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Cumulative cultural evolution (CCE) occurs among humans who may be presented with many similar options from which to choose, as well as many social influences and diverse environments. It is unknown what general principles underlie the wide range of CCE dynamics and whether they can all be explained by the same unified paradigm. Here, we present a scalable evolutionary model of discrete choice with social learning, based on a few behavioural science assumptions. This paradigm connects the degree of transparency in social learning to the human tendency to imitate others. Computer simulations and quantitative analysis show the interaction of three primary factors—information transparency, popularity bias and population size—drives the pace of CCE. The model predicts a stable rate of evolutionary change for modest degrees of popularity bias. As popularity bias grows, the transition from gradual to punctuated change occurs, with maladaptive subpopulations arising on their own. When the popularity bias gets too severe, CCE stops. This provides a consistent framework for explaining the rich and complex adaptive dynamics taking place in the real world, such as modern digital media.