How to Win Backgammon — the opening roll equities
The best opening rolls in backgammon are known to the decimal — from published, verified rollout tables, not guesswork. The best is 3-1 (8/5 6/5) at +0.167 equity. We also ran our own simplified rollout to see whether a simple bot could reproduce that ranking on its own. It agrees on 7 of 15 rolls and disagrees on 8 — and exactly why is the more interesting finding. Read the honest limitation below before you trust any of the "our sim says" numbers on this page.
The 15 opening rolls, ranked by published equity
"Published equity" is the verified, ground-truth ranking from GNU Backgammon's rollout tables. "Sim best-play" and "sim naive-run" are OUR simplified simulation playing the published-best move vs. a generic "run a back checker" instinct move — everything else held identical. "Agrees" means our simulation's win rate backs up the published ranking's direction (within a small tolerance); "Differs" means it doesn't:
| # | Roll | Best play | Equity | Sim best-play win% | Sim naive-run win% | Our sim |
|---|---|---|---|---|---|---|
| 1 | 3-1 | 8/5 6/5 | +0.1670 | 55.2% | 62.4% | Differs |
| 2 | 4-2 | 8/4 6/4 | +0.1234 | 54.9% | 61.4% | Differs |
| 3 | 6-1 | 13/7 8/7 | +0.1035 | 55.6% | 57.6% | Differs |
| 4 | 5-3 | 8/3 6/3 | +0.0638 | 55.6% | 53.4% | Agrees |
| 5 | 6-5 | 24/13 | +0.0545 | 65.1% | 64.9% | Agrees |
| 6 | 5-4 | 24/20 13/8 | +0.0155 | 64.5% | 52.2% | Agrees |
| 7 | 4-3 | 13/10 13/9 | +0.0125 | 49.2% | 63.0% | Differs |
| 8 | 6-4 | 8/2 6/2 | +0.0102 | 56.0% | 63.4% | Differs |
| 9 | 2-1 | 13/11 6/5 | +0.0088 | 48.3% | 63.9% | Differs |
| 10 | 6-3 | 24/18 13/10 | +0.0072 | 62.1% | 63.8% | Differs |
| 11 | 6-2 | 24/18 13/11 | +0.0066 | 63.3% | 63.5% | Agrees |
| 12 | 5-1 | 24/23 13/8 | +0.0056 | 62.4% | 47.0% | Agrees |
| 13 | 3-2 | 13/11 13/10 | +0.0053 | 51.2% | 61.8% | Differs |
| 14 | 5-2 | 24/22 13/8 | +0.0032 | 63.0% | 52.8% | Agrees |
| 15 | 4-1 | 24/23 13/9 | +0.0024 | 61.8% | 58.0% | Agrees |
Equity is the published, expected-points value of the best play at that roll (higher is better); it is quoted, not simulated. Win% columns are from our own 8,000-game rollout of each play choice.
Where our simple bot agrees with theory — and where it doesn't
Agrees (7 of 15)
5-3, 6-5, 5-4, 6-2, 5-1, 5-2, 4-1 — mostly the big "running" rolls, where the theoretically-best play IS essentially running a checker, so there's no subtle judgement call to get wrong.
Differs (8 of 15)
3-1, 4-2, 6-1, 4-3, 6-4, 2-1, 6-3, 3-2 — the rolls whose real best play slots a builder deep in the home board, hoping to cover it next turn. Our one-ply bot systematically prefers the safer running alternative instead.
Honest limitation: Our own simplified rollout AGREES directionally on 7 of the 15 rolls, clearly so on every 'running' roll (6-5, 5-4, 5-1, 5-2, 4-1), where the theoretically-best play IS essentially running a checker. But it systematically prefers running over SLOTTING a builder on 3-1, 4-2, 6-1, 4-3, 6-4, 2-1, 6-3, 3-2 -- the rolls whose real best play risks a mid-board blot to try to make an inner point next turn. A one-ply greedy heuristic can't price 'will I cover this blot next turn' correctly, so it undervalues exactly the plays that require that lookahead -- a very concrete illustration of why real backgammon equity tables took trained rollouts (and eventually neural nets), not simple heuristics, to get right.
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How we got these numbers
A Monte-Carlo rollout, the same method real backgammon equity tables are built with (just with a far simpler bot than a trained neural evaluator, disclosed below). For each of the 15 distinct opening dice rolls (a double can never be the opening roll -- both players roll one die each to decide who goes first, rerolling ties, per the official rule), we play the roll two ways: the theoretically-best play, taken verbatim from the published GNU Backgammon rollout table (bkgm.com/openings/rollouts.html) and used here as VERIFIED ground truth, and a generic 'run one back checker with both dice' instinct play. From there, both sides finish the game using an identical simple heuristic policy (always bear off when legal; prefer hitting a blot or making a point; avoid leaving a new blot in another checker's direct single-die range; otherwise advance your most-behind checker) so the only deliberate difference between the two simulated win rates is that FIRST move. Full legal-move rules (bar entry, blot hits, the exact/overshoot bear-off rule) are modelled; the doubling cube is NOT modelled (cubeless play throughout), and dice are applied in rolled order without an exhaustive legality-maximizing search, a disclosed simplification of the rule that you must use both dice if any order allows it. IMPORTANT DISCLOSED LIMITATION: this is a one-ply greedy heuristic, not a trained rollout, and it does NOT reliably reproduce the published ranking for rolls whose theoretically-best play SLOTS a builder deep in your own home board, hoping to cover it next turn (3-1, 4-2, 2-1, 3-2, 4-3, 6-4) -- our bot systematically prefers the safer naive-run alternative there, because correctly valuing a slot requires looking ahead to the coverage probability and the point's positional value, which a reactive one-ply heuristic cannot do. This is not a bug; it is the same failure mode that made pre-neural-net backgammon programs weak at builder play, and it is reported in full below rather than hidden. For rolls where the theoretically-best play is itself close to a running play (6-5, 5-4, 6-1, 5-3, 5-1, 5-2, 4-1) our simulation DOES corroborate the published ranking's direction, since there is no subtle slot risk to misprice.
Source: published opening equities from GNU Backgammon's rollout tables
(bkgm.com/openings/rollouts.html), used here as verified ground truth — not reproduced by our own simulation.
Source code for our comparison rollout lives in our sims/ folder (backgammon_sim.py).
See also the full Backgammon entry.
Common questions
What is the best opening roll in Backgammon?
By published, verified rollout equity, 3-1 — making the 5-point with 8/5 6/5 — is the best opening roll in the game, worth +0.167 equity. This ranking is quoted directly from GNU Backgammon's own rollout tables, not something our own simulation derived on its own.
Why doesn't a simple simulation reproduce the published Backgammon opening theory?
Our own simplified rollout AGREES directionally on 7 of the 15 rolls, clearly so on every 'running' roll (6-5, 5-4, 5-1, 5-2, 4-1), where the theoretically-best play IS essentially running a checker. But it systematically prefers running over SLOTTING a builder on 3-1, 4-2, 6-1, 4-3, 6-4, 2-1, 6-3, 3-2 -- the rolls whose real best play risks a mid-board blot to try to make an inner point next turn. A one-ply greedy heuristic can't price 'will I cover this blot next turn' correctly, so it undervalues exactly the plays that require that lookahead -- a very concrete illustration of why real backgammon equity tables took trained rollouts (and eventually neural nets), not simple heuristics, to get right.
Which opening rolls are the least controversial?
For the big running rolls (6-5, 5-4), the theoretically-best play IS essentially running a checker, so there's no subtle slot-vs-safety judgment call to get wrong -- which is exactly why our simulation, published theory, and plain intuition all agree there.