Can You Solve This System of First Order PDEs in Game Theory?

[UpperGround]

Hello,

I have been struggling at solving what I think is a system of 1st order PDEs. Here is what I have:
$$\frac{dy1}{dt1}$$ = y1*F1(t1,t2) + F2(t1,t2)
$$\frac{dy2}{dt2}$$ = y2*F1(t2,t1) + F2(t2,t1)

These equations have been obtained after modeling a problem using the game theory. More specifically, I want the Nash equilibrium to equal the Pareto optima by giving the players additional money if they cooperate (and thus achieving Pareto).

Any tips on how to solve this system of PDE ?

Note : the number of equations equals the number of players. For now, I limit the model to 2 players, but in the future, N players should be considered.

 

[UpperGround]

After review, these equations are linear 1st order PDE.

I have done my ODE & PDE courses 5 years ago and it is still very fuzzy in my head so any input on how to resolve this (either analytically (unlikely) or numerically) would be VERY appreciated.

I am using matlab.

 

[Crosson]

Do you have a functional form for F1 and F2 ? Or at least a table of values?

Although your classification is broad enough to include this set of equations, I think they will be easier to solve if we think of them as separate PDEs, since y1 does not appear in the equation for y2 and y2 does not appear in the equation for y1.

 

[UpperGround]

Thanks for your answer.

Because t1 and t2 are linked through a variable A, I have managed to find the solution by replacing the t1 , t2 variables with A. Therefore I had a system of ODEs :
$$\frac{dy1}{dA}*\frac{dA}{dt1}$$
$$\frac{dy2}{dA}*\frac{dA}{dt2}$$

All is good, everything is working perfectly!