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%
% Example to illustrate reaction curves, infinite games, and
% Nash equilibria.
%
% Author: K. Passino
% Version: 2/5/02
%
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clear all
% Define payoff (cost) functions for each player
theta1=-4:0.05:4; % Ok, while we think of it as an infinite game computationally
% we of course only study a finite number of points.
m=length(theta1);
theta2=-5:0.05:5;
n=length(theta2);
% A set of cost functions, J1 for player 1 and J2 for player 2
for ii=1:length(theta1)
for jj=1:length(theta2)
J1(ii,jj)=-1*(exp( (-(theta1(ii)-2)^2)/8 + ((-(theta2(jj)-4)^2)/2)));
J2(ii,jj)=-1*(exp( (-(theta1(ii)-1)^2)/1 + ((-(theta2(jj)+1)^2)/6)));
end
end
% Next, compute the reaction curves for each player and will plot on top of J1 and J2
for j=1:length(theta2)
[temp,t1]=min(J1(:,j)); % Find the min point on J1 with theta2 fixed, for all theta2
R1(j)=theta1(t1); % Compute the theta1 value that is the best reaction to each theta2
end
for i=1:length(theta1)
[temp,t2]=min(J2(i,:)); % Find the min point on J2 with theta1 fixed
R2(i)=theta2(t2); % Compute the theta2 value that is the best reaction to each theta1
end
figure(1)
clf
subplot(121)
contour(theta2,theta1,J1,10)
hold on
contour(theta2,theta1,J2,10)
hold on
plot(theta2,R1,'k-')
hold on
plot(R2,theta1,'k--')
xlabel('\theta^2')
ylabel('\theta^1')
title('(a) J_1, J_2, reaction curves R_1 (-) and R_2 (--)')
hold off
% Compute the Nash equilibria (with the view that it is a bimatrix game):
flag=0; % Flag for saying if there is no Nash equilibria
for i=1:m
for j=1:n
if J1(i,j)<=min(J1(:,j)) & J2(i,j)<=min(J2(i,:)), % Conduct two inequality tests
display('Nash equilibrium and outcome:') % If satisfied, then diplay solution
i
theta1(i)
j
theta2(j)
J1(i,j)
J2(i,j)
flag=1; % Indicates that there was one Nash equilibrium (or more)
end
end
end
if flag==0
display('There were no Nash equilibria')
end
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% Another possible set of cost functions:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for ii=1:length(theta1)
for jj=1:length(theta2)
J1(ii,jj)=-2*(exp( (-(theta1(ii)-2)^2)/5 +(-4*(theta1(ii)*theta2(jj))/20) + ((-(theta2(jj)-3)^2)/2)));
J2(ii,jj)=-1*(exp( (-(theta1(ii)-1)^2)/4 + (5*(theta1(ii)*theta2(jj))/10) + ((-(theta2(jj)+1)^2)/2)));
end
end
% Next, compute the reaction curves for each player and will plot on top of J1 and J2
for j=1:length(theta2)
[temp,t1]=min(J1(:,j)); % Find the min point on J1 with theta2 fixed, for all theta2
R1(j)=theta1(t1); % Compute the theta1 value that is the best reaction to each theta2
end
for i=1:length(theta1)
[temp,t2]=min(J2(i,:)); % Find the min point on J2 with theta1 fixed
R2(i)=theta2(t2); % Compute the theta2 value that is the best reaction to each theta1
end
figure(1)
subplot(122)
contour(theta2,theta1,J1,10)
hold on
contour(theta2,theta1,J2,10)
hold on
plot(theta2,R1,'k-')
hold on
plot(R2,theta1,'k--')
xlabel('\theta^2')
ylabel('\theta^1')
title('(b) J_1, J_2, reaction curves R_1 (-) and R_2 (--)')
hold off
% Compute the Nash equilibria (with the view that it is a bimatrix game):
flag=0; % Flag for saying if there is no Nash equilibria
for i=1:m
for j=1:n
if J1(i,j)<=min(J1(:,j)) & J2(i,j)<=min(J2(i,:)), % Conduct two inequality tests
display('Nash equilibrium and outcome:') % If satisfied, then diplay solution
i
theta1(i)
j
theta2(j)
J1(i,j)
J2(i,j)
flag=1; % Indicates that there was one Nash equilibrium (or more)
end
end
end
if flag==0
display('There were no Nash equilibria')
end
figure(2)
clf
contour(theta2,theta1,J1,10)
hold on
contour(theta2,theta1,J2,10)
hold on
plot(theta2,R1,'k-')
hold on
plot(R2,theta1,'k--')
xlabel('\theta^2')
ylabel('\theta^1')
title('J_1, J_2, R_1 (-), R_2 (--), and iteration trajectory')
hold off
%-------------------------------------
% End of program
%-------------------------------------