People differ in intelligence, cognitive ability, personality traits, motivation, and similar valued and, to a large degree, inherited characteristics that determine success and achievements. When does individual heterogeneity lead to a fair distribution of rewards and outcomes? Here, we develop this question theoretically and then test it experimentally for a set of structural conditions in a specific interaction situation. We first catalogue the functional relationship between individual endowments and outcomes to distinguish between fairness concepts such as meritocracy, equality of opportunity, equality of outcomes, and Rawl's theory of justice. We then use an online experiment to study which of these fairness patterns emerge when differently endowed individuals can share their resources with others, depending on whether information about others' endowments and outcomes is available. We find that while visible outcomes lessen inequality by decreasing the statistical dispersion of outcomes across the group, endowments need to be visible for better equality of opportunity for the most disadvantaged.