---
title: Weighting the bad outcomes at the root
family: fair-expectimax
summary: Instead of judging a column by its average outcome, judge it partly by its worst outcomes, and find that the resulting policy dies much sooner.
status: rejected
evidence: ledger-recorded
reads: public
---

Instead of judging a column by what happens on average, judge it partly by what
happens in its worst cases. This experiment mixed three parts average with one
part bad-tail into the value of every root move, and measured whether the more
cautious player survived longer.

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

## The intuition

Drop7 scores are heavy-tailed and every game eventually ends. A policy that
maximises the average can be quietly buying a big average with a small chance
of dying immediately, and dying immediately costs everything that would have
come after. So it is natural to ask the search to be a little afraid: value a
column not by the mean over the seven futures it examines, but by a blend of
that mean with the outcomes at the bottom of the distribution.

The statistician's name for "the average of the worst slice" is *conditional
value at risk*; the experiment used the worst 25%. With seven sampled outcomes
that slice gives full weight to the worst outcome and three-quarters weight to
the second worst, then divides by the 1.75 outcomes of tail mass.

The reason this can go wrong is the mirror image of the reason it can go right.
Being afraid of the worst sampled future is *also* being afraid of building
anything — chain structure is temporarily fragile before it fires. The recorded
result is a clean example of that trade going badly.

## How it works, step by step

1. At the root, take every legal column.
2. For each, construct exactly seven scenarios from a hash of the public
   position. A scenario fixes only the **immediate** chance outcome: the next
   disc and any reveal.
3. Once that outcome is on the board it is observable, so from there the
   unchanged fair search completes two more of the player's moves with the
   normal five chance samples at every chance point. No later decision ever
   sees the scenario tape, the game's seed, the score, the level, or the move
   count.
4. Combine the seven scenario values as `0.75 × mean + 0.25 × worst-25%`
   instead of the plain mean. The coefficient and the scenario count were fixed
   in advance and never retuned.
5. Play the column with the best blended value.

## What happened

The cautious policy was decisively worse. On the eight
fresh games of its screen it scored about 87,000 points and survived 61.5 moves,
against about 177,000 points and 114 moves for the unchanged search. It lost
seven of eight games on both score and survival. Its clear and reveal rates per
move both fell, and its average longest chain fell from 7.9 to 5.1 — the
signature of a player who stops assembling anything.

It disagreed with the ordinary fair recommendation on about 28% of its
decisions, so this is a real behavioural change, not a rounding difference. The
gate required both averages to improve; both regressed by a wide margin, so the
confirmation cohort was never opened.

<TechnicalDetails title="The technical record">

**Source.** `approaches/fair-expectimax/root-risk/fair-root-risk.cpp`.
[Experiment index](/docs/research/experiment-index): **rejected —
ledger-recorded**, "score, survival, and flow all regressed."

Ledger section "Fair root CVaR ablation". Fresh screen `0x3e9d0000...007`, eight
games, **historical 7,000-point Sequence scoring**; the comparator is the
confirmed fair depth-3 search with five chance samples.

| Arm | Mean score | Mean moves | Clears / move | Reveals / move | Mean max chain |
| --- | ---: | ---: | ---: | ---: | ---: |
| Fair depth 3 (reference) | 176,819.75 | 114.125 | 2.043 | 1.141 | 7.875 |
| Root risk blend | 87,112 | 61.5 | 1.730 | 0.945 | 5.125 |

Paired change −89,707.75 points and −52.625 moves; one win and seven losses on
both measures; neither cohort censored. The policy switched away from the local
fair recommendation on 28.25% of trajectory decisions. Policy work 53,423 units
per move versus 55,674 for the reference (the optional switch audit added
another 50,333 and is not part of the deployable policy). Peak resident memory
3,948,544 bytes; 30.7 wall seconds. Because both admission means regressed,
`0x3e9e0000...00f` was not read.

Both means regressed in the same direction, so this is one of the ledger
rejections that a rescoring to corrected Hardcore rules would strengthen rather
than reverse ([audit 03](/docs/exploratory/audit-03-claim-arithmetic) checked
every 7,000-point paired comparison for a sign change and found only two, of
which this is not one).

**Verification.** Strict `-Werror`, ASan/UBSan, frozen fair-search parity,
deterministic scenario values, fractional-tail arithmetic, reflection, metadata
blindness, game-seed exclusion, legality, and resource-bound tests all passed.

</TechnicalDetails>

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

This rejects one specific risk rule: a fixed 75/25 blend of mean and worst-25%,
applied at the root only, over seven immediate-outcome scenarios, on top of the
depth-3 fair search. It rejects it firmly: the regression is large, consistent,
and mechanistically legible in the flow numbers.

What it does not settle is whether the reference search is *already* risk-averse
in a way nobody chose. An audit of the reference points out that its penalty for
a modelled game over is a flat −1,000,000, equal to about 59 row rises or 294
moves of survival, and 3.24× the entire recorded 64-game mean. One modelled
death among five samples therefore swamps the differences between the surviving
options, which makes the search minimise modelled death probability first and
maximise the board evaluation second
([audit 02](/docs/exploratory/audit-02-fair-d4), H2). If that reading is right,
adding explicit downside weighting on top was pushing an already-cautious search
further in the direction it was already leaning. Calibrating that one constant
is the open experiment; it has not been run.

## Sources

- `approaches/fair-expectimax/root-risk/fair-root-risk.cpp`: one bounded
  risk-sensitive ablation over the reference depth-3 evaluator; seven fixed
  public root scenarios differing only in the immediate chance transition, then
  the exact non-clairvoyant fair continuation for two more moves.