# Do I have this two-person zero-sum game linear program set up properly?

cvxopt (which has a Python interface to the GLPK optimizer) has a unique input form for its linear programs:

$$\min c^Tx\: s.t\\\begin{array}{rcl}Ax & = & b\\Gx + s & = & h\\s&\geq& 0\end{array}$$

So any inequality constraints are input into $G$ and $h$, using slack variables $s$, whereas any equality constraints are input into $A$ and $b$.

I have the following two-person zero sum game:

$$\left[\begin{array}{rrrr}-5&3&1&8\\5&5&4&6\\-4&6&0&5\end{array}\right]$$

Question: am I setting up the linear program properly?

$$\min c^T\left[\begin{array}{rrrrrr}0&0&0&0&1&-1 \end{array} \right ]\cdot x\left[\begin{array}{r}y_1\\y_2\\y_3\\y_4\\v^+\\v^-\\\end{array}\right]\: s.t.\\\begin{array}{lcrcrcl}G\left[\begin{array}{rrrrrr}-5&3&1&8&-1&1\\5&5&4&6&-1&1\\-4&6&0&5&-1&1\end{array}\right] & \cdot & x\left[\begin{array}{r}y_1\\y_2\\y_3\\y_4\\v^+\\v^-\\\end{array}\right] & + & s\left[\begin{array}{r}s_1\\s_2\\s_3\end{array}\right] & = & h\left[\begin{array}{r}0\\0\\0\\\end{array}\right] \\ A\left[\begin{array}{rrrrrr}1&1&1&1&0&0 \end{array} \right ] & \cdot & x\left[\begin{array}{r}y_1\\y_2\\y_3\\y_4\\v^+\\v^-\\\end{array}\right] & & & = & b\left[\begin{array}{r}1\\\end{array}\right] \\ &&&&s&\geq &0 \end{array}$$

I ask because GLPK is saying that I have an unbounded solution. Note that I split the value of the game into $v^+$ and $v^-$ so that the code can work even if I get a matrix with all negative payoffs. BTW, the optimal value is 4 and $y_3 = 1$.

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