The recent international agreement on a minimum effective corporate tax rate marks a profound change in global tax arrangements. The appropriate level of that minimum, however, has been, and remains, extremely contentious. This paper explores the strategic responses to a minimum tax, which--the policy objective being to change the rules of tax competition game--are critical for assessing the design and welfare impact of, and prospects for, this fundamental policy innovation. Analysis and calibration plausibly suggest sizable scope for minima that are Pareto-improving, benefiting low as well as high tax countries, over the uncoordinated equilibrium.