To innovate, firms often need to draw from, and collaborate with, a large number of actors from outside their organization. At the same time, firms need also to be focused on capturing the returns from their innovative ideas. This gives rise to a paradox of openness-the creation of innovations often requires openness, but the commercialization of innovations requires protection. Based on econometric analysis of data from a UK innovation survey, we find a concave relationship between firms' breadth of external search and formal collaboration for innovation, and the strength of the firms' appropriability strategies. We show that this concave relationship is stronger for breadth of formal collaboration than for external search. There is also partial evidence suggesting that the relationship is less pronounced for both external search and formal collaboration if firms do not draw ideas from or collaborate with competitors. We explore the implications of these findings for the literature on open innovation and innovation strategy.