cs.LGAug 29, 2026
SavePokaiTrainer: Scaling Equilibrium Search to Competitive Pokémon VGC
Organizations: Independent Researcher
Abstract
Decision-time equilibrium search carried poker to superhuman play, but it has so far relied on tractable subgames: a handful of actions per decision, chance confined to card deals, one player moving at a time. Competitive Pokémon in its official doubles format (VGC) breaks all three assumptions at once. Both players act simultaneously from joint menus in the hundreds, each joint action resolves to hundreds of stochastic outcomes, and the opponent's reserves and stat allocations are hidden. No prior Pokémon agent performs equilibrium search, and whether it scales to this regime was open; we show that it does, and report what it took. PokaiEngine, our Rust battle engine, enumerates a joint action's full weighted outcome distribution in one pass, at parity with Pokémon Showdown and a fraction of the cost of sampling it. PokaiTrainer adapts Student of Games to this scale and trains it by self-play over hundreds of human teams. Each decision is solved by counterfactual regret minimization as a Bayesian matrix game over public belief states, subgames grow under an explicit compute budget, and value targets are harvested from the interior of every solve and grounded by realized outcomes. The strength is in the search. The network's policy alone loses even to a shallow heuristic search. PokaiTrainer is, to our knowledge, the first VGC agent rated on the live Showdown ladder. Under open team sheets it wins 59% of 150 best-of-three sets against a human field averaging Elo, holds a 1350-1400 Elo band, and at its peak reached 1492 Elo, entering the format's top 500.
Figures & tables
| Chess / Go | HUNL poker | Scotland Yard | No-press Diplomacy | VGC doubles | |
|---|---|---|---|---|---|
| Simultaneous actors | — | — | — | 7 powers | 2 players 2 slots |
| Hidden information | — | hands | 199 stations | — | 15 brings spreads per Pokémon |
| Chance | — | card deals only | — | — | – outcomes per joint action |
| Actions per decision | 35 / 250 | a few bet sizes | 10 | up to per power; a few dozen searched | per player, joint |
| Horizon | 80 / 150 plies | 4 bet rounds | 24 rounds | open-ended | 5–20 turns |
Table 1: VGC compared to the domains decision-time equilibrium search has mastered: the four of Student of Games ( Schmid et al., 2023 ) and no-press Diplomacy ( Gray et al., 2021 ) . The spread grid matters less than its size suggests ( Section 4.4 ).
Figure 1: One attack’s three chance events under the two engine modes. Compute rolls each event and continues a single timeline. Search forks the state and carries every branch, holding the 16 damage rolls inside one state as an HP distribution split only where it straddles a knock-out.
| Method | Mean time (ms) | Mass covered | Mass misallocated |
|---|---|---|---|
| Search , exhaustive | 9.9 | 100% | 0% (reference) |
| Search , 32-branch cap | 1.0 | 99.5% | 0.7% |
| Compute / Showdown sampling, | 1.3 / 36.0 | 94.3% / 93.0% | 10.0% / 10.2% |
| Compute / Showdown sampling, | 4.9 / 132.0 | 98.4% / 98.2% | 4.8% / 4.8% |
| Compute / Showdown sampling, | 19.4 / 509.0 | 99.7% / 99.4% | 2.8% / 2.6% |
Table 2: Recovering the outcome distribution of one joint action: Search -mode enumeration versus sampling, averaged over 500 random openings with random joint move actions. Outcomes are distinct when they differ in any end-of-turn discrete features. Coverage is the reference outcome mass the method observed, and misallocation is the total-variation distance from the reference weights. A 32-branch cap is simultaneously faster and more accurate than sampling at any tested budget.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: System architecture. The engine runs one rules code in Compute or Search mode, the latter exposed as the branch() API returning weighted outcome sets; the Showdown adapter reconstructs the same player-view BattleState the engine emits from protocol lines; and BattleEnv is one seat-view surface over all three backends. Python agents implement one Agent interface and run unchanged on any backend; the search player additionally calls branch() on hypothesized states.
| Component | Follows | Departure for VGC |
|---|---|---|
| Belief states, CFR solve, CV net at leaves | ReBeL, SoG | simultaneous-move matrix nodes; exactly enumerated chance nodes ( Section 4.2 ) |
| Incremental subgame growth | SoG (GT-CFR) | walk over a matrix game; admissions priced by compute cost, budget in cost units ( Section 4.3 ) |
| Value targets | SoG (solver values TD(1) rows) | per-row -mix, grounded on the realized world’s slot only ( Section 4.6 ) |
| Extra supervision | SoG (re-solved leaf queries) | interior nodes of the same solve, reach-sampled and bootstrap-weighted |
| Off-line coverage | SoG (recursive queries) | hypothetical augmentation, grounded by play |
Table 4: Where PokaiTrainer follows its lineage and where it departs.
Figure 4: Anatomy of one budgeted solve. Left: the subgame rooted at the public belief state ; every decision node holds a matrix game, chance nodes are the engine’s exactly enumerated outcome batches , leaves are priced by , and the red path is one PUCT expansion walk ( Equation 3 ) admitting a new continuation. Right, zooming into a turn node: one payoff matrix per world, weighted by the belief and solved jointly by CFR ( Equations 1 and 2 ); each cell resolves through one Search -mode engine pass of , and its successors either recurse or read the leaf value .
| Decision | Chooses | / | Hidden | Solve size |
|---|---|---|---|---|
| Team preview | both | bring-4 lead pair | none ( ) | |
| Turn | both | move-or-switch per slot | root; interior | |
| Forced switch (after faints) | both | replacement | ||
| Mid-turn switch (pivot) | one seat | replacement (one slot) | , opponent’s uncommitted actions |
Table 5: The decision types of a VGC battle as instances of the Bayesian matrix game ( Section 4.2 ), with solve sizes as rows columns worlds. is seat ’s menu after shortlisting by the policy prior (root solves keep our full menu; values in Section 5.1 ); replacement menus are bounded by the two-Pokémon bench; before the leads are revealed and after.
| Team | Elo | Pokémon |
|---|---|---|
| mb486 | 1676 | Charizard , Garchomp, Venusaur, Incineroar, Toxapex, Annihilape |
| mb355 | 1579 | Charizard , Floette-Eternal , Kingambit, Whimsicott, Basculegion, Garchomp |
| mb520 | 1531 | Tyranitar , Staraptor , Excadrill, Gholdengo, Sinistcha, Milotic |
| mb300 | 1523 | Raichu , Basculegion, Garchomp, Kingambit, Whimsicott, Ninetales-Alola |
| mb443 | 1520 | Charizard , Aerodactyl , Garchomp, Sylveon, Farigiraf, Kingambit |
| mb210 | 1516 | Froslass , Scovillain , Lycanroc-Dusk, Kingambit, Basculegion, Sneasler |
Table 6: The eval16 evaluation pool, sorted by strength. Each team is a public paste from the VGCPastes repository ( VGCPastes, 2026 ) (linked; author credit on the paste page), identified by its corpus index. Elo: Bradley–Terry fit over 2,940 internal head-to-head games among the 16 (mean 1500). Italics mark Pokémon holding a Mega Stone.
| # | Team | Win% | cRk | Pokémon |
|---|---|---|---|---|
| 1 | mb1 | 76.3 | 11 | Tyranitar , Staraptor , Excadrill, Basculegion, Sinistcha, Raichu |
| 2 | mb194 | 69.9 | 6 | Froslass , Incineroar, Gholdengo, Basculegion, Kingambit, Arcanine-Hisui |
| 3 | mb537 | 68.8 | 19 | Charizard , Mawile , Whimsicott, Basculegion, Farigiraf, Garchomp |
| 4 | mb345 | 65.6 | 13 | Charizard , Garchomp , Venusaur, Farigiraf, Incineroar, Sylveon |
| 5 | mb198 | 64.5 | 20 | Swampert , Floette-Eternal , Whimsicott, Pelipper, Archaludon, Basculegion |
| 6 | mb423 | 63.4 | 7 | Raichu , Tyranitar , Meowscarada, Basculegion, Kleavor, Talonflame |
Table 7: Full standings of the 32-team elite round-robin at the deepened deployment shape. Win% is over 93 games per team; cRk is the team’s rank within the 32 by the corpus-wide fit. Links and italics as in Table 6 .
| Arm | Change | Result | Verdict |
|---|---|---|---|
| no grounding | , | 50.0% (256/512) h2h, same deploy | wash at 2 rounds; calibration drifts |
| decoupled + zero-sum | re-priced targets, moment loss, | 42.6% (109/256) vs sibling r3 | loses; dropped |
| hypothetical forks, 3/game | augmentation only | 48.8% (125/256) compute-matched | wash per unit compute |
| hypothetical forks, 1/game + PUCT | 52.0% (133/256) on half the rows | kept | |
| full- on hypo rows | on augmented rows | 40.6% (104/256) vs control | loses |
| entry continuations | replacement + preview roots | 52.0% r5, 57.0% r7 | kept |
Table 8: Target and expansion ablations on the v7 deep line; each row is a head-to-head against its round-matched sibling unless noted. Most levers are within a 256-game cell’s pp of parity: the binding constraint is the value network, not the target construction.
| Agent’s solver | Agent’s play | Best-response win% [95% CI] | Gain / decision | |
|---|---|---|---|---|
| CFR | argmax (eval cells) | 77.7 | [72.2, 82.4] | 0.097 |
| CFR | (ladder) | 52.3 | [46.2, 58.4] | 0.027 |
| CFR | (solved mix) | 49.6 | [43.5, 55.7] | 0.0003 |
| decoupled UCT | argmax | 74.6 | [68.9, 79.6] | 0.193 |
| decoupled UCT | 72.7 | [66.9, 77.8] | 0.110 | |
| decoupled UCT | 83.6 | [78.6, 87.6] | 0.058 | |
Table 9: An informed local best response, seated second, against the round-32 agent under each solver and play mode (eval16, 256 games per cell). Gain is the best response’s in-model advantage per decision over the agent’s expectation, in payoff units on . The two solvers are at parity against fixed opponents (51.7%, 2,048 games) and 34pp apart here.