The Optimal Mean: A Monotone Arithmetic-Geometric Gap Correction Weighted by the Herfindahl-Hirschman Index

Composite indicators require aggregation rules that balance compensation, sensitivity to imbalance, and robustness at the lower boundary. The arithmetic mean is stable but fully compensatory, whereas the geometric mean is balance-sensitive but collapses when any component approaches zero. This paper proposes an Optimal Mean based on a Herfindahl-weighted arithmetic-geometric gap correction. For a positive commensurable vector x, let Ma and Mg denote the arithmetic and geometric means, and let HH be the Herfindahl-Hirschman index of the normalized component shares. The proposed mean is M*(x) = Ma(x) − [(1 − HH(x)) / 2][Ma(x) − Mg(x)] Equivalently, M*(x) = [(1 + HH(x)) / 2]Ma(x) + [(1 − HH(x)) / 2]Mg(x) The rule can be derived from the normalized Herfindahl non-concentration family Mδ = Ma − δDHH(Ma − Mg), where DHH = (1 − HH)/(1 − 1/n), by choosing the largest monotonicity-preserving coefficient δ = (n − 1)/(2n). The resulting expression is a simple monotone aggregation rule that combines arithmetic robustness, geometric balance sensitivity, and endogenous Herfindahl concentration without an externally tuned parameter. Monte Carlo diagnostics, monotonicity stress tests, and applications to the Human Development Index and Worldwide Governance Indicators show that the rule is stable for balanced profiles and most consequential when a near-boundary component would make the geometric mean too severe. Propagating the published WGI standard errors leaves the M* ranking at least as reproducible as the geometric ranking, and a head-to-head comparison shows that M* and unbalance-penalty indices move imbalanced profiles in opposite directions: the latter penalize a weak component further, whereas M* protects it.