The rapid development of COVID-19 vaccines had a crucial role in facilitating the global recovery from the pandemic. Countries around the world adopted different vaccine prioritisation strategies, but this variation and its underlying causes are poorly understood. Drawing on Oxford COVID-19 Government Response Tracker's data on vaccine prioritisation plans and eligibility, we describe which population groups were prioritised, when, and in which order across countries. We create a new measure of vaccination plan granularity, which captures the degree to which vaccination strategies targeted specific groups. We use correlations and regressions to examine the relationship between granularity and country-level factors-in particular, democracy and state capacity. In simple correlations, more granular vaccination strategies are associated with higher vaccine uptake, and both democracy and state capacity go hand-in-hand with greater granularity. Once potential confounding factors are controlled for, however, of the two, only democracy emerges as a key predictor. We also find that, in line with the World Health Organization's recommendations, older age groups, healthcare workers, and clinically vulnerable people were highly prioritised in vaccination campaigns. The link between granular vaccination plans and democracy highlights the importance of institutional factors in shaping policy design during public health crises like COVID-19.