Section 5.2.5: Mixed strategies for matrix games (LP formulation)
randn('state',0);
n = 12;
m = 12;
P = randn(n,m);
fprintf(1,'Computing the optimal strategy for player 1 ... ');
cvx_begin
variables u(n) t1
minimize ( t1 )
u >= 0;
ones(1,n)*u == 1;
P'*u <= t1*ones(m,1);
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Computing the optimal strategy for player 2 ... ');
cvx_begin
variables v(m) t2
maximize ( t2 )
v >= 0;
ones(1,m)*v == 1;
P*v >= t2*ones(n,1);
cvx_end
fprintf(1,'Done! \n');
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: ');
[t1 t2]
disp('They are equal as expected!');
Computing the optimal strategy for player 1 ...
Calling sedumi: 25 variables, 13 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 13, order n = 27, dim = 27, blocks = 2
nnz(A) = 180 + 0, nnz(ADA) = 169, nnz(L) = 91
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.24E+00 0.000
1 : -3.40E-01 3.71E+00 0.000 0.4499 0.9000 0.9000 2.34 1 1 6.9E+00
2 : 1.50E-02 1.60E+00 0.000 0.4309 0.9000 0.9000 4.69 1 1 1.0E+00
3 : 3.90E-02 4.27E-01 0.000 0.2673 0.9000 0.9000 1.49 1 1 2.2E-01
4 : 4.34E-02 8.53E-02 0.000 0.1997 0.9000 0.9000 1.12 1 1 4.3E-02
5 : 4.47E-02 6.94E-03 0.000 0.0814 0.9900 0.9900 1.03 1 1 3.7E-03
6 : 4.48E-02 5.72E-05 0.000 0.0082 0.9990 0.9804 1.01 1 1 4.5E-05
7 : 4.48E-02 1.45E-11 0.000 0.0000 1.0000 1.0000 1.00 1 1 2.2E-11
iter seconds digits c*x b*y
7 0.0 10.0 4.4840422224e-02 4.4840422219e-02
|Ax-b| = 3.6e-12, [Ay-c]_+ = 1.8E-12, |x|= 8.5e-01, |y|= 4.4e-01
Detailed timing (sec)
Pre IPM Post
0.000E+00 3.000E-02 1.000E-02
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.43441.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.0448404
Done!
Computing the optimal strategy for player 2 ...
Calling sedumi: 25 variables, 13 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 13, order n = 27, dim = 27, blocks = 2
nnz(A) = 180 + 0, nnz(ADA) = 169, nnz(L) = 91
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.24E+00 0.000
1 : -3.55E-01 3.64E+00 0.000 0.4423 0.9000 0.9000 2.39 1 1 6.7E+00
2 : -4.06E-02 1.65E+00 0.000 0.4525 0.9000 0.9000 4.70 1 1 1.1E+00
3 : -4.47E-02 4.55E-01 0.000 0.2760 0.9000 0.9000 1.52 1 1 2.4E-01
4 : -4.51E-02 8.72E-02 0.000 0.1916 0.9000 0.9000 1.13 1 1 4.4E-02
5 : -4.49E-02 1.66E-02 0.000 0.1900 0.9000 0.9000 1.03 1 1 8.5E-03
6 : -4.49E-02 6.66E-04 0.000 0.0402 0.9900 0.8430 1.01 1 1 5.0E-04
7 : -4.48E-02 1.32E-08 0.000 0.0000 1.0000 1.0000 1.00 1 1 1.6E-08
8 : -4.48E-02 3.35E-15 0.000 0.0000 1.0000 1.0000 1.00 1 1 5.7E-15
iter seconds digits c*x b*y
8 0.0 13.7 -4.4840422221e-02 -4.4840422221e-02
|Ax-b| = 4.0e-15, [Ay-c]_+ = 8.6E-16, |x|= 1.1e+00, |y|= 4.0e-01
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.69978.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.0448404
Done!
------------------------------------------------------------------------
The optimal strategies for players 1 and 2 are respectively:
0.2695 0.0686
0.0000 0.1619
0.0973 -0.0000
0.1573 0.2000
0.1145 -0.0000
0.0434 0.1545
0.0000 0.1146
0.0000 0.0000
0.2511 0.1030
0.0670 -0.0000
-0.0000 -0.0000
0.0000 0.1974
The expected payoffs for player 1 and player 2 respectively are:
ans =
-0.0448 -0.0448
They are equal as expected!