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

In summary, the conversation discusses a system of first-order PDEs that were obtained through game theory modeling. The goal is to achieve the Nash equilibrium and Pareto optima. The equations are linear and there are two players involved. Suggestions for solving the equations include using a functional form or table of values for F1 and F2, and considering the equations separately. The solution was eventually found by replacing t1 and t2 with the variable A and solving a system of ODEs.
  • #1
UpperGround
3
0
Hello,

I have been struggling at solving what I think is a system of 1st order PDEs. Here is what I have:
[tex]\frac{dy1}{dt1}[/tex] = y1*F1(t1,t2) + F2(t1,t2)
[tex]\frac{dy2}{dt2}[/tex] = 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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
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 :
[tex]\frac{dy1}{dA}*\frac{dA}{dt1}[/tex]
[tex]\frac{dy2}{dA}*\frac{dA}{dt2}[/tex]

All is good, everything is working perfectly!
 

1. What is a system of first order PDEs?

A system of first order partial differential equations (PDEs) is a set of equations that involve multiple unknown functions and their partial derivatives with respect to one or more independent variables. These equations are often used to model complex systems in various fields of science and engineering.

2. What is the difference between first order PDEs and second order PDEs?

The main difference between first order PDEs and second order PDEs is the highest order of derivative present in the equations. First order PDEs involve first-order derivatives, while second order PDEs involve second-order derivatives. This means that first order PDEs have one independent variable, while second order PDEs have two independent variables.

3. What are some common applications of systems of first order PDEs?

Systems of first order PDEs are used to model a wide range of phenomena in fields such as physics, engineering, and finance. Some common applications include fluid dynamics, heat transfer, electromagnetism, population dynamics, and option pricing in financial markets.

4. How are systems of first order PDEs solved?

Solving a system of first order PDEs involves finding the unknown functions that satisfy all of the equations in the system. This can be done analytically using methods such as separation of variables or by using numerical methods such as finite difference or finite element methods.

5. What are some challenges in solving systems of first order PDEs?

One of the main challenges in solving systems of first order PDEs is the high dimensionality of the problem. This can lead to complex mathematical expressions and make it difficult to find analytical solutions. Additionally, numerical methods may require a significant amount of computational resources and may be prone to errors if not implemented carefully.

Similar threads

First Order PDE Solution Method Issues
    • Differential Equations
Replies
1
Views
2K
RK4 for a system of 2 second order DEs
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
What Methods Can Be Used to Solve Coupled First Order PDEs?
    • Differential Equations
Replies
2
Views
3K
Unsuccessful attempts to solve a linear second order PDE system
    • Differential Equations
Replies
6
Views
4K
How do you solve a system of equations with 4 variables and 4 equations?
    • General Math
Replies
6
Views
2K
Solving First-Order Linear Differential Equations with Eigenvectors
    • Calculus and Beyond Homework Help
Replies
1
Views
4K
How Can Thick Lens Matrix Methods Locate Image Positions?
    • Introductory Physics Homework Help
Replies
2
Views
4K

Hot Threads

Recent Insights

Back
Top