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%
% Example of computation of minimax (and Nash) strategies for 
% bimatrix games. Note that it only computes one minimax 
% solution. How would you modify the program to have it compute
% all the minimax solutions?
%
% Author: K. Passino
% Version: 2/5/02
%
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clear all

% Set the number of different possible values of the decision variables

m=5; % If change will need to modify specific J1 value chosen below
n=3; 

% Set the payoff matrices J1(i,j) and J2(i,j):

% One set of payoff matrices that gives two Nash equilibria (1,3) and (4,1) is as follows. 
% Note that for these payoff matrices the minimax strategy is not a Nash solution.
J1 =[-1     5    -3;
    -2     5     1;
     4     3    -2;
    -5    -1     5;
     3     0     2]

J2 =[-1     2    -3;
     2    -3     1;
    -2     3     1;
    -1     1    -1;
     4    -4     1]

% For generating random bimatrix games:
J1=round(10*rand(m,n)-5*ones(m,n)) % Make it random integers between -5 and +5
J2=round(10*rand(m,n)-5*ones(m,n)) % Make it random integers between -5 and +5

% These can be used to find other equilibria. For instance, if you use the following two
% payoff matrices then you will get one (admissible) Nash equilibrium which is then the
% minimax strategy also.  This shows that for some games the minimax strategy 
% is quite good.
J1 =[4     1    -4;
    -2     5     3;
    -3     2    -1;
     4     4     4;
    -3    -5     2]

J2 =[2    -1     0;
    -2     4     0;
    -3     0    -1;
    -3     3     4;
    -3     0    -5]

	 
% Compute the minimax strategy (the name "minimax" clearly arises from the computations involved in 
% each player trying to minimize its maximum loss):

% First, compute the security strategy for player 1

[maxvals1,indexmax1]=max(J1'); % This is the max value for each row (note transpose)
[minimaxvalue1,secstratP1]=min(maxvals1) % Display the security value of P1 and its security strategy
                                         % (P1 loses no more than minimaxvalue1, no matter what strategy 
										 % P2 uses)


% Second, compute the security strategy for player 2 (note difference from computation of security
% strategies for matrix games since both payoff matrices are "loss" matrices, and note that computation of the
% minimax strategies is completely independent)

[maxvals2,indexmax2]=max(J2); % This is the min value for each column (note no transpose)
[minimaxvalue2,secstratP2]=min(maxvals2) % Display the security value of P2 and its security strategy

% The outcome of the game will be (with a minimax strategy)

outcome1=J1(secstratP1,secstratP2)
outcome2=J2(secstratP1,secstratP2)

% Compute the Nash equilibria:

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
			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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% Do another example:

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)=-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


% Compute the minimax strategy (the name "minimax" clearly arises from the computations involved in 
% each player trying to minimize its maximum loss):

% First, compute the security strategy for player 1

[maxvals1,indexmax1]=max(J1'); % This is the max value for each row (note transpose)
[minimaxvalue1,secstratP1]=min(maxvals1) % Display the security value of P1 and its security strategy
                                         % (P1 loses no more than minimaxvalue1, no matter what strategy 
										 % P2 uses)
theta1(secstratP1)

% Second, compute the security strategy for player 2 (note difference from computation of security
% strategies for matrix games since both payoff matrices are "loss" matrices, and note that computation of the
% minimax strategies is completely independent)

[maxvals2,indexmax2]=max(J2); % This is the min value for each column (note no transpose)
[minimaxvalue2,secstratP2]=min(maxvals2) % Display the security value of P2 and its security strategy

theta2(secstratP2)

% The outcome of the game will be (with a minimax strategy)

outcome1=J1(secstratP1,secstratP2)
outcome2=J2(secstratP1,secstratP2)



figure(1)
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--')
hold on
plot(theta2(secstratP2),theta1(secstratP1),'x')
xlabel('\theta_2')
ylabel('\theta_1')
title('J_1, J_2, R_1 (-), R_2 (--), "x" marks a minimax solution')
hold off

%-------------------------------------
% End of program
%-------------------------------------