Acceleration models are at the core of operational driving behaviors and include car-following models that capture interactions between a lead and following vehicles. The main assumption in these models is that the behavior of the following vehicle (e.g., change in acceleration) is related directly to a stimulus observed or perceived by the driver, defined relative to the lead vehicle (e.g., difference in speeds or headways). An important aspect missing from previous formulations pertains to the stochastic character of the cognitive processes used by drivers, such as perception, judgment, and execution while driving. A car-following model that reflects the psychological and cognitive aspects of the phenomenon and captures risk-taking behavior under uncertainty is explored and evaluated. In this model, Tversky and Kahneman's prospect theory provides a theoretical and operational basis for weighing a driver's different alternatives. The model is implemented and tested to assess its properties and those of the resulting traffic stream behavior.