For many queries in the Web retrieval setting there is an on-going ranking competition: authors manipulate their documents so as to promote them in rankings. Such competitions can have unwarranted effects not only in terms of retrieval effectiveness, but also in terms of ranking robustness. A case in point, rankings can (rapidly) change due to small indiscernible perturbations of documents. While there has been a recent growing interest in analyzing the robustness of classifiers to adversarial manipulations, there has not yet been a study of the robustness of relevance-ranking functions. We address this challenge by formally analyzing different definitions and aspects of the robustness of learning-to-rank-based ranking functions. For example, we formally show that increased regularization of linear ranking functions increases ranking robustness. This finding leads us to conjecture that decreased variance of any ranking function results in increased robustness. We propose several measures for quantifying ranking robustness and use them to analyze ranking competitions between documents' authors. The empirical findings support our formal analysis and conjecture for both RankSVM and LambdaMART.