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

[adam512]

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.

Thanks in advance!

 

[Jhero]

</code>It sounds like you are trying to solve a system of eight equations with eight unknowns. To do this, you will need to use a numerical method, such as Newton's method. Newton's method is an iterative process that works by taking an initial guess for the solution and then updating it using a series of steps. In each step, the derivative of the system of equations is calculated and used to update the guess for the solution. This process is repeated until the solution converges to a value that satisfies all of the equations.You can find out more about Newton's method in many textbooks on numerical methods. Additionally, there are many online resources, such as tutorials and videos, that can help you understand how to implement Newton's method. Finally, if you are using MATLAB, there are built-in functions for solving systems of nonlinear equations that will likely work better than your trial-and-error approach.