---
title: Fair expectimax reference (D3/D4)
family: fair-expectimax
summary: The reference search looks four moves ahead, averages the sampled chance outcomes, and uses a hand-tuned board evaluator at the bottom.
status: completed
evidence: ledger-recorded
reads: public
kind: strategy
technique: expectimax
featured: true
---

## The problem

Drop7 hides two things from the player at every move. The disc after the
visible next one is unknown, and the number under every gray disc stays hidden
until a clear beside it cracks it and a second clear reveals it. A player
weighing one column against another has to allow for both, and how that
allowance is made decides the move. Hope for the reveal you want and the
search builds boards that pay off one time in seven. Assume the reveal you
fear and it never builds anything.

Before any learned or novel policy can be judged, the program needs one
dependable player that handles that uncertainty honestly, reads only what a
person at the screen could read, and produces the same games every time it is
run on a cohort. Every other page on this site measures its candidate against
this one.

## Proposed solution

At each move the reference builds a look-ahead tree that alternates between
two kinds of node. At a choice node the player picks a column, and the search
keeps the best option. At a [chance node](/learn/concepts/chance-vs-choice)
the game deals a disc or reveals a hidden number, and the search takes the
average over what could happen. "Fair" means the average runs over every legal
next disc and every possible reveal. It doesn't hope for the best reveal or
plan for the worst.

The search imagines four of the player's own moves (its depth, written D4
below; the three-move version is D3) and then hands every board at the bottom
to a hand-written [leaf evaluator](/learn/glossary), described on
[the fair leaf evaluator page](/approach/fair-expectimax/fair-policy). The
policy reads only the visible board, the visible next disc, the moves until
the next rise, and whether the game is over.

<GameTreeFigure seed={0x5eed1002} moves={10} leafDepth={1} caption="The browser analogue of the reference search on a seeded game, with one further ply below each outcome. It is a playground demonstration of the mechanics; the cohort numbers on this page come from the native reference." />

## How it works

1. Read the position: the visible board, the visible next disc, the moves
   until the next rise, and whether the game is over.
2. For each legal column, drop the disc and let the rules engine resolve every
   clear, chain and reveal it causes. This is a choice node, and each column is
   one branch.
3. Below each branch the following disc and any reveals are unknown, so the
   search samples a fixed handful of representative outcomes, its
   [strata](/learn/glossary) (five in the reference), and gives each the same
   weight.
4. Repeat steps 2 and 3 until four of the player's moves have been imagined.
   Points scored along the way are counted exactly; the leaf evaluator scores
   the board at the end of each line.
5. Average at every chance node, take the best at every choice node, carry
   the values back up to the root, and play the column with the highest value.

## What happened

On the eight-game confirmation cohort, replayed under corrected scoring, the
depth-4 search averaged 400,675 points against the depth-3 search's 235,071
([ledger](/docs/research/history), "Corrected Hardcore scoring replay";
ledger-recorded). On the broader 64-game development cohort the depth-4
reference averages 308,296 points ([status](/docs/research/status)). Eight
paired games is a confirmation replay on already-used seeds, so the 64-game
mean is the number the rest of this site treats as the reference.

A fifth move did not help. Selective depth 5, full depth 5 and the
cycle-boundary variants all failed their gates; the sibling pages in
[this group](/approach/fair-expectimax) record each attempt. The reference
is the teacher behind most label-generation experiments and the fallback
inside most candidates, and its mean is far below the one-million-point mean
the program is after.

## What we learned

With this leaf, the fourth move of look-ahead is worth a lot and the fifth is
worth nothing anyone could measure. The later
[reveal-sampling work](/approach/lifetime-objective/reveal-sampling) put
that in a wider frame: a three-move search with a better chance model matched
the four-move reference at less work, so depth and the quality of the chance
average are two ways of buying the same thing. The reference's value is that
it is frozen, public, and reproducible game for game, which is what lets a
negative result on any other page mean something.

The open question is what, if anything, beats it on games it has never seen
at the same work per move.

<AgentContext summary="Records and provenance">

- The eight-game numbers are from the ledger section "Corrected Hardcore
  scoring replay" in [`docs/research/history.md`](/docs/research/history):
  after the mode audit restored the five-drop Hardcore award to 17,000, the
  qualified full-width fair-D4 policy was rebuilt and deterministically
  replayed on its already-consumed screen and confirmation seeds; no new
  gameplay seed was opened. Four-game screen: D3 318,528.75 points / 93 moves,
  D4 399,951.75 / 115, paired gain 81,423 points and 22 moves. Eight-game
  confirmation: D3 235,071.25 / 71, D4 400,675.25 / 116.375, D4 winning seven
  of eight pairs, paired gains 165,604 points and 45.375 moves; 1,000-move cap,
  neither arm censored. D4 used 1,351,112.595 logical work units per move,
  retained at most 36,105 cache entries, peaked at 30,441,472 RSS bytes, and
  ran at 0.744 moves per aggregate game-second. Replay artifact
  `/tmp/drop7-fair-only-depth4-hardcore17-replay.json`. The same table is in
  [`docs/strategies.md`](/docs/strategies), which adds that D4 changed about
  34% of the decisions D3 would have made.
- The 64-game figure is the current-status number in
  [`docs/research/status.md`](/docs/research/status) ("308,296 points" over 64
  games); the ledger's corrected-D4 bootstrap entry records it as 308,295.578
  points and 90.031 moves, and
  [finding-03](/docs/exploratory/finding-03-rollout-veto-17k) cites the same
  figure as 308,296 / 90.03.
- Builds: `fair-only-horizon.cpp` is the fair-only depth-3 evaluator;
  `fair-only-depth4.cpp` is the depth-4 reference build (`make native`
  produces `build/fair-depth4`). The TypeScript counterpart used by the
  benchmark registry and by the browser figure above is `evaluateMoves` in
  `src/core/typescript/solver.ts` with the combined leaf.

</AgentContext>

<AgentContext summary="Full results table">

| Configuration | Games | Mean score | Mean moves | Paired wins | Source |
| --- | ---: | ---: | ---: | ---: | --- |
| Fair D3 | 8 | 235,071 | 71.000 | 1 of 8 | ledger, corrected replay of the confirmation cohort |
| Fair D4 | 8 | 400,675 | 116.375 | 7 of 8 | ledger, corrected replay of the confirmation cohort |
| Fair D4, broad development cohort | 64 | 308,296 | 90.031 | | status.md; ledger corrected-D4 bootstrap entry |

All rows are corrected 17,000-point Hardcore scoring and ledger-recorded. The
eight-game rows are a deterministic replay on already-consumed seeds, so they
are the corrected-score baseline and cannot serve as confirmation evidence
again.

</AgentContext>

<AgentContext summary="Validity, gates and limitations">

- The ledger's own reading of the replay: it establishes the corrected-score
  baseline, and its confirmation mean is still about 40% of one million.
- A fifth move did not pass its gates in any form tried: selective D5, full
  D5, and the cycle-boundary variants all failed (see the sibling approaches
  in this family). The
  [chance-strata study](/approach/fair-expectimax/chance-strata) found the
  fifth ply unmeasurable by its design.
- One D4 game is reported at 1,246,684 points. That figure is task-record only:
  [audit-03](/docs/exploratory/audit-03-claim-arithmetic) records that no move
  count, seed, cohort, scoring mode or artifact exists for it anywhere, so it
  cannot be classified and must not be cited as evidence of anything. The
  million-point target requires the mean over a cohort to exceed one million;
  a single game is an anecdote.
- Every serious candidate in this program keeps an exact D4 fallback; D4 is a
  strong tactical fallback and teacher, and its average is far below the
  target.

</AgentContext>

<AgentContext summary="Scoring mode">

Corrected 17,000-point Hardcore award throughout. The replay followed the
mode audit; the TypeScript suite and a 256-seed native/TypeScript parity
sweep passed exactly across 6,852 transitions before the rebuild, and the
production score constant is regression-tested as 17,000. Ledger experiments
marked `levelBonus: 7000` are historical 7,000-point scoring, archival, and
none of their numbers appears on this page.

</AgentContext>