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

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Discussion Overview

The discussion centers around solving a system of first-order partial differential equations (PDEs) derived from a game theory model. The participant seeks assistance in finding solutions that relate to achieving Nash equilibrium and Pareto optimality in a two-player scenario, with potential extensions to N players.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant presents a system of first-order PDEs and seeks help in solving them, emphasizing the connection to game theory and the goal of aligning Nash equilibrium with Pareto optima.
  • Another participant identifies the equations as linear first-order PDEs and expresses uncertainty about solving them, requesting input on analytical or numerical methods, specifically mentioning the use of MATLAB.
  • A third participant inquires about the functional forms of F1 and F2 or a table of values, suggesting that treating the equations as separate PDEs might simplify the problem since each equation does not depend on the other variable.
  • The original poster later reports success in finding a solution by transforming the variables t1 and t2 into a single variable A, leading to a system of ordinary differential equations (ODEs) that they claim is functioning well.

Areas of Agreement / Disagreement

Participants express differing views on the approach to solving the system, with some suggesting separation of the equations while others explore variable transformation. The discussion remains unresolved regarding the best method for solving the original PDEs.

Contextual Notes

The discussion does not clarify the specific forms of F1 and F2, nor does it resolve the implications of the variable transformation on the original PDEs. There is also no consensus on the most effective solution method.

Who May Find This Useful

Readers interested in mathematical modeling in game theory, particularly those dealing with systems of PDEs or ODEs, may find this discussion relevant.

UpperGround
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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.
 
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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.
 
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.
 
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!
 

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