---
title: Looking five moves ahead
family: fair-expectimax
summary: Three attempts add a fifth move for selected columns, at row rises, or with fewer chance samples. None paid off.
status: rejected
evidence: ledger-recorded
reads: public
---

The reference search looks four of its own moves ahead. These three experiments
each try to buy a fifth — for the most promising columns only, only when a row
rise is about to happen, or everywhere at once by taking fewer chance samples
to pay for it.

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

Two were rejected on gameplay and one was stopped by its own runtime budget
before it could produce a result. None displaced the four-move search.

## The intuition, and the arithmetic that makes it hard

Every extra move of look-ahead multiplies the tree by the number of columns
times the number of chance outcomes considered: seven times five, or 35, in
the reference configuration. Four moves is already about 1.35 million unit
operations per decision. A fifth, done naively, is 35 times that.

So each of these experiments buys the fifth move by giving something up:

- **Search it only where it might matter.** Rank the columns cheaply, then
  search deeply only the best two or three. Everything the shallow ranking got
  wrong is now invisible to the deep search.
- **Search it only at the moment it might matter.** A row rises every five
  drops; a four-move search standing one move before a rise cannot see past it.
  Deepen only there.
- **Search it everywhere, but look at fewer futures.** Drop from five chance
  samples to three and spend the savings on depth. That makes the averaging over
  chance cruder: the thing the whole family is built on.

There is also a subtler trap, which the first experiment guarded against
correctly: if a deeper search runs out of its work budget it must fall back to
a shallower completed answer, so a "depth 5" policy can quietly be a depth-4
policy on most decisions. The recorded runs report their fallback rates
explicitly for that reason.

## How each one works, step by step

**Selective extension.** (1) Every legal column is searched at the root, full
width. (2) At internal decision points, an exact one-move fair evaluation using
the same five chance samples deterministically orders the legal columns.
(3) Only the leading two or three receive the deeper search. (4) A work cap and
cache bound every attempt, and an incomplete attempt returns the separately
completed four-move answer. The menu of configurations: five moves with the
top two columns, five with the top three, six with the top two — was frozen
before any data was read.

**Rise-boundary extension.** (1) The policy reads how many drops remain before
the next rise. (2) If exactly five remain, it uses the proved-complete
five-move, top-two-columns search. (3) Otherwise it uses the unchanged
four-move reference. Nothing is retuned. A fallback plan covering the last two
drops of the cycle existed but was only eligible if the primary configuration
failed its fit.

**Full-width fifth move with three samples.** (1) Three policies are compared on
the same games: the reference at five samples, the reference at three samples
as a sampling control, and the full-width five-move search at three samples.
(2) Every legal column is searched at every decision point. (3) The five-move
search carries a compile-time proof that it completes within 8,791,020 unit
operations per move without depending on cache retention. (4) A mandatory pilot
projects the total runtime from the first paired games before a second seed may
be read.

## What happened

**Selective extension** fitted well and then failed on fresh games. On the four
games used to choose the configuration, the five-move top-three-columns search
gained about 55,000 points and 32 moves. On the eight-game held-out pass it kept
a smaller gain. On the eight-game fresh screen that followed it lost — about
7,200 points and 3.5 moves — winning only three of eight games. The gain in the
earlier rounds had come mostly from single long games: remove the biggest
winning game from the screen and the average moves to −45,549 points. It cost
2.77 times the reference's work per move and fell back to the four-move answer
on 69% of decisions.

**The rise-boundary version** never got a statistical result. Its runner is
required to play one pair of games, project the cost of the full protocol from
that measurement, and stop if the projection exceeds 40 minutes. It did: the
projection came out at 44.8 minutes. On that single pilot pair the four-move
reference scored 231,290 in 150 moves and the candidate 107,668 in 75: a very
large loss, with both flow rates moving the wrong way as well. One pair is not
a rejection, but it gave no reason to override the time guard.

**The full-width fifth move with three samples** stopped the same way, and even
earlier: its mandatory pilot projected 46.7 minutes against a 45-minute limit,
so it paused after one paired triple. On that one game the reference scored
269,141 in 170 moves, the three-sample control 151,153 in 95, and the five-move
three-sample search 182,622 in 120. The fifth move beat the sampling control it
was paying for, but stayed well behind the plain reference, which is the shape
you would expect if chance-sample quality matters more than depth.

<TechnicalDetails title="The technical record">

**Selective extension**: `fair-selective-depth.cpp`; **rejected —
ledger-recorded** ("the screen lost score, survival, and flow"). None of the
three sections carries a scoring label; [audit 03](/docs/exploratory/audit-03-claim-arithmetic)
classifies every figure on this page as **historical 7,000-point Sequence
scoring** from the score identity. Rescoring does not change any of these
outcomes: score and survival moved together in all three.

| Stage | Cohort | Reference | Candidate | Paired change |
| --- | --- | --- | --- | --- |
| Fit (menu of 3) | 4 games `0x3dd00000...003`, 500-move cap | — | five moves / top three columns selected | +55,436 pts, +32.5 moves, robust utility +0.1848 |
| Training heldout | 8 games `0x3dd10000...007` | 120,580.75 / 82.875 | 133,977.125 / 89.75 | +13,396.375 pts, +6.875 moves |
| Fresh screen | 8 games `0x3ea70000...007`, 1,000-move cap | 182,041.125 / 115.875 | 174,808.375 / 112.375 | **−7,232.75 pts, −3.5 moves** |

Both held-out 95% lower bounds were negative (4/8 score wins, 3/8 move wins with
one tie), and removing one +201,765-point pair made the held-out means negative.
On the screen, removing the largest +260,980-point win moved the means to
−45,548.86 / −25.43, and only two of eight leave-one-pair-out means were
positive on both endpoints. Clears fell 2.0647 → 2.0378 per move, reveals
1.1672 → 1.1324. The candidate switched 76 of 899 same-position four-move
actions (8.45%) and fell back on 620 decisions (68.97%); 60.5% of switches
occurred in the last two drops of the rise cycle, which held 40.0% of decisions
— diagnostic, not causal. Work 3.839M per move versus 1.387M (2.77×); the cache
hit its 45,000-entry cap; 6,825.3 seconds for the gated run; no censoring.
Confirmation `0x3ea80000...00f` never read. Artifact SHA-256
`730caf83ad6dfb9b7f473a0a058bd10a4e29ed9306f9a56daec7b710125a2ab7`.

**Rise-boundary extension**: `fair-cycle-boundary-depth5.cpp`; **rejected —
ledger-recorded** ("the pilot was adverse and also missed its runtime gate").
Pilot pair `0x3de70000`: reference 231,290 / 150 moves, candidate 107,668 / 75,
change −123,622 points and −75 moves; clears 2.1200 → 1.9600, reveals 1.1933 →
1.1067; every deep decision completed without fallback. The pair took 256.071
seconds and projected 2,688.743 seconds against a fixed 2,400-second ceiling, so
the process exited paused before the second fitting seed, the fallback ablation,
the held-out, screen and confirmation ranges. The artifact marks this
pilot-only, with **no formal inference**. Source SHA-256
`b2f2f2987c63fe878e46a6d375f27c61c6c4fafc823909554b91145291d5e680`.

**Full-width fifth move, three samples**: `fair-depth5-s3.cpp`;
**runtime-paused: ledger-recorded**. Fitting gate (never reached) required the
candidate to improve both means against *both* controls while retaining 98% of
each control's clear and reveal throughput. First paired triple on
`0x3de40000`: reference (four moves, five samples) 269,141 / 170; three-sample
control 151,153 / 95; five moves, three samples 182,622 / 120 — behind the
reference by 86,519 points and 50 moves. Throughput 2.0750/1.1417 per move
versus 2.2000/1.2824 and 1.9895/1.0842. Work per move 1.521M / 0.226M / 3.904M.
The triple took 623.250 seconds and projected 2,804.627 seconds against a
45-minute limit, so the run paused before `0x3de40001`; **no gate conclusion may
be drawn from its single game**. Screen and confirmation ranges never read.
Artifact SHA-256
`c94a28dffe9d10c3a4a2026f2acdfe14b7e5f71bd4c6c0ef0289b7612848f448`.

All three passed their optimised `-Werror` and ASan/UBSan self-tests, including
exact parity with the reference search where required, completion proofs,
determinism, reflection safety, and metadata and game-seed blindness.

</TechnicalDetails>

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

The three runs rule out three specific ways of buying a fifth move inside the
frozen five-sample search: shallow-ranked selective extension, rise-boundary
extension, and paying for depth with coarser chance sampling. They also
established a practical habit that later work reused — projecting the cost of a
protocol from a measured pilot and stopping before consuming seeds you cannot
afford to finish.

They do not establish that a fifth move is worthless. Two open threads:

- **The chance estimator may be the reason.** A later exploratory experiment
  found that the fourth move is worth almost nothing at five chance samples and
  about 86,000 points at seven ([finding 05](/docs/exploratory/finding-05-chance-strata)).
  Every depth experiment in the ledger was run on top of the five-sample
  estimator, so "depth does not pay" and "depth does not pay *here*" are not
  distinguished by this evidence.
- **The cost projections that cancelled these runs were too pessimistic.** An
  in-progress exploratory measurement found that worst-case work bounds
  overestimate real work by roughly ten times at depth 5, because repeated
  positions collapse most of the tree, and that at depth 5 with seven chance
  samples the frozen 60,000-entry cache costs 2.19× more work than a
  200,000-entry one for **identical play**: a budget decision, not a
  scientific one. Its own interim
  five-move, seven-sample arm is nevertheless far behind the four-move
  comparator on the games finished so far — on a partial sample its author
  explicitly warns is biased against it
  ([finding 15](/docs/exploratory/finding-15-depth5-exact-estimator)).

## Sources

- `fair-selective-depth.cpp` — bounded, public-state-only selective extension;
  every root column stays full width.
- `fair-cycle-boundary-depth5.cpp`: the deeper search only when the visible
  phase is exactly one full five-drop cycle from the next rise.
- `fair-depth5-s3.cpp`: the reference, a three-sample control, and the
  full-width fifth move at three samples, at a cycle boundary.