Problems of inequality related to the disaggregation of Gini coefficients are examined. Measures of inequality can be decomposed so that in a grouped population, total inequality depends on inequality within and between groups. No such decomposition is available for the Gini coefficient, yet its direct relationship to the Lorenz curve has resulted in persistent attempts to derive a disaggregation that can be used in empirical work. The Gini coefficient can be interpreted in terms of the average expected gain from having the option of receiving the income of some other random individual. Variation within the components of the expression for the Gini coefficient contributes to overall inequality. The decomposition of the Gini coefficient further identifies the inequality within a population and may have particular relevance to studies of migration and discrimination. If migration includes movement from one income group to another or a change in educational status, the migration can be viewed in terms of expected gains for those individuals who are migrating. These expected gains for individuals can then be considered in terms of a statistical game framework. Statistical data are included. 12 references.