Weighting the bad outcomes at the root

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.

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, in plain English

The cautious policy was far worse — not marginally, decisively. 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.

The technical record

Source. approaches/fair-expectimax/root-risk/fair-root-risk.cpp. 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 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.

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, 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.