---
title: Exact search
family: heuristic-search
summary: Three attempts add independent chance estimates, depth, and voters to a search. None repairs a biased board evaluation.
status: rejected
evidence: ledger-recorded
reads: public
---

Three attempts to fix a search by giving it more — more independent chance
estimates, more depth, more voters, and the discovery that none of them
repairs a biased judgement of the board.

<EvidenceLabel status="rejected" evidence="ledger-recorded" reads="public" />

## The intuition

A search that looks ahead has two sources of error, and they need different
cures. It can be **noisy** — its handful of sampled chance outcomes happen to
flatter one column, or it can be **biased**: the evaluator at the bottom of
the tree is simply wrong about which boards are good. Noise averages out with
more samples; bias does not average out with anything.

These three programs are, in effect, a controlled experiment on that
distinction. Each one attacks noise, in a different way, while leaving the
evaluator alone:

- **Independent root quadratures.** Run the same complete search three times
  with three independent sampling salts and average the results. Pure variance
  reduction, no other change.
- **A fully completed fourth ply.** No shortlisting, no learned pruning: every
  legal action expanded at every depth, with the same chance sampler.
- **A three-member policy ensemble.** Let three complete searches *vote*, and
  when they split (no majority) keep the reference member's move rather than
  trusting a divided committee.

## How it works, step by step

1. **Read the public position** — board, next disc, rise clock.
2. **Expand every legal column at every choice node.** "Exact" here means
   full-width decision nodes; the gray-disc reveals are still sampled, and the
   ledger says so explicitly.
3. **Vary only the chance sampling** between members: each gets a different
   fixed, deterministic salt derived from the public state.
4. **Combine.** Average the root values across members, or count votes, or
   simply search one ply deeper.
5. **Fall back when the committee splits.** A split vote is treated as
   containing no robust direction, so the reference member's action is
   retained.

## What happened

The first two were rejected, and together they make a single argument that
shaped everything after them.

Averaging three independent chance estimates **passed a four-game screen and
then lost its eight-game confirmation decisively**, losing seven of the eight
paired games while spending about three times the work per move. The lesson
recorded in the ledger is sharper than "it did not help": reducing sampling
variance can *remove accidental regularisation* that one fixed quadrature was
providing. A noisy estimate of a wrong value was, by luck, better behaved than
a precise estimate of the same wrong value.

Completing a full extra ply lost its screen too, by 25,449 points and 18.75
moves, at about 24 times the cost. The ledger's conclusion is the one this
family is remembered for: *"a fully completed extra ply is therefore not the
missing ingredient: the phase heuristic at the horizon is the central source
of bias."*

The voting ensemble is a looser end. A small replicated gain was reported
against its own comparator, but it was never tested against the strongest
reference search, so it certifies nothing.

<Callout title="This is not the same result as the reference depth-4 search" tone="warn">
These programs use an **older leaf evaluator**. When the same fourth ply was
completed later with the *fair* leaf, it was a clear improvement — the
reference search in the
[fair expectimax](/approaches/fair-expectimax) family won seven of eight
paired games against depth 3. Depth helped there and hurt here, with the same
depth and different evaluators. That contrast is the point.
</Callout>

<TechnicalDetails title="The technical record">

**Independent root-quadrature ensemble: rejected, ledger-recorded.** Three
members, each expanding every legal action through three plies with five
stratified chance samples, differing only in a fixed public-state sampling
salt; root-Q arrays averaged before selection. A four-game screen passed
(128,858.75 points and 88.75 moves versus 96,304 and 67.5), and the gated
eight-game confirmation reversed decisively: **117,066.5 points and 80.625
moves versus 162,492.125 and 109.375**, losing **seven of eight** paired
games, at **157,047 work units per move versus 53,778**. Optimised `-Werror`
and ASan/UBSan self-tests passed determinism, reflection, public-state
isolation, legality, resource bounds, scoring and single-member action parity.

**Fully completed depth 4 (older leaf): rejected, ledger-recorded.** Every
legal decision action expanded with the same five-stratum public chance
sampler at depths 3 and 4, with compile-time worst-case bounds proving the
search fits in 3,134,950 work units and 45,430 cached nodes, and a hard
assertion that the requested iteration completed. Depth 3 averaged
**149,621.25 points and 103.75 moves**; depth 4 averaged **124,172.25 and 85** —
a paired difference of **−25,449 points and −18.75 moves** with a 2–2 record,
at **1,303,982 work units per move (about 24.1× depth 3)** and 0.943 moves per
second. It failed the screen, so confirmation seeds were not read.

**Both of those are historical 7,000-point Sequence-scored runs**, using the
7,000-point level award of the 30-drop rules instead of the 17,000-point award
of the mode studied here. Their absolute scores are not evidence about the
million-point target; the paired comparisons on the same games, the work
measurements and the win records remain valid
([mode-specific scoring correction](/docs/research/history)). Four and eight
games are small cohorts.

**Three-member exact policy ensemble: completed, task-record only.** The
[experiment index](/docs/research/experiment-index) records *"a small
replicated gain was reported against its comparator, but it was not qualified
against the strongest D3/D4 reference."* Task-record only means the comparison
lives in a research conversation and was never promoted into the ledger;
**no score, cohort or margin for it is retained in this repository.**
Repository-verified from the source: three or five members drawn from a fixed
table of five policy seeds, modes `vote`, `qmean` and `screen`, a strict
majority requirement (*"A split vote contains no robust direction; retain the
reference member"*), defaults of 4 games at a 1,000-move cap from
`0x3d707000`.

</TechnicalDetails>

## What this taught us, and what is still open

- **Precision is not accuracy.** Three independent estimates of a biased value
  gave a confidently wrong answer where one noisy estimate had been
  accidentally hedged. This is the cleanest demonstration in the repository
  that variance reduction is not a free improvement.
- **Depth is conditional on the leaf.** The identical extra ply was harmful
  with the older evaluator and helpful with the fair one. "Search deeper" is
  not a strategy; it is a multiplier on whatever judgement sits at the bottom.
- **A committee cannot invent information its members do not have.** Voting
  and majority fallbacks are safety mechanisms, in the same category as the
  conservative overrides the strategy catalog describes — useful for
  preventing damage, not for producing insight.
- **What is still open** is a properly powered comparison for the ensemble
  against the current reference search. As it stands it is an unqualified win
  over an unqualified comparator, which is not evidence of anything.