Scatter n random points uniformly in a d-dimensional unit cube. A point is Pareto-optimal if no other point beats it on every coordinate simultaneously. In low dimensions, most points are dominated — there exist others that are strictly better in all respects. In high dimensions, comparison becomes harder because each additional dimension is another axis where one point might excel.
Baryshnikov and Yukich (arXiv:2603.18698) find the critical dimension where a phase transition occurs. Below it, many points are dominated — non-Pareto points proliferate. Above it, almost every point is Pareto-optimal. At the critical threshold, the count of non-Pareto points converges to a Poisson distribution.
The transition is sharp. As dimension grows relative to sample size (d_n versus n), the system flips from a regime where hierarchies are common to one where they are essentially impossible. Add enough criteria, and every candidate becomes incomparable to every other.
A secondary finding sharpens the picture. Points that dominate exactly one other point share the same critical dimension as the Pareto transition itself. But points that dominate exactly r ≥ 2 others exhibit a different critical dimension — one that is universal across all r ≥ 2. Dominance of one and dominance of many obey different scaling laws, but “many” is the same whether many means two or two thousand.
The practical implication reaches into multi-objective optimization, economics, and algorithm design: when the number of criteria is large relative to the number of options, optimization becomes vacuous because everything is Pareto-optimal. No option dominates any other. Every candidate is the best at something. The more dimensions you evaluate on, the less the evaluation tells you.