A Minmax Theorem for Concave-convex Mappings with no Regularity Assumptions

We prove that zero-sum games with a concave-convex payoff mapping defined on a product of convex sets have a value as soon as the payoff mapping is bounded and one of the set is bounded and finite dimensional. In particular, no additional regularity assumption is required, such as lower or upper semicontinuity of the function or compactness of the sets. We provide several examples that show that our assumptions are minimal