Abstract: In many kinds of games with economic significance, it is very important to study the submodularity of functions. In this paper, we mainly study the problem of maximizing a concave function over an intersection of two matroids. We obtain that the submodularity may not be preserved, but it involves one maximal submodular problem (or minimal supermodular problem) with some conditions. Moreover, we also present examples showing that these conditions can be satisfied.

Key words: Matroid, Submodular function, Rank function, Convex closure, Game