Allocation Before the Clock: Graph, Set and Game Theory of Professional League of Legends Objective Setups

Zheng Gao, Zhenyu Liu, Yifan Yang, Xiaoyu Li

Before every neutral objective in professional League of Legends, each team decides which of its five players walk to the pit, which hold a lane, and when to commit. The leading analysis of such setups, TAGNet for Counter-Strike 2, clusters graph-network embeddings of player positions and reads cluster win rates as tactical quality. We treat a setup as an allocation on a walled map: formations are set partitions on the lattice $Π_5$, control is a geodesic Voronoi split of a navigation graph, and commitment is an allocation game solved as a quantal response equilibrium priced in win probability. On all 164 games of MSI and Worlds 2025 (2,994 team-setups, minimap tracks validated against Riot positions), trailing teams leave the pit empty far more often than leading teams, by +34.8 percentage points at the first baron and +30.4 percentage points at Atakhan. Our model, HASSE-QRE, finds six setup patterns that whole-model series-bootstrap refits reproduce (ARI 0.819, comparable to 0.813 for TAGNet's $K$-Means step alone) and indicates that teams price fights, travel and committed players, not the objective itself (weight 0.018; fixing it to one costs 59.9 log-likelihood points). The…