When we ask the coach in CUBE Backgammon whether to hit or make a point, generating the legal moves is only the beginning. The harder question is what those moves will eventually earn, after the opponent has replied and more dice have been rolled.
A position evaluator is a function that turns a description of the game into an estimate of its future result. In backgammon, equity expresses value under specified scoring conditions. For a simple game with a fixed one-point stake, it is the expected number of points won or lost. A good engine needs this judgment whenever it stops searching before the game ends.
What does an equity of +0.50 mean?
The easiest way to understand the number is to use a position whose entire future fits into one roll. Here, we can count the outcomes exactly before asking how a neural network would estimate a more complicated board.
Turning a 75% win chance into +0.50 points
Return to the small race in our article on dice and learning. Each player has fourteen checkers off; Ivory’s last checker is on 6 and rolls before Red, whose last checker is on their own 1-point. Ivory wins on 27 of the 36 rolls and loses on the other nine.
Calculate equity before the roll
| Outcome | Rolls | Points received by Ivory |
|---|---|---|
| Single win | 27 | +1 |
| Single loss | 9 | −1 |
(27 × 1 − 9 × 1) ÷ 36 = +0.50 points. This averages 27 points won and nine lost across the 36 possible outcomes. The 75% describes winning frequency; +0.50 describes expected points per game.
In a single game, this position pays either +1 or −1. Its equity before the roll is +0.50 because of the probabilities of those results. Nothing in the estimate predicts which roll is about to arrive.
If every win and loss were worth one point, equity would be twice the winning probability minus one. Gammons and backgammons prevent us from using that shortcut for ordinary backgammon positions.
Why our evaluator keeps six probabilities
The position evaluator described in our engine account estimates six separate outcomes: single wins, gammon wins, backgammon wins, and the three corresponding losses. These probabilities add up to one. Keeping them separate lets the scoring calculation value a gammon differently from a single win.
With a fixed one-point stake, gammons paying two and backgammons three, the calculation is the probability of each result multiplied by its payout, then added together: single wins plus twice gammon wins plus three times backgammon wins, minus the corresponding losses. The choice that wins most often can earn fewer points if it also loses too many gammons.
The cube and match score add context. A gammon that wins the match has a different consequence from one that provides points you do not need. Future cube offers can also end the game before the checkers finish their race. A useful value function therefore needs a declared objective and scoring model, not just a board without context.
Where do the estimates come from?
In a small endgame, a calculation or database can supply exact answers. For a broad range of positions, a learned evaluator uses numerical weights adjusted from examples. The inputs describe the position; the training target supplies an outcome or an analysis to learn from. The model is useful when it can apply that learning to positions absent from its training examples.
There are different ways to obtain targets. TD-Gammon learned through self-play, using changing predictions and completed games to improve its evaluator. Our published CUBE Backgammon account also describes learning from positions analysed by a stronger teaching engine, following the established idea of knowledge distillation. That training happens beforehand; the phone uses the learned weights while you play.
The target matters. An estimate learned from a particular player’s games or a particular teaching engine reflects how those games were played or analysed. It does not acquire the status of perfect-play truth because a network has learned it. Tests on unfamiliar positions and independent analysis help check what has actually improved.
How a value becomes a move
For each candidate play, the engine can evaluate the resulting position from the same player’s perspective and prefer the higher value. Searching further lets it test that first impression against possible dice and replies, then evaluate the positions reached later.
Chance branches are averaged with their probabilities. An opponent’s replies must account for the opponent choosing a strong response. Combining those future values produces a revised comparison at the original decision. If every available action already had an exact value for the objective being played, choosing the best would be straightforward; obtaining those values is the difficult work.
A larger network is one way to seek better estimates, while a quicker network can leave more time for search. We test the combination through the decisions it produces. In one development attempt, a network agreed more closely with its teacher but performed worse in a separate continuation test, so we restored the previous version.
The same role in chess and Go
Chess and Go also need to judge unfinished positions. Their rules contain no dice, yet a learned value network can estimate an expected outcome because the program has limited knowledge about how the game will unfold. A predicted winning probability is not evidence of randomness in the rules.
AlphaZero combined a value estimate with a policy over moves. The value judged positions; the policy helped guide search towards promising choices. The scoring objective and inputs differed by game, while the need to make useful decisions without finishing every possible continuation was shared.
At bridge, the evaluator must additionally respect the player’s information: a perfect value for an exposed deal may be useless to someone whose opponents’ cards remain hidden. At blackjack, fixed dealer rules can make direct expected-value calculations manageable. A neural network is one tool for approximating value, not a requirement for every game.
This is why evaluation matters so much in our difficulty ranking. Counting possible futures tells us how quickly search grows. Learning to value the positions where that search must stop helps explain whether a strong player can be built without exploring them all.
Sources: Gerald Tesauro, Temporal Difference Learning and TD-Gammon (1995); Silver and colleagues, the AlphaZero paper (2018); Hinton, Vinyals and Dean, Distilling the Knowledge in a Neural Network (2015); and the GNU Backgammon manual. The exact race reuses our original dice-learning position and its independently checked 27/36 winning-roll count. It values a fixed one-point, cubeless, win-only game. The CUBE Backgammon description follows our existing published engine account; this article introduces no new benchmark or model-size claim.