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The operationalization of algorithmic fairness comes with several practical challenges, not the least of which is the availability or reliability of protected attributes in datasets. In real-world contexts, practical and legal impediments may prevent the collection and use of demographic data, making it difficult to ensure algorithmic fairness. While initial fairness algorithms did not consider these limitations, recent proposals aim to achieve algorithmic fairness in classification by incorporating noisiness in protected attributes or not using protected attributes at all. To the best of our knowledge, this is the first head-to-head study of fair classification algorithms to compare attribute-reliant, noise-tolerant and attribute-unaware algorithms along the dual axes of predictivity and fairness. We evaluated these algorithms via case studies on four real-world datasets and synthetic perturbations. Our study reveals that attribute-unaware and noise-tolerant fair classifiers can potentially achieve similar level of performance as attribute-reliant algorithms, even when protected attributes are noisy. However, implementing them in practice requires careful nuance. Our study provides insights into the practical implications of using fair classification algorithms in scenarios where protected attributes are noisy or partially available.