ given by an implicit function, as described below.
We have $ i \in \{ 1, 2, ..., n \}$, $x_i \in [0, 1]$, $ \delta \in [0, 1]$, and $t_i > 0$. For each $x_i$ it should hold that $$ x_i = \frac{t_i (1-\delta) \left( 1-\sum_i x_i \right)}{\delta t_i + \left( \sum_i t_i \right) \left( 1-\sum_i x_i \right)}. $$ It could be that $t_i$ is the same for all $i$, but there might very well be a certain level of variance. Is there any possibility to analytically derive the equilibrium state $(x_1^*, x_2^*, ..., x_n^*)$ in which each element satisfies the above condition?