```% Boyd & Vandenberghe "Convex Optimization"
% Joëlle 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!');
```
```Computing the optimal strategy for player 1 ...
Calling Mosek 9.1.9: 21 variables, 11 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
Name                   :
Objective sense        : min
Type                   : LO (linear optimization problem)
Constraints            : 11
Cones                  : 0
Scalar variables       : 21
Matrix variables       : 0
Integer variables      : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00
Lin. dep.  - tries                  : 1                 time                   : 0.00
Lin. dep.  - number                 : 0
Presolve terminated. Time: 0.00
Problem
Name                   :
Objective sense        : min
Type                   : LO (linear optimization problem)
Constraints            : 11
Cones                  : 0
Scalar variables       : 21
Matrix variables       : 0
Integer variables      : 0

Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 11
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 22                conic                  : 0
Optimizer  - Semi-definite variables: 0                 scalarized             : 0
Factor     - setup time             : 0.00              dense det. time        : 0.00
Factor     - ML order time          : 0.00              GP order time          : 0.00
Factor     - nonzeros before factor : 66                after factor           : 66
Factor     - dense dim.             : 0                 flops                  : 2.07e+03
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME
0   1.8e+01  2.0e+00  2.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   4.0e+00  0.00
1   6.0e+00  6.7e-01  6.7e-01  2.98e+00   -6.260857828e-03  -3.318002505e-02  1.3e+00  0.01
2   1.5e+00  1.7e-01  1.7e-01  1.54e+00   -1.498451422e-02  -3.920997883e-02  3.4e-01  0.01
3   1.4e-01  1.6e-02  1.6e-02  1.22e+00   -2.819674705e-02  -3.036556243e-02  3.1e-02  0.01
4   2.2e-02  2.5e-03  2.5e-03  1.01e+00   -2.741266890e-02  -2.774982821e-02  4.9e-03  0.01
5   5.3e-04  5.9e-05  5.9e-05  1.01e+00   -2.786114217e-02  -2.786920623e-02  1.2e-04  0.01
6   1.4e-07  1.5e-08  1.5e-08  1.00e+00   -2.785588022e-02  -2.785588233e-02  3.1e-08  0.01
Basis identification started.
Primal basis identification phase started.
Primal basis identification phase terminated. Time: 0.00
Dual basis identification phase started.
Dual basis identification phase terminated. Time: 0.00
Basis identification terminated. Time: 0.00
Optimizer terminated. Time: 0.01

Interior-point solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: -2.7855880224e-02   nrm: 1e+00    Viol.  con: 2e-08    var: 0e+00
Dual.    obj: -2.7855882328e-02   nrm: 6e-01    Viol.  con: 0e+00    var: 2e-09

Basic solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: -2.7855878954e-02   nrm: 1e+00    Viol.  con: 3e-16    var: 0e+00
Dual.    obj: -2.7855882328e-02   nrm: 6e-01    Viol.  con: 0e+00    var: 4e-09
Optimizer summary
Optimizer                 -                        time: 0.01
Interior-point          - iterations : 6         time: 0.01
Basis identification  -                        time: 0.00
Primal              - iterations : 0         time: 0.00
Dual                - iterations : 0         time: 0.00
Clean primal        - iterations : 0         time: 0.00
Clean dual          - iterations : 0         time: 0.00
Simplex                 -                        time: 0.00
Primal simplex        - iterations : 0         time: 0.00
Dual simplex          - iterations : 0         time: 0.00
Mixed integer           - relaxations: 0         time: 0.00

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0278559

Done!
Computing the optimal strategy for player 2 ...
Calling Mosek 9.1.9: 21 variables, 11 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
Name                   :
Objective sense        : min
Type                   : LO (linear optimization problem)
Constraints            : 11
Cones                  : 0
Scalar variables       : 21
Matrix variables       : 0
Integer variables      : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00
Lin. dep.  - tries                  : 1                 time                   : 0.00
Lin. dep.  - number                 : 0
Presolve terminated. Time: 0.00
Problem
Name                   :
Objective sense        : min
Type                   : LO (linear optimization problem)
Constraints            : 11
Cones                  : 0
Scalar variables       : 21
Matrix variables       : 0
Integer variables      : 0

Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 11
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 22                conic                  : 0
Optimizer  - Semi-definite variables: 0                 scalarized             : 0
Factor     - setup time             : 0.00              dense det. time        : 0.00
Factor     - ML order time          : 0.00              GP order time          : 0.00
Factor     - nonzeros before factor : 66                after factor           : 66
Factor     - dense dim.             : 0                 flops                  : 2.07e+03
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME
0   1.8e+01  2.0e+00  2.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   4.0e+00  0.00
1   5.5e+00  6.1e-01  6.1e-01  3.11e+00   -2.671244704e-03  -4.496656348e-02  1.2e+00  0.01
2   2.4e+00  2.6e-01  2.6e-01  1.50e+00   -5.119488993e-03  -2.968767703e-02  5.2e-01  0.01
3   4.1e-01  4.6e-02  4.6e-02  1.15e+00   1.932696704e-02   1.544161683e-02   9.1e-02  0.01
4   7.0e-02  7.8e-03  7.8e-03  1.01e+00   2.782806398e-02   2.712182366e-02   1.6e-02  0.01
5   5.8e-03  6.5e-04  6.5e-04  1.02e+00   2.768917711e-02   2.762813376e-02   1.3e-03  0.01
6   2.1e-05  2.3e-06  2.3e-06  1.00e+00   2.785629421e-02   2.785601048e-02   4.6e-06  0.01
7   2.1e-09  2.3e-10  2.3e-10  1.00e+00   2.785587900e-02   2.785587897e-02   4.6e-10  0.01
Basis identification started.
Primal basis identification phase started.
Primal basis identification phase terminated. Time: 0.00
Dual basis identification phase started.
Dual basis identification phase terminated. Time: 0.00
Basis identification terminated. Time: 0.00
Optimizer terminated. Time: 0.01

Interior-point solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: 2.7855878996e-02    nrm: 1e+00    Viol.  con: 3e-10    var: 0e+00
Dual.    obj: 2.7855878967e-02    nrm: 7e-01    Viol.  con: 0e+00    var: 9e-12

Basic solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: 2.7855878954e-02    nrm: 1e+00    Viol.  con: 2e-16    var: 0e+00
Dual.    obj: 2.7855878967e-02    nrm: 7e-01    Viol.  con: 0e+00    var: 5e-11
Optimizer summary
Optimizer                 -                        time: 0.01
Interior-point          - iterations : 7         time: 0.01
Basis identification  -                        time: 0.00
Primal              - iterations : 0         time: 0.00
Dual                - iterations : 0         time: 0.00
Clean primal        - iterations : 0         time: 0.00
Clean dual          - iterations : 0         time: 0.00
Simplex                 -                        time: 0.00
Primal simplex        - iterations : 0         time: 0.00
Dual simplex          - iterations : 0         time: 0.00
Mixed integer           - relaxations: 0         time: 0.00

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0278559

Done!
------------------------------------------------------------------------
The optimal strategies for players 1 and 2 are respectively:
0.1804    0.0000
0.0000    0.3254
-0.0000    0.0924
0.1549    0.0000
0.1129   -0.0000
-0.0000    0.0264
-0.0000    0.4099
0.1003    0.0509
0.1474    0.0949
0.3040   -0.0000

The expected payoffs for player 1 and player 2 respectively are:

ans =

0.0279    0.0279

They are equal as expected!
```