I have studied particular ambiguous random dot stereograms where multiple matches (that are equally possible) are available at each point. The human visual system resolves these ambiguities in two qualitatively different ways. In some cases a few transparent surfaces are perceived corresponding to all the ambiguous matches. In other cases a single dominant opaque surface is perceived. The conditions under which each behavior occurs are described. Additional experiments, designed to explore whether a number of modified stereo matching algorithms can predict human perception, are described, and their theoretical implications are discussed.