We propose a new notion of uniqueness for the adversarial Bayes classifier in the setting of binary classification. Analyzing this concept produces a simple procedure for computing all adversarial Bayes classifiers for a well-motivated family of one dimensional data distributions. This characterization is then leveraged to show that as the perturbation radius increases, certain notions of regularity for the adversarial Bayes classifiers improve. Furthermore, these results provide tools for understanding relationships between the Bayes and adversarial Bayes classifiers in one dimension.