# Simultaneously optimal strategies for both players in zero-sum games

Given a matrix $A \in \mathbb{R}^{m \times n}$ that describes a zero-sum game. What are necessary and sufficient conditions for cases in which the k-th pure strategy of the row-player and the l-th pure strategy of the column-player are simultaneously optimal.

If I understood the definition of a saddle point correctly, the condition for the payoff matrix I'm looking for should be to have such a point.

A necessary and sufficient condition for a saddle point to exist is the presence of a payoff matrix element which is both a minimum of its row and a maximum of its column. Therefore some entry $a_{kl}$ of the matrix needs to have the property of being the minimum of the k-th row and the maximum of the l-th column.

So if my assumptions are correct, are there payoff matrices with at least one saddle point that do not provide simultaneously optimal strategies for both players? Or in other words, are there more necessary and sufficient conditions to be satisfied?

• Have you tried proving or refuting your conjecture? – Yuval Filmus Jan 20 '18 at 16:07
• I tried it with an example matrix. I predicted the optimal pure strategies for both players with my saddle point assumption and solved the corresponding LPs for both players. It worked for my examples but I'm not sure if it works in general and if this is the only necessary and sufficient condition. – balderdash Jan 22 '18 at 8:10