Outcomes in the cultural arena are due to many factors but are there general rules that can suggest what makes some cultural traits successful and others not? Research in cultural evolution theory distinguishes factors related to social influence (such as copying from the majority, or from certain individuals) from factors related to individual, nonsocially influenced, propensities such as evolved cognitive predispositions, or physical, biological, and environmental constraints. Here, we show, using analytical and individual-based models, that individual preferences, even when weak, determine the equilibrium point of cultural dynamics when acting together with nondirectional social influence in three out of four cases we study. The results have implications regarding the importance of keeping into account individual-level, nonsocial, factors, when studying cultural evolution, as well as regarding the interpretation of cross-cultural regularities, that must be expected, but can be product of weak directional forces, intensified by social influence.