Contests are a historically important and increasingly popular mechanism for encouraging innovation. A central concern in designing innovation contests is how many competitors to admit. Using a unique data set of 9,661 software contests, we provide evidence of two coexisting and opposing forces that operate when the number of competitors increases. Greater rivalry reduces the incentives of all competitors in a contest to exert effort and make investments. At the same time, adding competitors increases the likelihood that at least one competitor will find an extreme-value solution. We show that the effort-reducing effect of greater rivalry dominates for less uncertain problems, whereas the effect on the extreme value prevails for more uncertain problems. Adding competitors thus systematically increases overall contest performance for high-uncertainty problems. We also find that higher uncertainty reduces the negative effect of added competitors on incentives. Thus, uncertainty and the nature of the problem should be explicitly considered in the design of innovation tournaments. We explore the implications of our findings for the theory and practice of innovation contests.