Comparing possible futures to inform the present.
Intro
This is exploratory data analysis around ‘Backgammon Swing Rolls’, inspired by Mochy’s comments in the book Backgammon Super Genius Quiz by James Vogl. This review hopes to catch a glimpse of how the best players approach the game, and perhaps bring back a few lessons to help the rest of us.
What is Swinger Analysis
In a backgammon checker play decision we roll the dice and need to decide what to move. Let's call this Roll 0 and compare two alternative moves, let’s call those Move A and Move B.
Looking one level deeper at our opponent's next roll, let’s call that (Roll 1), we can compare the position after Move A or Move B and after some specific Roll 1. In some cases there is a big difference for us, and so we say there is a big ‘swing’ on that roll 1 in this case. This is tricky to describe in words, so let’s look at an example:
Black is considering Move A (5/1 4/1) or Move B (6/3 4/Off). What will happen in each case if White rolls 6,1 next? After Move A then White’s 6,1 will dance (good for black), but after Move B, White’s 6,1 will escape (good for white). So we can say there is a big swing on White rolling 6,1 on Roll 1 - and we can visualise this by graphing the difference in equity for each of white’s subsequent rolls.
We can even take this a level deeper, and consider the difference in equity after the subsequent Roll 2. Here we assume that Roll 1 is played perfectly, then compare all the average of all possible Roll 2 specific dice combinations.
Here’s an example of Roll 2 swinger analysis:
Black is considering Move A (24/23 7/1) or Move B (24/17), score is 7a7a. It is a very close decision, however let’s consider what happens on Black’s next roll. If Black makes the anchor this move, double 6’s will play very well next move, escaping both checkers. But if Black runs with (24/17), double 6’s on the next roll will not be so good - one checker will still be trapped behind a prime! So we can see there is a big swing on Roll 2’s double 6’s.
Now that we've defined swinger analysis, what is the point of describing position equities in this way? Let’s investigate some possible uses.
Making Sense from XG
Sometimes the computer’s recommended move causes even elite players to shrug their shoulders. Using swinger analysis, we humans might more easily interpret the sometimes bemusing word of god. Here is an error made in the UBC final:
Why does XG prefer (13/11 13/7) and why is (13/5) a blunder? We can certainly rely on proverb’s like Michi’s “Break the Mountain” to often find the right answer, but why is this true? Let’s try to compliment our heuristic thinking with swinger analysis:
Looking at this graph, we can make a statement like:
After Black plays the right move (13/11 13/7), regardless of White’s next roll, all of Black’s next rolls play better than after an initial (13/7), except 53 and 33. That is, flexibility is important in this position and we should perhaps consider return shots .
It seems that use of this tool can make more obscure ideas more readily apparent in some positions.
All Rolls Aren’t Created Equal
Over the board we would ideally evaluate all 36 of our opponent’s next (Roll 1) rolls and all 1296 combinations of the next two rolls (Roll 1 * Roll 2). Unfortunately, doing this exhaustively is just not feasible for us mere humans, so we can instead look for shortcuts to investigate the most valuable paths of inquiry first.
Here we collate the swinger analysis from each of the 61 significant checker errors (>0.040) from the 9 matches of the 2022 UBC final between Dirk and Mochy. Although this data set is not particularly large, it was chosen because it was conveniently already thoroughly analysed, and because any position which causes an error for an elite player is likely to hold some useful insight for the rest of us!
Calculating the variance (standard deviation) across each roll for each position, we find:
And for Roll 2:
The obvious pattern is that:
In difficult positions, the largest swings for both Roll 1 and Roll 2 tend to occur with doubles, and more prominently with double 6’s and double 5’s.
So we might say, for every decision consider ‘how does 66 and 55 play for my opponent after this roll - also - how does 66 and 55 play for me next roll’? This more manageable exercise may quickly prove the difference in finding the best play over the board.
Patterns in Positions with Primes
Let’s look at swinger analysis, specifically in positions where escaping from a Prime is a key theme.
Example 1
Example 2
Example 3
In each of these examples, we can notice that there are positive and negative swings for each Roll 1, but for Roll 2, the best move only shows positive swings.
Over the board we might expect positions with positive and negative swings to be difficult to evaluate. However this analysis suggests that:
In prime related positions, rather than considering what your opponent will roll (Roll 1), instead more focus could be placed on your next roll (Roll 2).
That is, choosing moves where all your numbers play better on your next roll may be a useful mechanism to finding the best move.
How was the Analysis performed?
Position equity data is produced via XG Dice Distribution function. Move A and Move B positions are compared and graphs produced using this sheet. Some further analysis performed with Python here.
Other Avenues of Exploration
Some loose ends not investigated here but may be worthwhile following:
Using swinger variance as a proxy for decision difficulty. If there is large variance between rolls (some very good and some very bad) then is this positions more ‘difficult’? Does this position produce a higher rate of human error?
More comprehensive position categorisation. Are there proverbs or rules which we can apply to position types which would become more apparent after swinger analysis?
Conclusions
This review has only scratched the surface of the potential insight which swinger analysis could bring. Although we’ve hardly uncovered any groundbreaking ideas, we have demonstrated that swinger analysis can be a useful tool to better understand computer analysis and the game as a whole. Additionally, some existing and new heuristics can be further developed and supported using this technique. Ultimately, swinger analysis seems most promising in making human understanding of the game more efficient.