The fair innings principle of health resource allocation gives priority to those who would live for fewer quality-adjusted life years (QALYs). However, it leaves scope for different interpretations and intensities of this prioritisation. We run a novel localised convex budget-set experiment, with a UK general public sample (n = 230, obs. = 20,700), to test support for different interpretations and to measure the intensity of support for the principle. We specify social welfare functions (SWFs) representing alternative interpretations, estimate participant-specific parameters within each SWF, and identify which SWF best fits each participant’s choices. We find overwhelming support for the fair innings principle, broadly defined. Prioritarian SWFs - that display continuously diminishing marginal welfare from QALYs - fit the data better than threshold SWFs - that are linear in QALYs with weights that decrease discontinuously on reaching a ‘fair innings’. We illustrate policy consequences of using the estimated SWFs in decision making, and identify welfare weights by quality-adjusted life expectancy.