---
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
---

The strongest dependable public-information policy in this repository. It is
the comparator every new candidate must beat, and the teacher behind most
label-generation experiments.

## The idea

At each move, fair expectimax builds a tree that alternates between **player
choices** (take the best) and **chance events** (take the average):

<svg viewBox="0 0 640 210" className="my-4 w-full max-w-2xl rounded-xl border border-zinc-800 bg-zinc-900/40" role="img" aria-label="Expectimax tree">
  <g fontSize="11" fill="#e4e4e7">
    <rect x="270" y="12" width="100" height="30" rx="6" fill="#1e3a8a" />
    <text x="320" y="31" textAnchor="middle" fontWeight="700">MAX: column</text>
    <rect x="60" y="72" width="130" height="30" rx="6" fill="#78350f" />
    <text x="125" y="91" textAnchor="middle">CHANCE: reveal/next disc</text>
    <rect x="250" y="72" width="130" height="30" rx="6" fill="#78350f" />
    <text x="315" y="91" textAnchor="middle">CHANCE: reveal/next disc</text>
    <rect x="440" y="72" width="130" height="30" rx="6" fill="#78350f" />
    <text x="505" y="91" textAnchor="middle">CHANCE: …</text>
    <rect x="40" y="142" width="80" height="26" rx="6" fill="#1e3a8a" />
    <text x="80" y="159" textAnchor="middle">MAX</text>
    <rect x="150" y="142" width="80" height="26" rx="6" fill="#1e3a8a" />
    <text x="190" y="159" textAnchor="middle">MAX</text>
    <rect x="280" y="142" width="80" height="26" rx="6" fill="#14532d" />
    <text x="320" y="159" textAnchor="middle">leaf value</text>
    <rect x="460" y="142" width="80" height="26" rx="6" fill="#14532d" />
    <text x="500" y="159" textAnchor="middle">leaf value</text>
    <g stroke="#3f3f46" strokeWidth="1.5">
      <line x1="300" y1="42" x2="135" y2="72" />
      <line x1="320" y1="42" x2="315" y2="72" />
      <line x1="340" y1="42" x2="495" y2="72" />
      <line x1="110" y1="102" x2="80" y2="142" />
      <line x1="140" y1="102" x2="190" y2="142" />
      <line x1="310" y1="102" x2="320" y2="142" />
      <line x1="500" y1="102" x2="500" y2="142" />
    </g>
  </g>
</svg>

<GameTreeFigure seed={0x5eed1002} moves={10} leafDepth={1} caption="The same tree, live: the browser analogue of this search on a seeded game, with one further ply below each outcome. It is a playground demonstration of the mechanics, not the native reference whose cohort numbers this page quotes." />

"Fair" means the chance nodes **average over what can actually happen** —
every legal next disc, and the possible gray-disc reveals. Instead of hoping
for the best reveal or fearing the worst. Depth 4 (D4) completes four full
max/chance layers; the leaf is a hand-tuned public evaluator.

## Why averaging matters

A hidden gray disc can become anything from a 1 to a 7. An optimistic search
assumes the reveal you want; a pessimistic one assumes the reveal you fear.
Both mis-rank moves, and the ledger confirms it: fair chance handling beats
optimistic, worst-case, and tiny reused reveal samples. The price is work —
the tree grows by roughly an order of magnitude per ply.

## What the evidence showed

| Configuration | Games | Mean score | Mean moves | Note |
| --- | ---: | ---: | ---: | --- |
| Fair D3 | 8 | 235,071 | 71.0 | comparison policy |
| **Fair D4** | 8 | **400,675** | 116.4 | won 7 of 8 paired games |
| Fair D4 (broad cohort) | 64 | 308,296 | 90.0 | the reference mean |

- Deeper is **not** automatically better: selective D5, full D5, and
  cycle-boundary variants all failed their gates (see the sibling approaches).
- One D4 game scored 1,246,684 points: an anecdote, not an average. The
  million-point target requires the **mean** to exceed one million.
- D4 is a strong tactical fallback and teacher, but its average is far below
  the target. Every serious candidate keeps an exact D4 fallback.

## Sources

- `fair-only-horizon.cpp`: the fair-only depth-3 evaluator.
- `fair-only-depth4.cpp`: the depth-4 reference build (`make native` produces
  `build/fair-depth4`).
- The TypeScript counterpart used by the benchmark registry is
  `evaluateMoves` in `src/core/typescript/solver.ts` with the combined leaf.