The literature on algorithmic fairness has examined exogenous sources of biases such as shortcomings in the data and structural injustices in society. It has also examined internal sources of bias as evidenced by a number of impossibility theorems showing that no algorithm can concurrently satisfy multiple criteria of fairness. This paper contributes to the literature stemming from the impossibility theorems by examining how informational richness affects the accuracy and fairness of predictive algorithms. With the aid of a computer simulation, we (...) show that informational richness is the engine that drives improvements in the performance of a predictive algorithm, in terms of both accuracy and fairness. The centrality of informational richness suggests that classification parity, a popular criterion of algorithmic fairness, should be given relatively little weight. But we caution that the centrality of informational richness should be taken with a grain of salt in light of practical limitations, in particular, the so-called bias-variance trade off. (shrink)