Grant allocation processes occasionally exhibit seemingly random outcomes: high-quality proposals are often rejected and weaker ones are sometimes funded. Existing explanations attribute such randomness to reviewer error, inconsistency, or institutional inattention. This paper presents a different interpretation. Even when project quality is perfectly observed, a funding agency may emph{optimally} introduce randomness into the allocation mechanism. Randomness broadens the applicant pool at the cost of average quality, and an agency that values breadth alongside quality may rationally accept it, particularly when high-quality projects possess positive outside options. I develop a simple equilibrium model in which the agency chooses a degree of priority for high-quality proposals, and applicants decide strategically whether to apply. Agency welfare is hump-shaped in the degree of randomness: pure lotteries raise adverse selection because high-quality applicants with valuable outside options opt out, while strict meritocracy maximizes quality but excludes the broad base of low-quality applicants. When the agency values participation sufficiently, the optimal mechanism involves an interior level of randomness, rationalizing partial lotteries, random tie-breaking, and other randomness observed in grant allocation.