The Match Equity Table
[From How to Play Tournament Backgammon, Kit Woolsey, 1993.]
In order to make intelligent cube decisions during a match, it is essential to know what your chances of winning the match are at various match scores. Backgammon theoreticians have been working on match equity tables for several years, attempting to improve them. Thanks to Hal Heinrich, we finally have a large database from which we can determine empirically vital information such as likelihood of gammons and the value of cube leverage to the trailer in the match. Using this data, I have constructed the following match equity table:
In this table, the numbers down the side represent the number of points Black has to go, and the numbers across the top represent the number of points White has to go. The number in the table where the appropriate row and column intersect represents Black’s probability, in percentage, of winning the match.
For example, suppose in a 21-point match Black is leading 14 to 8. Black has 7 points to go, White has 13 points to go. Looking at row 7 and column 13, we see that Black has a 78% chance to win the match, therefor White has a 22% chance. Note: All equities which involved either player one point away assume that we are in the Crawford game.
Constructing the Table
How is this chart constructed? Basically, the idea is to start from the smaller scores and work backward. Using results from the database and a little intelligent judgment we can take any match score, determine what the probabilities of various occurrences are, multiply these probabilities by the equities of the match scores resulting from these occurrences (which we have already worked out since we started with the smaller scores), and arrive at the desired match equity.
For example, let’s determine the equity with Black: 1 away, White: 2 away (Crawford). Obviously if we reach the score Black: 1 away, White: 1 away, then Black’s equity will be .50. Since we are in the Crawford game the cube cannot be turned, so three things can happen:
- Black can win the game. If this happens, Black wins the match for an equity 1.00.
- White can win a single game. If this happens, we are at double match point, so Black’s equity will be .50.
- White can win a gammon or backgammon. If this happens, White wins the match so Black’s equity will be 0.
The database shows us that Black (the leader) has a 51% chance to win the game. The reason he is better than even money is that White will take some extra risks to get the gammon he badly needs, so White will win a bit less often.
Also, the database shows us that in a one-way gammon situation (that is, one player can use a gammon while the other player cannot when the game must be played to conclusion), about 22% of White’s wins will be gammons. This figure has been disputed by some players, but it is consistent with general agreement and what the database shows. Thus, of White’s 49% wins, 22% of these or 10.78% will be gammons.
Therefore, we have the following results:
|Black wins game:||51.00%|
|White wins single game:||38.22%|
|White wins gammon:||10.78%|
Multiplying these probabilities by the match equities (which we already have determined) for each occurrence, we see that Black’s probability of winning is:
Rounding this down and putting it in the table, we see that Black’s match equity at this score is about 70%, and that is the figure we use in our table.
Let’s look at another example. Suppose we are trying to determine the equity for Black: 2 away, White: 6 away, and that we have already determined all the equities for scores with a total (for both players) seven or fewer points to go. The database gives us the following results with the leader having two points to go and the trailer having six or more points to go:
|Leader wins 2:||31%|
|Leader wins 1:||11%|
|Trailer wins 1:||27%|
|Trailer wins 2:||22%|
|Trailer wins 4:||9%|
As you can see, the trailer does very well under these conditions, winning 58% of the games for an average equity of .33 points per game. This, of course, is due to the tremendous cube leverage available to the trailer when his opponent has two points to go. If he doubles, the leader can never redouble, but if the leader doubles and the trailer takes, then the trailer will immediately whip it back to four.
In addition, gammons help the trailer if the cube is turned, but they don’t help the leader. Thus the trailer can make much more effective use of the cube than the leader, so he figures to have a considerably higher winning rate at this match score, and that is confirmed by the database.
Without the database, it would be anybody’s guess just how important the cube leverage actually is. (In fact, earlier attempts to construct a match equity tabled tended to undervalue this cube leverage.) But with the database we can get a good empirical grasp of the strength of the cube leverage. Anyway, if we assume that these precentages are correct, we would have the following possible results after the next game:
|0 away||6 away||100%||×||.31||=||.3100|
|1 away||6 away||90%||×||.11||=||.0990|
|2 away||5 away||75%||×||.27||=||.2025|
|2 away||4 away||68%||×||.22||=||.1496|
|2 away||2 away||50%||×||.09||=||.0450|
This is rounded to 81%, and that is the figure in the chart for 2 away, 6 away. This process is continued until the chart is complete.
Accuracy of the Table
How accurate is the match equity table? Not clear. Obviously there are some small errors since all the figures are rounded up or down to the nearest percent. In addition, some of the assumptions used to contruct the table may be incorrect, which will cause further errors. However, experience has shown the table to be quite accurate. It conforms very well with the large database of matches which have been surveyed. Thus, there is good evidence that none of the figures are off by more than 1%.
Since evaluating backgammon positions is usually an inexact science anyway, errors from misevaluating a position are likely to be far more serious than errors from using the match equity table. Therefore, we can assume for all practical intents and purposes that the table is accurate, and the rest of this booklet will be based on that premise.
[From rec.games.backgammon, Kit Woolsey (1999).]
I will be the first to admit that there may be errors in my table. The figures were not derived on a particularly scientific basis. They are a mishmash of empirical data, a program which was based on some assumptions which may not be sound, a lot of judgment, and some fudging. Other people have derived similar match equity tables, all of which are probably just as inaccurate.
[From rec.games.backgammon, Kit Woolsey (1995).]
When I first constructed my equity table I believe I used a gammon rate of about 21%. This was consistent with my own estimates as well as the results from a database of several hundred games which had to be played to conclusion due to the match score.
Currently I think the theoretically correct figure should be higher, perhaps 25%, with correct play. However, since most players are not sufficiently adept at gammon collection, the 21% figure appears to work out well in practice, and the match equity table is consistent with real life results.
Personally I consider Friedman’s 36% estimate way out to lunch. He has no evidence to back this up other than his own, probably biased, rollouts. All three computer programs (Expert Backgammon, TD-Gammon, and Jellyfish) come up with a gammon rate somewhere in the mid 20’s, and this would be what the majority of experts would agree with also, I think.
Keep in mind that I did not construct my equity table using a precise formula. Rather, I took a large database of empirical results, molded together some assumptions from these results, did the appropriate fudging, and out came my match equity table. I don’t claim it is mathematically precise, but it does have one very important thing going for it — it appears to work!