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 Π5, 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 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 p=0.05 (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 Π5 (up = fewer blocks) and its players at the pit. The choice: k -arrival times τ(k) and the LB-QRE action (k,m,τ) at the commitment time, with win probability W , fight probability p^ 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 Π5 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 W (out of fold, deciles). (b) Log-loss by game phase for three specifications. (c) Event values at W=0.5 : 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 qBH<0.05 .
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 s+60 , all first spawns) under three estimators; only the unweighted team-mean difference crosses p<0.05 (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 ( K=6 ). Each row is a pattern’s centre path μc,t on Π5 , coloured by the number of blocks (darker = more gathered), with rigidity θ and inertia ι 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 γn/γg 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 ν=1 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 λi 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 K (V layer). (b) BIC and ICL. (c) Held-out log-likelihood relative to K=3 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 s−60 , s and s+60 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 K=5 dictionary keeps the early-river and late patterns. Pattern dictionary at K=5 ( ξ=0.127 ).
Figure 20: The K=8 dictionary keeps the early-river and late patterns. Pattern dictionary at K=8 ( ξ=0.221 ).
Figure 21: Context effects are reported at K=5 as a robustness check. Context effects at K=5 .
Figure 22: Context effects are reported at K=8 as a robustness check. Context effects at K=8 .
Figure 23: Cross-validated loss is flat near the chosen penalty. Cross-validated log-loss and Brier score of W against the inverse penalty C for the three specifications.
Figure 24: TAGNet’s own criteria fall as K grows. TAGNet’s own cluster criteria on our setups, K=2..8 , 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 λ .
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 p^ (colour = taker: blue/red/none; shape = objective); bars end at the break-even π∗ .
Reasoning in large language models (LLMs) is often grounded in human text, human demonstrations, and human-generated rationales. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium. This discrepancy is amplified in games with mixed-strategy equilibria, where human data is heavily biased toward pure strategies. Consequently, conditioning LLMs on this data yields weak game strategies. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output. We propose Mixed-Strategy Decision Tree (MDT), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations. We instantiate this study on No-Limit Texas Hold'em by querying a solver oracle for over 250 million mixed-strategy decisions; MDT together with other techniques \textbf{reduces the ℓ1 distance to the equilibrium by 52.6%} across 8 different LLM configurations. A Route-only ablation tests the incremental contribution of the shadow-based contrast, while complete River-endgame and Liar's Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting.
Han Wang, Philippe Beardsell, Boning Li +4
Shanghai Jiao Tong University · GTO Wizard · Tsinghua University +2
The emergence of large language models (LLMs) has spurred economists to study how humans and LLMs behave in strategic settings. We organized a series of round-robin tournaments in the Colonel Blotto game. This game attracts game theorists' attention due to high-dimensional action space and the absence of pure strategy Nash equilibria. In the first tournament, more than 200 human participants competed against one another. In the second tournament, several popular LLMs were invited to submit strategies. In the third tournament, we matched the number of LLM strategies to the number submitted by humans. We find that humans more often employ better-calibrated intermediate-level allocation heuristics and outperform the simpler, more stereotyped strategies submitted by LLMs. Strategic sophistication is key to success if and only if the necessary level of reasoning depth is reached, while lower and higher levels of reasoning offer no clear advantage over the primitive strategies. Among humans, field of study weakly predicts success: participants with STEM backgrounds perform better in the first tournament. Surprisingly, humans almost do not adjust their strategies across tournaments with different sets of opponents. This result suggests that humans base their choices primarily on the game's rules rather than on the identity of their opponents, treating LLMs much like human competitors.
Dmitry Dagaev, Egor Ivanov, Petr Parshakov +2
HSE University · New Economic School · Central Economic Mathematical Institute, Russian Academy of Sciences +3
In competitive games, players frequently switch strategies after losing streaks, yet our analysis of 926,334 match records from 34,619 Clash Royale players reveals a counterintuitive pattern: switching frequency is inversely associated with the win rate, with effects that vary substantially across players and situational contexts. We attribute this to a limitation common in many prior recommendation systems, which evaluate strategies by expected quality while overlooking the behavioral cost of switching and individual differences in switching propensity. We refer to this implicit premise as the Zero Switching Cost Assumption. To address this, we reformulate strategy recommendation as a transition-level decision problem and instantiate it as TQP (Transition Quality Predictor), a three-stage pipeline structured as Who -> When -> What. PersonaGate suppresses recommendations for players whose strategic consistency is empirically associated with superior outcomes. TimingGate identifies moments when switching is likely to yield a net benefit over staying, using a subtype- and state-matched baseline to control for natural win-rate recovery. ScoreFusion ranks candidate strategies by combining an adoptability signal with predicted transition quality (delta WR). We further introduce SwitchGap, an evaluation metric that measures a policy's discriminative quality without treating observed player choices as optimal ground truth. This property is particularly important because the most frequent switchers record the lowest win rates. The full pipeline achieves a SwitchGap of +10.4 percentage points at a recommendation rate of 5.4%, and loss-triggered switchers, despite being the lowest-performing group, benefit the most from subtype-conditioned guidance.
Heeyun Heo, Huy Kang Kim
School of Cybersecurity Korea University Seoul, Republic of Korea