May a dominant firm refuse to share its intellectual property (IP) with its rivals? This question lies at the heart of a highly divisive, international debate concerning the proper application of the antitrust laws. In this short Essay, we consider a profound, yet previously unaddressed, incongruity underlying the controversy. Specifically, why is it that monopolists refuse to share their IP, even at monopoly prices? To resolve this issue, some have recommended compulsory licensing, which would require monopolists to license their IP in certain circumstances. This proposal, however, entails an inescapable contradiction, one rooted in the issue of monopolists’ seemingly inexplicable refusal to share their IP.