You are looking for something called a "simple tree ordering test". There are quite a few methods for this in the order restricted inference literature, though I am not sure how much of it is available in software. A quick Google search lead me to a paper that seems applicable.
Peddada, S. D., Prescott, K. E. and Conaway, M. (2001), Tests for Order Restrictions in Binary Data. Biometrics, 57: 1219–1227.
Based on the comments, it appears I misunderstood the original question. It is still a simple tree ordering, which is a topic of ordered restricted statistical inference, but the reference addresses the comparison of independent binomial probabilities. I have found another paper,
Nettleton, D (2009), 'Testing for the supremacy of a multinomial cell probability', Journal of the American Statistical Association. Vol. 104, Pages 1052-1059, that describes exactly your situation. I think the author has a preprint posted on his webpage, and you might be able to get most of the info from there.