cs.CVOct 24, 2026
SaveAllocation Before the Clock: Graph, Set and Game Theory of Professional League of Legends Objective Setups
Organizations: UNSW Sydney
Abstract
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 , 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 -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 equilibrium reproduces a quarter of the commitment gap, almost all of it through state variables rather than its strategic channels. Weighting a team-level contrast by exposure moves it across (0.046 against 0.052). We release the tracks, setups and code.
Figures & tables
Figure 1: The paper in one picture. One real setup (game 4 of the Worlds 2025 final, first baron, 60 s after the spawn). The map: walkable graph (white: walls) coloured by which team reaches each cell first, with each formation drawn as the blocks of a set partition. The setup: each team’s walk on the partition lattice (up = fewer blocks) and its players at the pit. The choice: -arrival times and the LB-QRE action at the commitment time, with win probability , fight probability and the objective’s value.
Figure 2: The measurement layer is validated end to end. (a) Last live clock reading minus official game length for all aligned videos. (b) Tracker position error against Riot v5 at held-out odd minutes, by position source; the dotted line marks 600 units. (c) Share of errors below 600 units by local crowding (other champions within 2,000 units). (d) Agreement of tracked formations with v5 formations: exact partition match and Rand index, by number of alive players.
Figure 3: The partition lattice of the five roles (T top, J jungle, M mid, B bot, S support). Rows are numbers of blocks; edges are covering relations (one merge). Node area is the share of V-layer team-snapshots at the spawn with all five alive.
Figure 4: Win probability prices every outcome. (a) Calibration of the outcome-only win surface (out of fold, deciles). (b) Log-loss by game phase for three specifications. (c) Event values at : win probability added over 180 s (main) and the total effect on the result.
Figure 5: Trailing teams stay away once the objective is up. Share of team-snapshots with nobody at the pit (top) and mean number of players at the pit (bottom), by gold state, from 120 s before to 180 s after the spawn (V layer, risk set, pit radius 1,500; bands: 95% intervals; cells with fewer than 30 setups omitted).
Figure 6: The contrast is post-spawn and shared across objectives. Trailing-minus-leading difference in the share of empty pits (left) and of three or more players at the pit (right), by objective group and time (V layer, radius 1,500). Dots mark headline cells: same sign at all three pit radii and .
Figure 7: Exact Riot frames confirm the baron contrast and expose two estimator artefacts. (a) Share of empty baron pits by gold state at the spawn and 60 s later, for three pit radii. (b) One pooled contrast (share of teams with three or more at the pit at , all first spawns) under three estimators; only the unweighted team-mean difference crosses (0.046 against 0.052; both within bootstrap noise of the line). (c) The first-dragon gradient of the kill-anchored pilot against the spawn-anchored risk-set contrasts.
Figure 8: The pattern dictionary ( ). Each row is a pattern’s centre path on , coloured by the number of blocks (darker = more gathered), with rigidity and inertia in Hasse steps; the boxed column is the spawn. Right: mixture weight of each pattern in the three objective classes.
Figure 9: Whole-model HASSE refits and TAGNet’s K-Means re-runs are comparably stable, while single TAGNet runs agree less than best-of-ten HASSE fits. (a) Agreement between repeated full-data fits: five HASSE fits, each the best of ten restarts, and ten single TAGNet runs (with and without its context vector). (b) Series-bootstrap ARI: HASSE refits the whole model on each resample, TAGNet re-runs only its K-Means step on fixed embeddings. (c) Cramér’s V between the assignment and the gold tercile when context is kept out of the pattern layer and when it is fed in.
Figure 10: The larger side rarely loses the next player. (a) Probability that the next death falls on the larger side, by local numbers within 2,000 units, against the stochastic square law. (b) The Tullock ratio under two count definitions.
Figure 11: The allocation game reproduces a quarter of the commitment gap, almost all of it from state variables rather than the four channels. (a) Trailing-minus-leading gap in the share of teams with three or more players at the pit at the commitment time: observed, LB-QRE with all channels, and with all channels off. (b) Shapley values of the four channels over 300 bootstrap replicates (ticks: specification variants). (c) Log-likelihood loss from imposing in each replicate (a nested model, so the loss is non-negative by construction; its size is the result).
Figure 12: Team fingerprints come with wide intervals. (a) Pattern usage of the ten teams with most setups (T1 in red) with series-bootstrap intervals. (b) Split-half reliability of team usage per pattern. (c) Team decision precision with 95% intervals.
Appendix figures & tables16 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 13: The evidence favours rich structure with diminishing returns. (a) Held-out log-likelihood per setup over five series-grouped folds against the number of patterns (V layer). (b) BIC and ICL. (c) Held-out log-likelihood relative to for the 7-snapshot grid, formation radii 1,500 and 2,500, and the R layer.
Figure 14: Context shifts pattern use modestly once time and objective class are held fixed. Average marginal effect on each pattern’s occupancy per standard deviation of each covariate, from a soft-label multinomial logit fitted after the patterns (cut design); blue: 95% interval excludes zero.
Figure 15: The first team to reach three players at the pit usually takes river objectives, but not Atakhan. (a) Share of spawns with a decided race in which the race winner took the objective within 180 s. (b) Time of the first three-player arrival relative to the spawn.
Figure 16: Most games track well; a few are outliers. Tracker quality per game. Per-game median error, share of errors below 600 units, and formation agreement with v5 at held-out minutes.
Figure 17: The lone player in a 4+1 formation is usually the top laner. Formation shapes on the R layer at , and by objective (left) and the role of the lone player in 4+1 formations against the uniform 20% (right). The top laner is the lone player in 49.5% [44.1, 54.9] of 4+1 formations, the mid laner in 29.7% [24.1, 35.1].
Figure 18: Dropping mostly-interpolated snapshots leaves the contrasts essentially unchanged. V-layer trailing-minus-leading contrasts computed on all snapshots and on snapshots with at most half of the positions held or interpolated (radius 1,500; red: headline cells).
Figure 19: The dictionary keeps the early-river and late patterns. Pattern dictionary at ( ).
Figure 20: The dictionary keeps the early-river and late patterns. Pattern dictionary at ( ).
Figure 21: Context effects are reported at as a robustness check. Context effects at .
Figure 22: Context effects are reported at as a robustness check. Context effects at .
Figure 23: Cross-validated loss is flat near the chosen penalty. Cross-validated log-loss and Brier score of against the inverse penalty for the three specifications.
Figure 24: TAGNet’s own criteria fall as grows. TAGNet’s own cluster criteria on our setups, , with and without the context vector.
Figure 25: Gaps and Shapley signs hold across the six non-degenerate specification variants. LB-QRE specification variants: observed and model commitment gaps (left) and the four Shapley values (right).
Figure 26: Team parameters overlap widely. Team cost and valuation parameters with 95% intervals (T1 in red), teams ordered by .
Figure 27: Faker arrives earlier, but T1 behaves like the rest of the league. T1 and Faker in numbers. (a) A3: Faker minus other starting mid laners on seven position measures in pre-spawn snapshots (distance in thousands of units). (b) A5: commitment share by gold state, T1 against all other teams. (c) A8: T1’s commitment share at MSI and Worlds 2025.
Figure 28: The ledger lists every spawn of the Worlds 2025 final. A7: every spawn of the Worlds 2025 final. Markers: blue’s fight probability (colour = taker: blue/red/none; shape = objective); bars end at the break-even .