Presenting different images to either eye can induce perceptual switching, with alternating disappearances of each image--a phenomenon called binocular rivalry. We believe that disappearances during binocular rivalry can be driven by a process that facilitates visibility near the point of fixation. As the point of fixation is tied neither to a particular stimulus nor to a specific eye, indifference to both would be an essential characteristic for the process we envisage. Many factors that influence disappearances during binocular rivalry scale with distance in depth from fixation. Of these, here we use blur. We break the links between this cue and both eye of origin and stimulus type. We find that perceptual dominance can track a better focused image as it is swapped between the eyes and that perceptual switches can be driven by alternating the focus of images fixed in each eye. This implies that, as a determinant of suppression selectivity, blur is functionally independent from both eye of origin and stimulus type. Our data and theoretical account suggest that binocular rivalry is not an irrelevant laboratory curiosity but, rather, that it is a product of a functional adaptation that promotes visibility in cluttered environments.