Geometric Coherence via Weighted Matching for 3D Heterogeneous Multi-Agent Reach-Avoid Games

We study assignment quality in 3D heterogeneous multi-agent reach-avoid games and identify a recurring failure mode of cardinality-only matching in geometrically structured scenarios, which we term Geometric Sprawl. In these cases, multiple maximum-cardinality assignments are available, but some induce spatially incoherent pairings and inefficient pursuit trajectories. Building on the evasion-space framework of Yan et al., we introduce a cardinality-first weighted sequential matching method in which the Hamilton--Jacobi--Isaacs interception value z I(s,j) is used as a secondary assignment weight. Each sequential stage is solved with a min-cost max-flow backend, while the unweighted baseline uses the same solver with the weight term removed. We evaluate both methods on a deterministic 35-scenario benchmark (7 families & 5 initialization variants) under two regimes: a diagnostic stationary-unmatched setting and a hybrid saddle-point setting with goal-directed unmatched evaders. On this benchmark, the weighted method resolves 13 of 15 geometric stress-test instances that cause repeated timeouts for the unweighted baseline in the diagnostic regime, and under the hybrid regime improves mean captures from 3.20 to 3.91 (22% increase) and mean interception height from 3.87 to 5.36 (40% increase). We also report lower path tortuosity and lower angular-effort proxy values, suggesting smoother pursuit trajectories in this first-order simulation model. We release the simulator and benchmark suite at github.com/Prajwal-Vijay/geometric-coherence-weighted-matching to support reproducible evaluation of assignment strategies for 3D reach-avoid games.