It just makes… cents

Why on earth do we use millipoints to describe equity? Here, we have a proposal for a new unit, the cent, with some discussion on what makes it better than the de facto standard.

How weird would it be to be given directions in lightyears? It would sound something like “continue on that highway for 3 millionths of a millionth of a parsec.” How about scheduling your day — would you ever use “millimonths”? Those go for about 40 minutes, they’re just about the right length for an appointment. These options sound absurd when we can use just miles or hours, but this is exactly how we discuss equity in backgammon. Just as we organically choose the appropriate unit of time or length for the context in real-life situations, we should start doing so in the game we love.

A position’s equity is an essential quantity for the backgammon player, used to determine both checker play and cube decisions. It represents the fair value of a backgammon position. If the position has an equity of 0.5, and you are playing for $1 a point in a money game, you would expect to win 50 cents starting from this position when averaging across all variations of wins, gammon wins, and losses. The cubeless equity, which presumes that every game is played to its completion (i.e., no doubling out), is calculated as a function of expected wins (W), gammon wins (WG) backgammon wins (WBG), losses (L), gammon losses (LBG) and backgammon losses (LBG) as:

Equity = 3WBG + 2WG + W − 3LBG − 2LG − L

Powerful modern analysis tools, such as XG and GnuBG, allow us to make accurate estimates for each of these quantities, and therefore of the equity, for any position at every match score. As an example, consider the decision of what to play following an opening roll of 65, shown in the position below:

Here, you have two reasonable options: you can commit to a racing game plan and make Lover’s Leap, playing 24/13 and moving a back checker to the midpoint; alternatively, you can make a more balanced play, splitting and unstacking your midpoint, 24/18 13/8. Which play is better?

Even a beginner player will know that the leap is the preferred play, and the bots agree! But how much better is it? GnuBG (3-ply) says that our equity after Lover’s Leap is 0.095, whereas, after the balanced play, it’s 0.035. It turns out that the so-called “balanced play” would be considered an error by our standards since we would be giving up over 0.03 units of equity.

Well… what are the units of equity? These differences in equity represent the number of points we stand to gain or lose. Here, the difference is 0.060 points. Often, to trim the leading zeroes, this quantity will be quoted as 60 millipoints.

This, right here, is the moment in every equity discussion that makes my stomach turn. A millipoint? Why? It’s an extremely unnatural unit, like “millimonths” would be, except we’ve against all odds adapted to it as the de facto standard. This arbitrary choice is especially painful when there is a natural candidate for a replacement term waiting in the wings.

Here is my proposal: the community should use cents as a unit of equity instead of millipoints. 1 cent corresponds to 0.01 units of equity, or 10 millipoints. Cents arise naturally when valuing each game at a dollar or a euro (we’ve already unwittingly used it in this fashion in this article!) Therefore, players should have an intrinsic understanding of the unit the moment they learn of it. Two plays that are basically even would be within 1 cent (−0.010); an error costs you 3 cents (−0.03) and a blunder 8 cents (−0.08). It even comes ready with a symbol (¢) that can be rendered in any web browser. This unit descends from the heavens, fully formed and ready for consumption. Nobody should have any trouble adapting to this change.

It features multiple additional benefits. For starters, we never really need to be much more accurate than a cent — what would you do differently if you knew an error cost you −0.040 equity instead of −0.034 — and so it would simplify discussions by eliminating the extra information. If you really need to quantify a difference between decisions that is smaller than 1¢, you can certainly still use decimals. The decimal separator organically acts as a signal to the reader that the difference is negligible to all but the most expert players. Consider the following position:

The opening 21 can be played by slotting 21$ (13/11 6/5) for an equity of +0.003, or by splitting 21S (24/23 13/11) for an equity of −0.001. The difference between these two plays is 0.4 ¢ (−0.004). It’s instantly obvious why it’s considered up to personal preference which you should play — the “wrong” play will cost you less than half a cent.

It even takes less time to pronounce. What bots called a “−0.060 error” above could be read as six-tee-mi-lee-points (5 syllables) or six-cents (2 syllables). Which would you rather say?

So to summarize — cents are shorter than millipoints, are less distracting, are more intuitive, and don’t sacrifice any valuable precision. It’s time to make the switch! It makes too much sense not to.

This article was originally published on the Opengammon blog, complete with interactive diagrams. Reprinted with permission.