Match Equities |
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Hal Heinrich’s Database
As many readers are undoubtedly aware, Hal Heinrich has stored on computer a database of over 1000 matches, most of which involve at least one top-level player. To my knowledge, this is the most extensive such collection in existence. Recently, he has done some analysis of the data, which yields very interesting results. He examined the statistics of all match scores and printed the results. A typical printout is as follows:
2. | 5 | occurs | 140 | 13.40% | ||
Leaders wins 75.30% | ||||||
0. | 5 | occurs | 32 | 22.86% | ||
1c. | 5 | occurs | 28 | 20.00% | ||
2. | 0 | occurs | 2 | 1.43% | ||
2. | 1c | occurs | 9 | 6.43% | ||
2. | 3 | occurs | 24 | 17.14% | ||
2. | 4 | occurs | 45 | 32.14% |
Translation: With the score 2-away 5-away (the leader needs 2 points to win, the trailer needs 5 points), the leader won the match on the next game 22.86% of the time; the leader reached 1-away 5-away 20% of the time; the trailer won a single game 32.14% of the time; the trailer won two points 17.14% of the time, and so forth. As you can imagine, this kind of data lends itself to empirical analysis of gammons, match equities, and cube decisions, which has not been previously possible.
I have done some analysis of particularly interesting areas. In this article, I’ll discuss three: (1) calculating the frequencey of gammons, (2) creating a new (and much more accurate) match equity table, and (3) examining the taking percentage on initial doubles.
Gammon Frequency in One-Way Gammon Situations
By “one-way gammons,” I mean situations when a gammon only matters to one side. This is a very important figure for any match equity table, since the match equity of all Crawford and post-Crawford games is clearly dependent on this number.
When I constructed my first match equity table, I used a gammon estimate of 18% (that is, 18% of the wins for the side that can benefit from a gammon are in fact gammons), but I have since been convinced that this estimate was a bit low.
Recently Roy Friedman has claimed to have rolled it out several thousand times; he came up with a gammon rate of 35%. This is considerably higher than anyone else’s estimate and — if correct — would clearly call for some changes in the match equity tables.
I found this estimate hard to believe and tried playing out a few hundered games, playing as aggressively as possible for the gammon for the side which needed it and as aggressively as possible for the win for the side that would be hurt by the gammon.
The number of gammons from my rollouts should be higher than in real life. However, even with this approach, my gammon results came to 25%, which indicated to me that the correct figure is closer to 20%. Of course, my sample was small and there is the possibility of some kind of bias in my rollouts.
It would be better if we had a sample using a variety of different players, so that the playing style of one individual would not create a bias. From Hal’s database, we have such a sample. I examined all one-way gammon situations which, of course, had to fall into one of the two following areas:
- The match is in the Crawford game, and the trailer has an even number of points to go.
- The Crawford game has passed, and the trailer has at least three points to go.
My analysis did not take into account any effect the free drop may have had, but I’m pretty sure it would be minimal. The results were as follows:
Leader wins: | 212 | 52% |
Trailer wins single: | 153 | 38% |
Trailer wins gammon: | 41 | 10% |
Total: | 406 | 100% |
So there you have it. Only 21% of the trailer’s wins were gammons. This is much more in line with generally accepted theory and indicates there may have been some kind of bias in Friedman’s rollouts.
While 406 is admittedly not the largest sample in the world, it is a reasonable size. The sample represents a cross-section of many different players and the results are consistent with the general consensus, so until proven otherwise I would be inclined to accept these results and reject Friedman’s rollouts.
One further question: Why did the leader win 52% of the games? I would have expected him to win less than half of the games, since the trailer can play all out for the win (i.e., stay back until the bitter end when playing an ace-point game). After all, it doesn’t matter to the trailer if he is gammoned.
There are several possible explanations, all of which may have some influence:
- The first is luck. As mentioned, 406 games is not that large a sample. Assuming the trailer and the leader had equal chances of winning the game, the results are less than one standard deviation away from the mean, meaning the results could have easily been pure chance.
- Another possible influence is that the trailer is playing too aggressively for the gammon. (On the other hand, this may be his correct strategy — we just don’t know).
- The free drop is another possibility. Clearly the free drop has some value to the leader. A total of 100 of the 406 games were played under free drop conditions (post-Crawford, trailer has an odd number of points to go). In addition, some of the games were undoubtedly played using the Holland rule, which makes the free drop even more valuable. It is hard to say just what this is worth to the leader, but I would not be surprised if it adds 1% to his chances.
- Last but not at all least is the skill factor. While these were generally top-level matches, many of them involved a world class player against a weaker opponent. Under these conditions, it is more likely that the better player would be the leader when the crucial situation is reached — simply because he is the better player. Given this, he would have better than a 50% chance in the game.
Match Equities
Back around 1980, I first attempted to construct a decent match equity table. Using various questionable presumptions, I made a model for my computer which then calculated the match equities at each score under these presumptions. At the same time, Bill Robertie and Danny Kleinman had done the same thing. We had all done this totally independently with no knowledge of the work of the other two.
I collected all three tables and found that they were very similar. In no case was there a discrepancy of more than 2%. I compiled an average for each match score and published the first match equity table in the Backgammon Times newsletter in 1981.
This table, with various modifications and augmentations (see Bill Robertie’s table in his book Lee Genud vs. Joe Dwek, for example), has been pretty well accepted as correct through the years, and various formulas such as the Underwood sequence have been generated using this table as a basis. However, we have not been able to test the table for accuracy due to lack of data. This is no longer a problem because Heinrich’s database should give us some good hard evidence.
2 | 3 | 4 | 5 | 6 | 7 | 8 | |
1 |
70 70.25 (79) |
75 74.99 (76) |
83 83.82 (65) |
85 85.09 (79) |
90 92.62 (42) |
91 88.42 (32) |
94 93.26 (23) |
2 |
59 60.34 (198) |
70 72.07 (155) |
76 75.30 (140) |
82 81.16 (101) |
86 86.31 (79) |
89 88.20 (60) | |
3 |
60 61.05 (196) |
67 67.52 (206) |
74 73.66 (140) |
79 76.58 (131) |
84 82.37 (84) | ||
4 |
58 58.78 (197) |
66 65.06 (145) |
72 70.72 (138) |
78 74.99 (105) | |||
5 |
58 57.24 (174) |
64 61.92 (187) |
70 65.47 (138) | ||||
6 |
58 55.79 (207) |
64 60.24 (141) | |||||
7 |
57 55.33 (187) |
The numbers down the left-hand side of the table represent the number of points needed by the leader in the match. The numbers across the top represent the number of points needed by the trailer. In each case, the first number in a cell represents the original figure in the table published in Backgammon Times (in percent equity for the leader). The second figure represents the results of the database. The third figure represents the number of samples for that match score in the database.
So how did we do? For 5-point matches (that is, the trailer having 5 or fewer points to go), it is clear that the match equity table has stood the test of time. Virtually all the percentages were extremely close, as close as could possibly be expected considering the relatively small samples.
The one exception — 2 away, 4 away — can be seen as a clear anomaly. Of the 155 matches with that score, the trailer won the match on that game only once! This is obviously way off as we would expect the trailer to win a 4-game at least a dozen times out of 155 matches. If this had happened, no doubt the results would have been in line with the table.
It seems reasonably safe to assume for short matches that the table is approximately correct. For other scores, the table doesn’t appear to be as accurate. It seems to consistently make the leader a bigger favorite than the actual results show him to be.
I had always suspected that the methodology I used tended to favor the leader too much for lopsided scores. The problem lies in the tremendous cube leverage the trailer has available to him. My presumption attempted to take this into account, but it is now clear that this cube leverage was undervalued.
To get an idea of just how strong it is, let’s take a look at all games where the leader had two points to go and the trailer had six or more points to go. The results were as follows:
Leader wins 2: | 122 | 31% |
Leader wins 1: | 42 | 11% |
Trailer wins 1: | 104 | 27% |
Trailer wins 2: | 84 | 22% |
Trailer wins 4: | 35 | 9% |
Total: | 387 | 100% |
If you carry through the arithmetic, you will see that on average the trailer wins .33 points per game! Similar (although not as extreme) results occurred when the leader had more points to go but had a significant lead in the match. Maybe it shouldn’t be that way in Utopia, but that’s the evidence in matches involving the best backgammon players in the world. So we have to accept that cube leverage is quite high for lopsided scores.
So, back to the drawing board. Using these results to give me a likely distribution at various scores, I rewrote the computer program to generate new match equity figures.
The following table gives the trailer considerably greater chances at lopsided scores than previous tables (compare with Robertie’s table in Lee Genud vs. Joe Dwek, for example), but I have good reason to believe it is more accurate. The empirical cube leverage is taken into account and the results conform both with Heinrich’s database and my own intuition much better than the previous tables.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
1 | 50 | 70 | 75 | 83 | 85 | 90 | 91 | 94 | 95 | 97 | 97 | 98 | 98 | 99 | 99 |
2 | 30 | 50 | 60 | 68 | 75 | 81 | 85 | 88 | 91 | 93 | 94 | 95 | 96 | 97 | 98 |
3 | 25 | 40 | 50 | 59 | 66 | 71 | 76 | 80 | 84 | 87 | 90 | 92 | 94 | 95 | 96 |
4 | 17 | 32 | 41 | 50 | 58 | 64 | 70 | 75 | 79 | 83 | 86 | 88 | 90 | 92 | 93 |
5 | 15 | 25 | 34 | 42 | 50 | 57 | 63 | 68 | 73 | 77 | 81 | 84 | 87 | 89 | 90 |
6 | 10 | 19 | 29 | 36 | 43 | 50 | 56 | 62 | 67 | 72 | 76 | 79 | 82 | 85 | 87 |
7 | 9 | 15 | 24 | 30 | 37 | 44 | 50 | 56 | 61 | 66 | 70 | 74 | 78 | 81 | 84 |
8 | 6 | 12 | 20 | 25 | 32 | 38 | 44 | 50 | 55 | 60 | 65 | 69 | 73 | 77 | 80 |
9 | 5 | 9 | 16 | 21 | 27 | 33 | 39 | 45 | 50 | 55 | 60 | 64 | 68 | 72 | 76 |
10 | 3 | 7 | 13 | 17 | 23 | 28 | 34 | 40 | 45 | 50 | 55 | 60 | 64 | 68 | 71 |
11 | 3 | 6 | 10 | 14 | 19 | 24 | 30 | 35 | 40 | 45 | 50 | 55 | 59 | 63 | 67 |
12 | 2 | 5 | 8 | 12 | 16 | 21 | 26 | 31 | 36 | 40 | 45 | 50 | 54 | 58 | 62 |
13 | 2 | 4 | 6 | 10 | 13 | 18 | 22 | 27 | 32 | 36 | 41 | 46 | 50 | 54 | 58 |
14 | 1 | 3 | 5 | 8 | 11 | 15 | 19 | 23 | 28 | 32 | 37 | 42 | 46 | 50 | 54 |
15 | 1 | 2 | 4 | 7 | 10 | 13 | 16 | 20 | 24 | 29 | 33 | 38 | 42 | 46 | 50 |
Taking Percentage on Initial Doubles
It has always been my belief that most backgammon players pass too much and double too little. With proper play, I believe that at least two thirds of initial doubles should be taken, but in practice the figure is a lot lower.
How much lower? That’s what we have the database for. In order to eliminate any possibility of bias due to the match score, I restricted my survey to the following four situations:
- Leader has 15 or more points to go — any score.
- Leader has 11–14 points to go — only if trailer is within 2 of leader.
- Leader has 7–10 points to go — only if trailer is within 1 of leader.
- Leader has fewer than 7 points to go — don’t count it.
Since we have over 1000 matches and many of them are long matches, there was no shortage of samples. Survey says:
Initial double taken: | 1501 | 40.5% |
Initial double passed: | 2210 | 59.5% |
Total: | 3711 | 100.0% |
This is a large sample size, so there is every reason to believe that these figures are meaningful.
The real situation is actually worse than this. If a player played for an undoubled gammon and got it, the game would show up as a two-point win on the printouts, so I would have counted it as an accepted initial double (not being able to tell the difference). Therefore, it seems safe to say that at most 40% of initial doubles are taken. However, there is one strange bias in these results.
Consider the following situation: We have two different 25-point matches. One of them is between two fast and loose players; the other between two players who are tight with the cube. The fast and loose players figure to have more taken doubles, and therefore on average they will play fewer games in the 25-point match than the tight players. Tight players, by their nature, are hogging more than their fair share of the sample! Thus there is some reason to believe that the true figure for average number of initial doubles taken is actually higher than the survey shows, but I don’t think it will make all that much difference.
There is no doubt about it: The average good player is much tighter with the cube than I think he should be.
To check if I was imagining all this, I played through all my personal matches in the collection so see what my own percentages were. In order to have a decent sample size, I was more liberal regarding which match scores I would accept. My conditions were as follows:
- Leader has 9 or more points to go — any score.
- Leader has 7 or 8 points to go — trailer must be within 3 of leader.
- Leader has 5 or 6 points to go — trailer must be within 1 of leader.
- Leader has 4 or fewer points to go — do not count.
The results were as follows:
Opponent takes: | 67 | 48% |
Opponent passes: | 72 | 52% |
Total: | 139 | 100% |
Woolsey takes: | 90 | 59% |
Woolsey passes: | 63 | 41% |
Total: | 153 | 100% |
Well, at least I appear to practice what I preach, for better or worse. My take percentage is way above the national average. The only reason why it isn’t higher, in my opinion, is that my opponents aren’t doubling enough. My opponents’ take percentages aren’t as high as I think they should be but they are much higher than the national average, which indicates I am also doubling faster than most players.
Observant readers will have noticed an anomaly. If I am such a fast doubler compared to other players, why is it that my opponents get in the initial double more frequently than I do?
Perhaps I was unlucky in these matches? No, my results for them was 29–21, certainly respectable.
Is my opening play really so bad that I get into trouble more often than my opponents? Maybe, but it is hard to believe I could survive in top-flight backgammon if that was the case. I have some ideas on the subject but, for now, I’ll leave it for you to think about.