Let \(X\subset {\mathbb{R}}^ k\), \(Y\subset {\mathbb{R}}^ m\) be compact convex sets. A martingale \(Z_ n=(X_ n,Y_ n)\), \(n=1,2,...\), with values in \(X\times Y\) is called a bi-martingale if for each n either \(X_ n=X_{n+1}\) or \(Y_ n=Y_{n+1}\) (a.s.). A subset of \(X\times Y\) is called bi-convex if each of its x- and y-sections is convex. Suppose a bi-martingale starts in \(z\in X\times Y\), (i.e. \(Z_ 1=z\) (a.s.)) and consider the limit \(Z_{\infty}=\lim_{n\to \infty} Z_ n\) (a.s.). For a given set \(A\subset X\times Y\), the authors describe the set of starting points z for which \(Z_{\infty}\in A\) (a.s.). For this purpose, three notions of bi-convex hulls are introduced which, under certain conditions, equal the set of possible starting points of the bi- martingale. In contrast to the convex case, the various bi-convex hulls turn out to be different, in general. The concepts are motivated by applications in the analysis of repeated games of incomplete information.