%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Example of computation of Stackelberg strategies for % bimatrix games. % % Author: K. Passino % Version: 2/5/02 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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): % 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 uniqueness in the response of P2. 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] % One set of payoff matrices that for i=4 by P1 gives two possible responses from the follower P2 % that both achieve minimum loss for P2. 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] % Compute the Stackelberg strategy: % First, find follower reactions (note that this just finds the min loss by P2, so it can % be that more than one P2 strategy results in this loss): for i=1:m minJ2(i)=min(J2(i,:)); % Finds the minimum loss of P2 given the leader P1 chooses i end P1loss=-inf*ones(1,m); % Initialization so that it will pick the first min value above it jstar=0*ones(1,m); % Initialization to nonvalid values (valid ones picked next) for i=1:m for j=1:n if J2(i,j)==minJ2(i) % Test that j corresponds to a min point for P2, for a given i if J1(i,j)>=P1loss(i) % Tests if it is above any previously stored value for the loss % (this finds the security value against all possible P2 rational reactions) P1loss(i)=J1(i,j); % Keeps the value to compare to any other possible P2 reactions later jstar(i)=j; % Save the index of the reaction of P2 that results in worst loss for P1 if it uses i end end end end % Specify the Stackelberg strategy: [stackelbergcost,istar]=min(P1loss) % Display the P1 Stackelberg strategy and cost jstar(istar) % Display the P2 follower strategy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 Stackelberg strategy: % First, find follower reactions (note that this just finds the min loss by P2, so it can % be that more than one P2 strategy results in this loss): for i=1:m minJ2(i)=min(J2(i,:)); % Finds the minimum loss of P2 given the leader P1 chooses i end P1loss=-inf*ones(1,m); % Initialization so that it will pick the first min value above it jstar=0*ones(1,m); % Initialization to nonvalid values (valid ones picked next) for i=1:m for j=1:n if J2(i,j)==minJ2(i) % Test that j corresponds to a min point for P2, for a given i if J1(i,j)>=P1loss(i) % Tests if it is above any previously stored value for the loss % (this finds the security value against all possible P2 rational reactions) P1loss(i)=J1(i,j); % Keeps the value to compare to any other possible P2 reactions later jstar(i)=j; % Save the index of the reaction of P2 that results in worst loss for P1 if it uses i end end end end % Specify the Stackelberg strategy: [stackelbergcost,istar]=min(P1loss) % Display the P1 Stackelberg strategy and cost jstar(istar) % Display the P2 follower strategy theta1(istar) theta2(jstar(istar)) figure(1) clf contour(theta2,theta1,J1,14) hold on contour(theta2,theta1,J2,14) hold on plot(theta2,R1,'k-') hold on plot(R2,theta1,'k--') hold on plot(theta2(jstar(istar)),theta1(istar),'x') xlabel('\theta^2') ylabel('\theta^1') title('J_1, J_2, R_1 (-), R_2 (--), "x" marks Stackelberg solution') hold off %------------------------------------- % End of program %-------------------------------------