Critical points of system of ODE in MATLAB - Game theory and poker

1. Nov 2, 2013

Hello there,

I hope I'm posting in the right section.

I have been doing some work on evolutionary game theory and poker. I will give a brief description of how I got here.

I have eight strategies $i = 1, 2, \ldots, 8$ and the eight proportions of the population playing each strategy is $x_1, x_2, \ldots , x_8$.

I also have a fitness function $F_{x_i}$ for each strategy.
I have the average fitness of the population $\bar{F} = \sum_{i = 1}^8{x_i(F_{x_i})}$.
Finally I have the replicator equations $\frac{dx_i}{dt} = x_i(F_{x_i}-\bar{F})$.

I now want to find the critical points of the system containing the eight replicator equations. So I set $\frac{dx_i}{dt} = 0$ for all $i = 1, 2, \ldots ,8$. I have tried this using MATLABs function solve, and after much trial and error the best result was a parametric solution in 6 of the eight variables $x_1, x_2, \ldots x_8$ and my instructor was quite confided that I would only find a single non-trivial solution.

If anyone have advice on how I can solve the problem it would be great. I am not that good with MATLAB. I also would like to specify that $x_i > 0$ for all $i$.

OBS: I have all the fitness functions, and they are all functions of all eight variables $x_i$, and they are all linear, but the average fitness $\bar{F}$ and the replicator equations aren't.