In desirability rating tasks, decision makers evaluate objects on a continuous response scale. Despite their prominence, full process models of these rating tasks have not been developed. We investigated whether a preference accumulation process, a process often used to model discrete choice, might explain ratings as well. According to our model, attributes from each option are sampled and evaluated stochastically. The evaluations are integrated over time, forming a preference. Preferences for options compete with each other, and accumulated preferences can decay. The model makes precise predictions regarding the statistical distribution of desirability ratings, as well as their dependence on deliberation time and on context. We test and confirm these predictions in two experimental studies. Additionally, quantitative model fits indicate that participants are better described by our proposed model, relative to a model without dynamism, competition, or stochastic attribute sampling. Our results show that the descriptive power of models of preference accumulation extends beyond discrete choice, and that the assumptions of this framework accurately characterize the core cognitive processes at play in the construction of preference and the evaluation of objects.