“Let’s look ahead one more turn” is the board-game equivalent of “let’s just add a database.” It is sometimes exactly right, and it has also launched a thousand regrettable weekends.

OverGrid’s first two-ply experiment did not fail because replies are unimportant. It failed because we asked the reply search to overrule everything.

A worst-case opponent is not necessarily an opponent

The first version was spite minimax: after a candidate move, it selected the reply that was worst for the acting player. On six paired 30-turn games, that policy lost 405 score and 6,035 AUC against one-ply while taking 2.83× the mean compute.

That is a useful failure. The supposedly “worst” reply often asked the opponent to throw away its own position merely to hurt us. A real opponent is not a malicious unit test. They are trying to win the board too.

Ranking replies by the opponent’s own evaluator removed much of the self-destructive theatrics, but it still underperformed: −80 score, −1,580 AUC, and 2.61× compute on the same locked games. A deeper search was now being rational in the wrong places.

The narrow question that worked

The productive version did not replace the shallow evaluator. It let the shallow evaluator finish first, then asked for a reply check only when the root had a genuine near-tie.

At most eight constructions within 3% of the one-ply best were eligible. For each, the engine advanced through the canonical reducer to the next hand exposed by the rules at that decision boundary, sampled 60 opponent constructions, and compared the least favorable of the opponent’s four strongest replies. A hard 30,000 validator-call ceiling preserved the completed one-ply choice whenever deeper work could not finish.

On the 30-game map panel, this near-best tie-breaker gained +565 score, +5,425 AUC, +290 height, and +22 captures at 1.33× compute. That is a promising directional result, not a win-rate claim: the panel ended at a 20-turn horizon.

The longer completion probe was appropriately humbling. Of 12 games run to 120 turns, only three finished naturally. Selective search won two and one-ply won one; the remaining nine caps are too much unresolved game to call that a solved contest.

The useful visual model

complete bounded one-ply search
              │
              ├─ clear best move ───────────────► play it
              │
              └─ near-best alternatives
                     │
                     ├─ next rules-exposed hand
                     ├─ bounded rational replies
                     └─ worst surviving outcome ─► break the tie

This is less glamorous than a universal minimax tree. It is also the right shape for a responsive game opponent: calculate deeply where the calculation can change a close choice, not where it only adds a larger invoice to an obvious move.

Board sequenceA useful reply model advances both players' goals.

Follow the marked cells from position to reply to result.

Your territoryRival territoryObjectiveYour lineTheir reply
  1. State 1Create the threat
    Create the threatBlue prepares a capture chain that will harden the marked pocket next turn.
    If ignored
    +30 next turn

    Blue prepares a capture chain that will harden the marked pocket next turn.

  2. State 2Reply is forced
    Reply is forcedRed abandons its preferred height route to occupy the only closing cell.
    Red’s preferred route
    delayed

    Red abandons its preferred height route to occupy the only closing cell.

  3. State 3Spend the tempo
    Spend the tempoBlue uses the stolen turn to advance a separate objective while red remains tied to defense.
    Initiative
    blue keeps it

    Blue uses the stolen turn to advance a separate objective while red remains tied to defense.

Read this asRed's modeled reply improves Red by blocking the threat. Blue's move is valuable only because the forced defense leaves a separate public objective available.

Public information is a hard boundary

The reply search may use the public board, current scores, the acting player’s hand, committed move history, and the next hand exposed by the rules after advancing the reducer. It may not peek at a third hand, an RNG cursor, a future deal, a hidden opponent hand, or an opponent’s character label.

That boundary is not merely a fairness slogan. It makes the move explainable. If Ender says a route is vulnerable, we should be able to reconstruct the public reply that demonstrates it—not tell a post-game story about what the seed secretly had waiting.

<details> <summary>Why we kept the promotion language deliberately cautious</summary>

The tie-breaker was provisionally adopted in the early solver because it passed reducer parity, bounded-work, and map-panel checks while preserving the one-ply fallback. But natural resolution was sparse: only 3 of the 12 long games finished, and every short comparison was horizon-capped. Later work tightened terminal rules, role crossing, and fair evaluation before treating arena outcomes as strength evidence. See the AI Lab’s research trajectory.

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The enduring lesson is not “two ply wins.” It is: lookahead earns its cost when it resolves ambiguity. When it tries to replace the whole evaluation with a cartoon villain’s reply, it mostly gets in the way.

For the playable version of that idea, read The reply is part of the move.