On Frank-Wolfe and Equilibrium Computation (Supplementary)

The celebrated minimax theorem for zero-sum games, first discovered by John von Neumann in the 1920s [14, 10], is certainly a foundational result in the theory of games. It states that two players, playing a game with zero-sum payoffs, each have an optimal randomized strategy that can be played obliviously – that is, even announcing their strategy in advance to an optimal opponent would not damage their own respective payoff, in expectation. Or, if you are not fond of game theory, the statement can be stated quite simply in terms of the order of operations of a particular two-stage optimization problem: letting ∆n and ∆m be the n− and m−probability simplex, respectively, and let M ∈ Rn×m be any real-valued matrix, then we have