We identify a universal mathematical structure in microscopic deterministic traffic models (with identical drivers), and thus we show that all such existing models in the literature, including both the two-phase and three-phase models, can be understood as special cases of a master model by expansion around a set of well-defined ground states. This allows any two traffic models to be properly compared and identified. The three-phase models are characterized by the vanishing of leading orders of expansion within a certain density range, and as an example the popular intelligent driver model is shown to be equivalent to a generalized optimal velocity (OV) model. We also explore the diverse solutions of the generalized OV model that can be important both for understanding human driving behaviors and algorithms for autonomous driverless vehicles.