Metrics/Methods for deciding duration of video retention for on-demand websites
Clustering can be an ideal choice here. From the question, seems the data will most probably be in continuous format. Essentially, clustering is a method of finding groups of similar objects. The similarity between the objects is determined by the type of distance measure being used.
With this background, you can start by finding the videos that are most liked or disliked by people. Measurements like mutual information gain can be helpful.