Patterns of Time Usage During Games

In this installment of Chess Analytics, I've decided to investigate the question: What are the distinct patterns of time usage during 10-minute online chess games? In doing so, we will also attempt to determine just how many distinct patterns we can distinguish.

Cluster Analysis

To attempt to answer this question, we can use cluster analysis. Cluster analysis is possibly also known as "segmentation analysis" or "taxonomy analysis", though I am not familiar with those terms.

Through cluster analysis, we can group chess games into similar (homogeneous) subgroups, based on how similar the games are on certain variables. The goal of cluster analysis is to obtain games that are very similar within groups, and very different between groups (if that's at all possible). So to examine patterns of time usage during online chess games, we can calculate the time taken at each move by the players, and perform a cluster analysis on those time values. This will give us subgroups of games that are similar in the time taken at each move.

If you're familiar with machine learning, you'll know that cluster analysis is a type of "unsupervised" learning algorithm: The algorithm will categorize the chess games in different groups without having known beforehand which group the games belong to (we just don't know what they are yet!). The term unsupervised is used to contrast with "supervised" learning algorithms, in which the algorithm learns to categorize observations based on a sample of observations that have already been categorized into groups.

The Data

To look at the patterns of time usage in chess games, I used the first 200 MB of the rated games played in October 2017 on the internet chess server lichess, retaining only those games for which the computer evaluation of each move was available (this was not necessary for this particular analysis, but this same dataset could be used for other analyses as well). In total, this gave us 7,387 games. However, because the current investigation is about time usage, I restricted the analysis to those games whose time control was 10 minutes in total, with or without increment (for example, 10+0, or 4+9), and which lasted at least 25 moves, so as to look at the patterns in the first 25 moves of the game. So in the end, a total of 843 games fit those criteria. The goal of this analysis is to group those 843 games into homogeneous subgroups of games that are similar in their patterns of time usage.

To give you an idea of what the dataset looks like, here's a screenshot of the first few rows (this is a Pandas DataFrame in Python):

Because of the likely relationship between the time taken at each move by both players during the game, here we focus on the time pattern for White players (but the results were similar when looking at the Black side instead). The first column, clocks_diff_31, indicates the difference in the remaining time of the White player (in seconds) between the third ply (i.e., White's second move) and the first ply (i.e., White's first move). In other words, the first column gives the number of seconds White spent on their second move. Analogously, the second column, clocks_diff_53, indicates the time taken by White between the third and fifth ply (i.e., during White's third move), and so on, up to the 25th move. Note that on lichess, the clock only starts ticking at the second move, so we don't have access to the time taken during the first move for these games.

Number of Patterns & Hierarchical Clustering

Perhaps the most important decision that goes into a cluster analysis is choosing the number of clusters to detect, or in this case, the number of distinct patterns of time usage we can distinguish in the games. One data-driven way to do this is through hierarchical clustering, where chess games are grouped two by two based on the similarity of their pattern of time usage, then these new pairs are grouped with another pair again based on similarity, and so on, until only one big cluster--containing all the games--is created. In this way, clusters are formed sequentially, and we can decide where to stop the clustering.

We can visualize this hierarchical clustering through a dendrogram (don't do like me and misspell it "dendogram" every time!). A dendrogram shows which games are being grouped together due to their similar time-usage patterns at each iteration of the process, starting with no clusters at the bottom, and one giant cluster that includes all games at the top.

Here is a simple example of a dendrogram (courtesy of StackExchange):

In this dendrogram, 10 observations (or if you prefer in our case, 10 chess games) are being sequentially grouped. The horizontal x axis indicates the game index (which is arbitrary), and the vertical y axis indicates the dissimilarity between games within a cluster (i.e., the "distance" between games). Starting from the bottom of the dendrogram, we can see that games #2 and #10 are most similar in their pattern (the distance between them is the smallest at < 0.2) and are grouped together first. Next games #5 and #8 are grouped; then games #7 and #6; then game #9 is most similar to the "average" of games #5 and #8 and so is grouped with that cluster; and so on until the cluster formed by games #3, #7, and #6 is combined with the cluster formed by the other games.

Choosing the number of clusters means choosing a maximum difference (or distance) between games within each cluster, as represented by the y axis. So in the simple example above, we could decide that 4 clusters is appropriate by drawing a horizontal line somewhere around a distance of 0.6, thereby yielding the following clusters: (1) game #3; (2) games #7 and #6; (3) games #1 and 4; and (4) games #2, 5, 8, and 10.

Evidently, the current analysis--with 843 chess games rather than 10--yields a much more dense dendrogram (you might want to zoom in or open the picture in a new window):

Based on this dendrogram, a reasonable choice for the number of patterns seems to be 3, and if the three-cluster solution yields patterns that are difficult to interpret, then another reasonable choice is 2. (The Python library that generated this dendrogram seems to suggest 2 clusters, given the color coding.) The dendrogram might also lead us to conclude that 4 clusters is not very reasonable, as the distance between the third and fourth clusters would be very small.

The good news is that nothing prevents us from trying out several alternatives. A convenient way to choose the number of clusters is to try a few reasonable numbers and choose one that yields groupings that make sense given the context (here, time usage patterns) all the while providing groups with a large-enough size (a cluster that includes only a handful of games is not very useful). In some applications, obtaining clusters of approximately equal sizes is desirable, but that's not really the case here. So we can try out 3 clusters, and also examine the two-cluster solution in case it ends up being more interesting or interpretable.

Three Patterns of Time Taken During a 10-Min Game

The graph below shows the three distinct patterns of time usage detected in a three-cluster cluster analysis. The graph shows the median time taken at each move, so that 50% of the games of a given group took less time than is shown on the graph, and 50% of the games took more time than is shown. In total, 158 games were categorized in pattern #1; 438 games were categorized in pattern #2; and 247 games were categorized in pattern #3.

Let's try to describe these patterns. It looks like they are not too surprising and make sense. Pattern #3 (green line at the bottom) represents games that are played rather quickly throughout, with perhaps a slight increase in move time as the game goes on. Pattern #1 (blue line at the top), on the other hand, represents games that are played much more slowly, especially starting around move 7-8, with some peaks a little bit later in the game; these games also seem to pick up the pace again after move 15, but it's difficult to say given that our analysis ends at move 25. Finally, Pattern #2 (orange line in the middle) represents games that are in-between the faster-paced and slower-paced games of Patterns #1 and #3.

Note that the graph showing the average time taken at each move (rather than the median time) is nearly identical to this one.

The three patterns of time usage found above are easy to interpret and make sense. To make sure that we're not missing out on a better solution, let's do as promised and look at the two-pattern solution. We get the following graph, which again shows median time taken at each move in each pattern:

Pattern #2 (orange line on top) is exactly the same as Pattern #1 above (represented by the same 158 games as before). This is because this pattern is the pattern that is most different from the other patterns (see the games in green on the left of the dendrogram above). With the two-pattern solution, the two faster patterns are now combined into Pattern #1 (blue line at the bottom), with a total of 685 games (i.e., the sum of 438 + 247 games from Patterns #2 and #3 in the three-pattern solution). These two patterns are also simple and easy to interpret, but nothing is gained from the three-pattern solution; in fact, we've lost the distinction between the faster-paced games in terms of their pattern of time taken. So I prefer the three-pattern solution here.

Conclusion from the Cluster Analysis on Time Patterns

Overall, from the cluster analysis above we could draw the conclusion that there are 3 general patterns that 10-minute online chess games follow in terms of time usage:

1. Games that are faster-paced in the opening but that slow down noticeably at the end of the opening (around move 8; this is Pattern #1 in the three-cluster cluster analysis);

2. Games that are relatively fast throughout, though slightly longer after the opening (this is Pattern #3 in the three-cluster cluster analysis);

3. Games that are neither fast nor slow, and whose moves also tend to slow down slightly after the initial opening moves.

What's Next?

This analysis was limited to some 800 online games played in October 2017 on lichess.org. I doubt that patterns would change much over time (say if we were to do this analysis again in October 2018), but perhaps patterns do change by chess servers (maybe players are more or less impatient on chess.com, for example?), and my guess is that patterns are different in over-the-board games rather than online games. (If you have a good dataset for OTB games, please let me know.) This analysis also doesn't show what happens beyond the 25th move, for example during endgames.

Another unanswered question that seems interesting is the relationship between pattern of time taken and game outcome. However, not only might the effect of time usage on game outcome vary from player to player, but there probably is a very strong relationship between time taken at each move by White and time taken by Black. Nonetheless, one (odd) possibility is that some patterns are associated with more wins for White, while other patterns are associated with more wins for Black... I haven't checked, but I do think it would be very odd!

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