We analyze an innovation game between one incumbent and a large number of entrants. In the first stage, firms compete to develop innovations of high quality. They do so by choosing, at equal cost, the success probability of their R&D approach, where a lower probability goes along with a higher value in case of success—that is, a more radical innovation. In the second stage, successful entrants bid to be acquired by the incumbent. We assume that entrants cannot survive on their own, so being acquired amounts to a ‘prize’ in a contest. We identify an equilibrium in which the incumbent chooses the least radical project. Entrants pick projects of pairwise different success probabilities, and the larger the number of entrants, the more radical the most radical project becomes. Generally, entrants tend to choose more radical R&D approaches and are more likely to generate the highest value innovation. Thus, the need of entrants to be acquired yields an entirely new explanation of why radical innovations tend to come from entrants. We illustrate our theoretical findings by a qualitative empirical study of the Electronic Design Automation industry, and derive implications for research and management.