I am using a statistical mechanics framework to solve an evolutionary game theoretical problem where different individuals can be in a +1 state (cooperate) or 0 state (defect). Using an Ising-model-inspired mean field approach, I obtain the following equation for the average state:
$$ \bar{\sigma} = \frac{1}{N}\sum_{i=1}^N\frac{1}{1+e^{-\beta(C_i \bar{\sigma}-l)}} $$
Where $C_i$ is a constant which describe certain properties of each individual agent, $l$ is a property of the system, and $N>>1$ is the number of agents. The distribution of $C_i$ is known.
Is there a way of solving this equation analytically, or do I have to use a numerical approach?
Many thanks in advance for your time and help.
