I am reading Computing the Optimal Strategy to Commit to by Conitzer and Sandholm recently.
And I have trouble in understanding how to use linear programming to find an optimal mixed strategy for 2-player normal-form games. The authors said in Theorem 2. "In 2-player normal-form games, an optimal mixed strategy to commit to can be found in polynomial time using linear programming."
I have no trouble in understand the statement and the proof of Theorem 2. But actually confused by how to solve the linear program below.
Specifically, how to transform the constraints condition $\sum_{s\in S} p_s u_f(s,t) \ge\sum_{s\in S} p_s u_f(s,t') $ to a detail inequality. Like $a_1x_1 + \cdots a_nx_n \le b$ in canonical form (Standard form) (https://en.wikipedia.org/wiki/Linear_programming) of linear programming. Then maybe I can solve it by simplex algorithm.
Thanks in advance!