Equity in backgammon

#Equity is the value of a position

Equity is what a position is worth to the player on roll, in points, on average. It answers one question: if this exact position were played out a very large number of times, how many points per game would you come out ahead or behind?

An equity of +0.200 means you expect to win a fifth of a point per game from here. An equity of 0 means an even game, and -1.000 means you are as badly off as if you had already lost a single game. Every analysis, every move list and every cube decision in HedgeHog is written in this one number, which is what makes a move and a cube action comparable at all.

#Built from six outcomes

A game of backgammon ends in one of six ways: you win or lose a single game, a gammon or a backgammon, worth one, two or three points. An engine estimates the chance of each, and equity is the sum of each outcome's value times its chance.

Engines report the chances in a cumulative form. Win is the chance of winning at all, gammon the chance of winning a gammon or better, and backgammon the chance of winning a backgammon, with the same three for losing. In that form the cubeless equity is:

equity = (win - loss) + (gammon won - gammon lost) + (backgammon won - backgammon lost)

Each term counts one extra point: a gammon is counted once as a win and again as a gammon, and a backgammon a third time.

#A worked example: the opening 3-1

The best opening roll is 3-1, played 8/5 6/5 to make the 5-point. HedgeHog's rollout of that position, 2592 games played to the end, gives the player who made it:

So the equity is (0.5532 - 0.4468) + (0.1732 - 0.1170) + (0.0077 - 0.0053), which is 0.1064 + 0.0562 + 0.0024 = +0.165. More than a third of that comes from gammons: making the 5-point wins a few more games, and it wins them bigger. Every roll's numbers are in Opening moves.

#Cubeless and cubeful equity

The formula above is cubeless equity: the value of the position if the game were played to the end at its present stake, with nobody allowed to double. It is what a neural network estimates directly, and the right number for comparing two plays in most positions.

The doubling cube changes the value. Owning the cube is an asset: it lets you double when the position turns in your favour and never be doubled out while it is yours. So a position is worth more with the cube in your hands than in the middle, and more in the middle than on the other side. Cubeful equity includes that. HedgeHog prices it from the cubeless chances with Janowski's model of how efficiently a cube is used, or by searching the cube decisions ahead, and its analysis reports errors in cubeful equity by default.

The two can disagree about which play is best. A play that keeps the position volatile, when you own the cube, can be worth more cubeful than a safer play with the higher cubeless number, because a big swing in your favour turns into a double the opponent cannot take.

On a cube decision, equity is written relative to the stake before the double: Double/Pass is always +1.000, the one game the opponent concedes, and Double/Take is what the game is worth once the opponent accepts: its cubeful value with the cube on their side, times two, because the stake is now twice as high. Reading the three rows is in The doubling cube.

#Equity in a match

In a match, points are not what you want: you want to win the match. The real value of a position is your match winning chance (MWC), and a gammon is worth whatever it does to that chance, which can be much more than a single game or nothing at all. At double match point (both players one point away) a gammon counts for nothing, and the equity collapses to 2 × win - 1.

To keep one scale across money and matches, HedgeHog converts match winning chances back into EMG, equivalent to money game: at the current score and cube, losing a single game is -1 and winning one is +1, and every other result falls where its match winning chance puts it. An error of 0.050 EMG therefore means about the same thing at any score, which is what lets a match's errors be added up into one error rate. The chances themselves come from the match equity table.

#Why errors are measured in equity

A mistake costs the difference between the equity of the best play and the equity of the play you made. Measured that way, a mistake is a quantity, not a label: choosing a play worth 0.100 less is the same size of error whether it happened in the opening or the bearoff, in a checker play or a cube action. That is the foundation of the error symbols and the error rate, described in Backgammon error rates and PR.