This paper addresses the theoretical foundations of pedestrian models for crowd dynamics. While the topic gains momentum, widely different mathematical approaches are actually in use, even if we only consider continuous agent-based models. To clarify their underpinning, we first rephrase the common hierarchical decomposition into strategic, tactical, and operational levels and show the practical interest in preserving the continuity between the latter two levels by working with a floor field, rather than way-points. Turning to local navigation, we clarify how three archetypical approaches, namely, reactive models, anticipatory models based on the idea of times to collision (exemplified by the recently proposed ANDA model), and game theory, differ in their extrapolation of future trajectories, and insist on the oft-overlooked distinction between processes pertaining to decision-making and mechanical effects in dense settings. The differences are illustrated with a comparison of the numerical predictions of instances of these models in the simple scenario of head-on collision avoidance between agents, by varying the walking speed and the reaction times, notably.