How can I find the vector that simultaneously maximizes my two functions?

Hi everyone, I am new in using matlab, I am using it for my own work, this is the Code:

fa=0:1; %[1:100] ho cambiato per uguagliare dimensione vettori
fh=0:4;
% Frequenze di viaggio
%Fa=1:4;   
Fa=1;
%Fh=1:4;  %[1:10] ho cambiato per uguagliare dimensione vettori
Fh=1;
%Tempi di viaggio
Ta=2;
Th=1;
diffTemp=Ta-Th;
%VOT e costi
lVOT=1;
uVOT=0;
diffVOT=uVOT-lVOT;
ca=1;
Ca=1;
ch=0;
Ch=0;
%for i=2:length(fa);
%    j=2:length(fh);
%   Pa(i,j) = (fa(i)./diffVOT)*((fh(j)-fa(i))/diffTemp)-lVOT-Fa.*ca-Ca
%end
% Calcolo di Pa senza l'uso di cicli
%Pa = (fa./diffVOT).*((fh - fa)./ diffTemp) - lVOT - (Fa.*ca) - Ca
for i=1:length(fa)
    for j=1:length(fh)
        Pa(i,j) = (i/diffVOT).*(((j-i)/diffTemp) - lVOT) - (Fa*ca) - Ca
        Ph(j,i) = ((j/diffVOT).*(uVOT - ((j-i)/diffTemp))) - (Fh*ch) - Ch        
    end 
end

In this way i'm able to calculate the value of the function for the different combinations of (fa,fh), but my problem consists in the following: I have two players "a" and "h", each of them has its own function that expresses the profit "Pa , Ph"; all values in the function are constants, except for "fa" and "fh" which are expressed as vectors of different sizes.

I need to find an x(fa*,fh*) vector made from the value of "fa" and "fh" which simultaneously maximizes the "Pa" and "Ph" profit functions of both players, but I don’t know how to implement this on matlab.

Can anyone help me?

[Walter Roberson]

I suggest you use Problem Based Optimization https://www.mathworks.com/help/optim/problem-based-basics.html

Answer 1

[Hassaan]

% Clear workspace, close all figures, and clear command window
clear; close all; clc;

% Define constants
diffTemp = 2 - 1; % Ta - Th, example values
lVOT = 1;
uVOT = 0;
diffVOT = uVOT - lVOT;
Fa = 1;
Fh = 1;
ca = 1;
Ca = 1;
ch = 0;
Ch = 0;

% Define optimization variables
fa = optimvar('fa', 'LowerBound', 0, 'UpperBound', 1); % Adjust bounds as necessary
fh = optimvar('fh', 'LowerBound', 0, 'UpperBound', 4); % Adjust bounds as necessary

% Define the objective functions using anonymous functions
objPa = @(fa, fh) (fa./diffVOT).*((fh - fa)./ diffTemp) - lVOT - (Fa.*ca) - Ca;
objPh = @(fa, fh) ((fh./diffVOT).*(uVOT - ((fh - fa)./ diffTemp))) - (Fh.*ch) - Ch;

% Combine the objectives into a single function (for demonstration)
totalObjective = objPa(fa, fh) + objPh(fa, fh);

% Create the optimization problem
prob = optimproblem('Objective', totalObjective, 'ObjectiveSense', 'maximize');

% Solve the optimization problem
[sol, fval, exitflag, output] = solve(prob);

% Display the solution
disp('Solution:');
disp(sol);
fprintf('Combined Profit: %f\n', fval);
fprintf('Exit Flag: %d\n', exitflag);
fprintf('Solver Output:\n');
disp(output);

• The interaction between two players into a single optimization problem for illustrative purposes. In real-world scenarios, especially in game theory, interactions might require more sophisticated approaches to find equilibria or optimal strategies.
• The effectiveness and applicability of combining objectives like this depend on the nature of the interaction between fa and fh. In cases where their interests are opposed, a more nuanced approach to find a Nash equilibrium or Pareto optimality might be necessary.
• The solve function automatically selects an appropriate solver based on the problem type. For more control or for large-scale problems, consider specifying solver options.

Note:

• It's an initial attempt and may need adjustment as per your requirements.

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[Guglielmo]

Thx for the help, i'm doing this work for my Master Degree Thesys, i need to find the Generalized Nash Eq (GNE) in this game, where the competition is between 2 different mode of transport (a=air, h=high speed rail) for a common path.

I'm quite new to Game Theory and Matlab, and for that reason i'm finding problems, so thanks a lot for ur time

Answer 2

[Torsten]

Usually, you can't find a vector that maximizes both of your functions simultaneously. You can compute the pareto-front for your two functions:

https://uk.mathworks.com/help/gads/pareto-front-for-two-objectives.html