Asymmetric simple exclusion processes with two species of particles, driven in opposite directions, can be interpreted as simple traffic models describing fast and slow cars, or pedestrian encounters in narrow hallways. At first sight, these models appear to exhibit remarkably different behaviors, depending on the number of ''lanes'' in the model. Strictly one-dimensional (''one-lane'') models are clearly disordered, while simulations of ''two-lane'' models show a large, macroscopic jam.
In this talk, I will describe this puzzle and propose a resolution. Some other peculiar features, such as a region with negative response in a model with lane preference, will also be discussed.