Consider a zero-sum game between Player A and Player B. The m×n payoff matrix for Player A isAwhere the element aijin row i and column j represents the payoff that Player A receives when Player A chooses strategy Ai and Player B chooses strategy Bj. The indexes i = 1,...,m and j = 1,...,n where m and n are positive integers.

Let the optimal mixed strategy for Player A be p∗ and let the optimal mixed strategy for Player B be q∗. Assume the value of the game v is greater than zero.

- (i) Write down the Linear Programming (LP) problem for each player. (Proofs are

not required. However, you should give clear definitions of any extra notation used.)

- (ii) Reconsider the payoff matrix with m= n. Assume matrix A is non-singular. Using the LP problems of Part (b)(i), show that

p∗ = v(AT)^{−1}u and q∗ = vA^{−1}u

where u ∈ Rn is the vector whose entries are all 1.

I need help with the second part of this question,

for part i, I get

$ minimise \sum_{i=1}^{m} x_{i} = \frac{1}{v} subject to \sum_{i=1}^{m}a_{ij} x_{i} \geq 1 $