matrix_games.m
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matrix_games.m
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			% Section 5.2.5: Mixed strategies for matrix games
			% Boyd & Vandenberghe "Convex Optimization"
			% Jo?le Skaf - 08/24/05
			%
			% Player 1 wishes to choose u to minimize his expected payoff u'Pv, while
			% player 2 wishes to choose v to maximize u'Pv, where P is the payoff
			% matrix, u and v are the probability distributions of the choices of each
			% player (i.e. u>=0, v>=0, sum(u_i)=1, sum(v_i)=1)

			% Input data
			randn('state',0);
			n = 10;
			m = 10;
			P = randn(n,m);

			% Optimal strategy for Player 1
			fprintf(1,'Computing the optimal strategy for player 1 ... ');

			cvx_begin
			variable u(n)
			minimize ( max ( P'*u) )
			u >= 0;
			ones(1,n)*u == 1;
			cvx_end

			fprintf(1,'Done! \n');
			obj1 = cvx_optval;

			% Optimal strategy for Player 2
			fprintf(1,'Computing the optimal strategy for player 2 ... ');

			cvx_begin
			variable v(m)
			maximize ( min (P*v) )
			v >= 0;
			ones(1,m)*v == 1;
			cvx_end

			fprintf(1,'Done! \n');
			obj2 = cvx_optval;

			% Displaying results
			disp('------------------------------------------------------------------------');
			disp('The optimal strategies for players 1 and 2 are respectively: ');
			disp([u v]);
			disp('The expected payoffs for player 1 and player 2 respectively are: ');
			[obj1 obj2]
			disp('They are equal as expected!');