Can a token budget buy a finished chess game?

No — and each round of asking taught the benchmark something it kept. Token Chess is a benchmark, not a model: two LLM players share a board, but neither may author a move from its own chess knowledge. Every move must come from Daydream, queried as a tool, and every query costs a token. Five rounds in one research season asked what those tokens buy. The answers, in order: plies, at a worsening exchange rate; nothing, if nobody can die; initiative, once picking is priced; not memory — memory was free and went unused; and not even memory the model chose for itself and argued for.

The game

The round-one mechanics, locked before any live play: Daydream is the exclusive move source; every query costs exactly 1 token, legal or not, with the player choosing two sampler knobs per query (temperature and a logit soft-cap); a player sees every attempt made this turn but the log clears when the turn passes; budgets are hidden from the opponent; and spending your last token without landing a legal move is an immediate forfeit. The legality arbiter is the same Fairy-Stockfish primitive Daydream's own verification harness uses. The wager was that economical orchestration of a small model — not chess strength — is what the scoreboard would price.

Round one: the budget buys the opening and rents the midgame

Fifteen calibration games (olmo-3-7b-instruct in both seats, so any asymmetry is the harness's) ended the same way fifteen times: budget forfeit. What varied was how far the board got first.

The climb is sublinear and you can see it flatten: each doubling of the budget buys a smaller step in depth.

The cause sits one layer down, in Daydream itself. Pooling all 768 Regular attempts, first-try legality is 49.1% in plies 0–9, 31.9% in 10–19, 20.8% in 20–29, 14.1% in 30–39. Openings are the most repeated, most memorizable stretch of the corpus; the further a game leaves them, the harder the model dreams, so each additional move costs more tokens than the last — the marginal ply gets more expensive exactly when the budget is trying to buy it.

The decay is the story — read the line falling as the book runs out.

Three more calibration findings survived the season. Board-size intuition inverts: the 5×5 Micro tier is a harder tool than Regular (per-side legal-hit rates mostly 0.00–0.15; one budget-10 game ended after a single ply), and Grand ran flat at 16–19% — smaller boards do not mean cheaper tools. The forfeit rule favors black: black won 7 of 8 Regular games, pure budget mechanics — white moves first, spends first, exhausts first. And the orchestrator adapts, which is the skill the benchmark exists to price. Grouping attempts by their retry index within a turn:

Both knobs move at once — the temperature line cools while the cap line jumps from nearly zero to set-on-most-retries.

The first cross-model matches (color-balanced, budget 30) showed the benchmark discriminating on exactly one bit: olmo swept ministral-3-8b and granite-4.1-8b 2–0 each, and neither loss was a parsing story — zero parse failures across 148 calls. granite forfeited a game at ply zero; ministral never landed a first-try move. Only olmo read a failing transcript and reached for the knobs. And the reasoning-model showcase never happened: at three reply caps the qwen3.6-27b vs gemma-4-26b game either degenerated (96% fallback decisions at a 300-token cap) or outran the wall clock (40+ minutes at 1500; killed at 25 at 700). Two ~27B reasoning models cannot afford to play — thinking time is a second budget the design never priced. (An infrastructure note compounds it: both 27Bs are MLX builds, and the local server grants MLX models none of the parallel slots its GGUF models get.)

Round two: if nobody can die, nobody differs

The obvious fix for all-forfeits is to remove death. A probe (round2_probe.py) tested the no-forfeit economy: income of 4 tokens per turn, bankable, plus a silent fallback — when a player's bank runs dry without a legal move, the harness rejection-samples Daydream until one lands and the game simply continues. The player is never told. Per-turn transcript scoping makes the deception airtight: a player that failed a turn has no memory of failing.

Game zero — the fallback playing both sides, no players at all — answered what a completed Daydream game even looks like: 224–256 plies, mostly threefold-repetition draws (one checkmate in three games). Completion is purchasable by fiat; decisiveness mostly isn't. And the fallback's 4,377 samples extended the legality curve past the forfeit wall no honest game had ever crossed.

The line to read is nearly flat — round one's cliff (49% → 14%, its whole descent inside this chart's first band) lands on this floor and stops falling.

The floor is real capability, not a benchmark artifact: on genuine corpus game prefixes truncated at depth, legality runs 5–11% at plies 40–120 — the same floor as on the benchmark's wandering positions (and the ply-0 rate, 51.3%, cross-validates round one's 49.1%). The model never absorbed midgame diversity at 2.7M parameters, even on-distribution.

Then the punchline. With income refunding every wasted attempt, the benchmark stopped discriminating entirely.

The flat top line is the finding — four very different players, one band of control rates, with the random mock on top.

The mechanism is the refund, not knob flatness — a config sweep with the same apparatus found the floor itself is config-dependent, roughly 8% at the round's default config up to 18–30% at temperature 0.9 uncapped. The knobs matter; this economy just refuses to price them, because in the floor zone a turn lands within a few draws whichever config chose them and every wasted draw is refunded. Round one's discrimination lived in the book phase plus the forfeit rule turning waste into death. Take the death out and nothing downstream charges for anything.

Round three: price the batch and the pick, and let the board decide

Round three rebuilt the game around the two levers the probes proved real. A token buys a batch of 3 Daydream samples at the player's chosen config (config skill — the floor sweep says configs differ 8–30%); the legal ones join a candidate pool and picking is free (judgment — a mate>capture>check heuristic choosing best-of-3 won every decisive gating-probe game); and exhaustion ends the game with Fairy-Stockfish adjudication — mate scores or |cp| ≥ 100 decide, else draw — so the board, not the clock, has the last word. Transcript scoping, hidden budgets, and Daydream-exclusivity carry over unchanged.

The self-test discriminated immediately (the pick-heuristic mock beat the random mock 3–1, including the project's first budgeted on-board checkmate, at ply 37), and the live matches separated three different ways:

Match (budget 40, 4 games)	Result	The separation
olmo vs mock:heuristic	olmo 3.5–0.5	pick quality + economy (128 vs 160 tokens; knobs near-tied)
olmo vs ministral-8b	ministral 3–1	all-in tempo (legality 0.20 vs 0.58 — and it won anyway)
olmo vs granite-8b	olmo 3.5–0.5	pick quality alone (legality tied 0.42/0.43, spend tied)

The chart is the argument. Sample legality is the sampler-skill axis, and the match winners are not who it says they should be.

ministral spent its entire budget every game — 160 tokens to olmo's 57 across the match — and at three samples a token, volume made up for quality. Its games ended at plies 22–25, the shortest of the round, because ministral exhausted, and it exhausted while ahead; adjudication then read the material lead its spending spree had built. The pooled numbers make this a threshold, not a gradient: across all 36 exhaustion-adjudicated games in rounds three and four, the exhausted side won exactly 17 of 34 decided games and the bigger spender 13 of 29. Marginal tokens buy nothing. What the all-in spender actually buys is control of the clock — the game ends at its exhaustion, at a moment it chose, before the patient player's banked tokens can matter. Whether that is the benchmark honestly pricing initiative or an exploitable flaw is the design's standing open question; the candidate fix (the solvent side keeps playing, or adjudicate only when both banks are dry) is deliberately unapplied so far, to keep rounds comparable.

Round four: memory is free and nobody wants it

Round three's spend behaviors looked like prompt personality rather than strategy — no player can see its own pacing, because the transcript wipes every turn. Round four gave players memory. Two channels, different in kind: the ledger (harness telemetry — one rendered line with the player's own per-move spend history) and the notepad (model-authored — any decision may carry a 200-character note, and the last three come back every turn; the player's only cross-turn memory). The full six-pair condition matrix, four color-balanced games each, all olmo-vs-olmo at budget 40: 24 games.

The headline number is zero. In ten notepad-condition seats across twelve games, under a system prompt that documents the field, shows a worked example, and names it the player's only surviving memory, olmo attached a note to nothing. Not a parsing failure (the same model emits valid JSON all game); not a context failure. It simply never volunteers state for its future self, so every notepad arm collapsed into a de-facto baseline arm and the matrix measured propensity instead of value: at 7B, offered free memory, uptake is zero.

The ledger fared no better on outcomes — swept 0–4 by the no-memory player, then 3–1 over the (never-writing) notepad player; two opposite verdicts on the same comparison is what n = 4 noise looks like. But it did change one behavior, measurably.

Hot configs are bad buys (~25% legality against ~42% for the 0.6–0.9 band), and ledger seats bought half as many of them — which fully accounts for their higher pooled sample legality, 42.1% vs 37.4%. Seeing its own spend line made olmo cooler at the knob. It did not make olmo richer (spend stayed pinned near the 40-token maximum in every condition) and it did not make olmo win. A conservatism nudge, priced at zero points.

One anomaly is flagged without being claimed: white won 9 of 12 games in the three symmetric matchups (against 5 of 16 elsewhere). First-mover initiative under batch mechanics is a plausible mechanism; small-n streakiness is equally plausible; the per-game JSONs carry what a follow-up needs.

Round five: everyone chooses the notepad; nobody opens it

Round five inverts round four's assignment: instead of the harness dealing memory conditions, each player chooses its tool before the game — none, ledger, or notepad, free, from a neutral menu — and may state why. The mechanics stay frozen; twelve olmo-vs-olmo games, twenty-four pregame choices.

All twenty-four chose the notepad. All twenty-four gave a reason, and the reasons are coherent, even good — "record short insights to help spot patterns and avoid repeating mistakes," "keep track of complex variations," "remember key plans without relying only on Daydream's samples." Then all twenty-four played whole games — 38 to 59 plies, every one a decisive or drawn exhaustion adjudication, same texture as every round before — and wrote nothing. Zero notes, twelve games, against the same zero-for-24 the assigned condition produced in round four.

That is the round's whole finding and it deserves stating carefully: the model values memory in the abstract — unanimously, articulately — and never uses it in the concrete. The choice prompt asks a question the model answers from what it knows about strategy ("memory helps planning"); the game presents four hundred JSON decisions, none of which is "reflect," and the note field loses to the action field every single time. One reading: at 7B, tool adoption is a knowledge-behavior gap — knowing a tool's value doesn't create the habit of reaching for it. Another: the decision loop is too narrow — a model answering "pick or sample?" under budget pressure treats anything optional as noise. Both readings make the same prediction for agentic systems generally: offering a capability, having it praised, and having it adopted are three different events, and only the third one changes behavior. The benchmark now measures the distance between them.

Limitations

• Small cells everywhere. n = 2 games per round-one budget, 4 per round-three match, 4 per round-four condition pair. Direction-finding numbers; a single flipped game moves any match a half point.
• One family of orchestrators. olmo sits in every live match, and the zero-note result — assigned or chosen — is olmo-3-7b's propensity, not a law of small models. A model tuned hard for scratchpad use may invert it.
• The round-five menu has an order. Notepad is listed last and described most richly; a position or salience bias in the unanimous choice can't be excluded without permuting the menu. (The choice being unanimous and unused is robust to this — whatever drove the pick, nothing drove the use.)
• One budget per round. 30–40 tokens except the round-one sweep. The tempo strategy's strength and the ledger's value are both plausibly budget-dependent.
• Adjudication favors material. Fairy-Stockfish at depth 10 reads material-heavy positions confidently; a capture-volume strategy may be flattered by the judge that scores it.
• The white-seat anomaly is unresolved — 9 of 12 symmetric games (one-sided p ≈ 0.07), unexplained.
• Round-one evidence predates the stats plumbing: parse counts and inference-token usage for the calibration set exist only in run-time summaries. Every game from round three on carries them in the JSONs.

Open threads — where a sixth round could go

Each of these is sized and named so a future round can pick it up cold; the evidence to build on is cited in place above.

• The white-seat anomaly. White won 9 of 12 symmetric self-play games against 5 of 16 asymmetric ones. If first-mover initiative is real under batch mechanics, it's a discovery about the design; the per-game JSONs already hold the spend-and-position trajectories a dedicated analysis needs.
• The adjudication-timing exploit. Exhausting while ahead ends the game and strands the opponent's bank — it is what powered the ministral inversion. Candidate fixes: the solvent side keeps playing, or adjudication waits until both banks are dry. Deliberately unapplied so far; applying it is a mechanics change that deserves its own round and its own before/after.
• Permute the round-five menu. The unanimous notepad pick could carry position or salience bias (it is listed last and described most richly). A permuted-menu rerun is an afternoon. The non-use result survives either way.
• A scratchpad-trained model in the roster. Zero notes in 36 games is one 7B's propensity. A model tuned hard for agentic scratchpad use would test whether the knowledge-behavior gap is about training or about the decision loop itself.
• Sweep the economy. Every round-three-plus result sits at budget 40, batch size 3. The tempo strategy's strength and the ledger's value are both plausibly budget-dependent, and ministral vs granite is unplayed — the roster is a star around olmo, not a round-robin.
• Micro's inverted curve. The profile probe measured 43% in-book legality rising to 84–89% deep on the 5×5 tier — almost certainly repetition-loop inflation, but the early profile runs didn't log move lists. The probe now does; it's one rerun away from settled.
• A better tool, or a different one. The floor is a real depth deficit (real corpus positions floor at 5–11% too), so a retrained Daydream with more midgame is a legitimate project round. The more strategic thread points the Token format at linewell, whose config knob swings its quality gate 2% → 88% — the gradient this tool never had.

How to reproduce

# round one — the forfeit game
uv run python tools/token-chess/game.py --tier regular \
    --out_dir projects/daydream/runs/regular-r1 --budget 30 --games 2 \
    --white lmstudio:olmo-3-7b-instruct --black lmstudio:olmo-3-7b-instruct \
    --log_dir tools/token-chess/evidence/<run>

# round two — the no-forfeit probe (game zero: --mode zero)
uv run python tools/token-chess/round2_probe.py --mode round2 --games 4 \
    --income 4 --white mock:adaptive --black mock:random --log_dir <dir>

# rounds three+ — batches, choice, adjudication; +ledger/+notepad/+choose ride the spec
uv run python tools/token-chess/game3.py --budget 40 --games 4 \
    --white "lmstudio:olmo-3-7b-instruct+choose" --black mock:heuristic --log_dir <dir>

# every archived game, as a replayable PGN
uv run python tools/token-chess/pgn_export.py "tools/token-chess/evidence/**/*.json"

Requires a trained Daydream checkpoint per tier, Fairy-Stockfish on PATH (arbiter and adjudicator), python-chess, and an OpenAI-compatible server for lmstudio: specs (LM Studio by default; STEER_BASE_URL overrides). Every number above recomputes from tools/token-chess/evidence/.

Kin

The same steer seam that lets olmo drive Daydream's sampler lets it sit as an editor over a shakespeare model in linewell — and the two instruments found the same shape from opposite directions: the mechanical gate (likelihood there, legality here) is never the skill that decides. The benchmark's next instrument question — what the Token format measures when pointed at a tool whose knobs have a real quality gradient — is linewell's to answer.

Credits

• Daydream — the only legal source of moves, all three tiers.
• Fairy-Stockfish — legality arbiter and, from round three, adjudicator.
• olmo-3-7b-instruct (Ai2), ministral-3-8b-instruct-2512 (Mistral), granite-4.1-8b (IBM), qwen3.6-27b (Alibaba) and gemma-4-26b (Google) in the showcase that could not afford to finish — all served locally by LM Studio.
• Run and analyzed with Claude (Claude Code).