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.