Fair Agents Can Form Unfair Systems: Sharp Limits of Fairness Composition

Demographic parity of individual agents does not imply demographic parity of their joint decision. We characterize the largest system disparity compatible with local rate audits and the utility cost of controlling it. By marginal duality, this disparity equals a regularized affine-approximation error, which yields sharp, attained bounds for every voting threshold: under exact local parity, every threshold rule over at least two agents admits a gap of at least $1/2$. Conversely, any monotone coordinator with worst-case gap at most $\rho$ lies within $\rho /(1+\rho )$ of an agent–constant mixture that meets the same budget. This bounds the utility advantage of nonlinear coordination, and under an explicit tight-budget condition a mixture is exactly optimal. Matching all pairwise moments does not reduce odd majority's worst-case gap. Experiments separate fairness on the current distribution from robustness to dependence. In a three-LLM committee, the observed majority gap is $0.0417$ and the largest observed individual gap is $0.0031$. At larger budgets, observed accuracy differences between full coordination and fitted mixtures range from $0.74$ to $1.87$ percentage points in explicit-cue LLM comparisons. Public-data differences are about $0.1$–$0.2$ percentage points, while observed and worst-case disparities differ substantially. Together, the results quantify what local fairness audits leave uncontrolled and how much predictive flexibility must be given up to control it.