Human choice behavior exhibits many paradoxical and challenging patterns. Traditional explanations focus on how values are represented, but little is known about how values are integrated. Here we outline a psychophysical task for value integration that can be used as a window on high-level, multiattribute decisions. Participants choose between alternative rapidly presented streams of numerical values. By controlling the temporal distribution of the values, we demonstrate that this process underlies many puzzling choice paradoxes, such as temporal, risk, and framing biases, as well as preference reversals. These phenomena can be explained by a simple mechanism based on the integration of values, weighted by their salience. The salience of a sampled value depends on its temporal order and momentary rank in the decision context, whereas the direction of the weighting is determined by the task framing. We show that many known choice anomalies may arise from the microstructure of the value integration process.