On Buchholz and Sonneborn-Berger coefficients.

The practice of chess tournaments provides two traditional metrics that are used to rank participants beyond their mere scoring. Their names are the Buchholz coefficient and the Sonneborn-Berger coefficient (often called just Berger). They are frequently used as tie-breakers in chess events, however I arrived to completely different application for them for the National Hockey League seasons.

1. The Buchholz coefficient

The Buchholz coefficient is simply the sum of the points of your opponents.

B = Σ_n=1^N P_n

So, if you played five games, and your opponents currently have 5, 3, 8, 6 and 6 points, your Buchholz value will be 28. Please note, that the current number of points is always used, not the number of points at the moment of meeting. The outcome of the game does not matter (for that one see the Sonneborn-Berger).

At first, the usefulness of such a criteria would prompt a raise of the eyebrow. However, it's not used in round-robin all-play-all tournaments as a final tie-break, because, naturally, the coefficient would be the same for all tied parties. It's used in a special format of chess events called the Swiss Tournament, not very popular outside of the realm of board games for purely logistic reason. But then, consider, first, an NFL season. The list of opponents every team plays there over the 16-game season may be quite different. And, whoever would end up with a larger Buchholz coefficient, clearly would've had stronger opposition on the way.

Now let's go back to hockey. First of all, at the end of the season, although everyone has played everyone, they did so a different number of times. Thus, the sum of opponents' points at the end of the season could be different between teams - including within the same division, if they had a different schedule. So, this could still be a very valid tiebreak. Secondly, the season is so long (82 games, unlike a chess Swiss which is rarely longer than 11 rounds), and that gives us a lot of midway points in time, when the all-play-all has not been completed yet! Here the Buchholz coefficient can clearly show, who has had the stronger opposition up until a certain moment.

Then, if we look at the remainder of the schedule for each team, and for every game we add the opponent's points we get an excellent remaining schedule strength estimator.

Wait... there's a caveat.

Unlike in a chess tournament, where every round occurs for everyone at the same time, and barring very rare circumstances, every participant played an equal amount of games at any point of the tournament, there may be a significant difference in the number of games played by different teams, so summing the opponents up will not work very well. And these opponents also played a different number of games, so their total amount of points is not a very good indicator.

Fortunately, it's not a big deal. Instead of totals, let's operate with per-game numbers. So the NHL Buchholz Coefficient for a team after N games becomes:

B = (Σ_n=1^N PPG_n)/N. 

Same applies for the remaining schedule strength, where the per-game numbers of the remaining opposition are summed an averaged.

So, if the team played three games against opponents who currently are:
A) 6 points in 4 games, B) 3 points in 3 games, C) 2 point in 5 games, then the team's Buchholz value would be (6/4 + 3/3 + 2/5) / 3 = 2.9/3 ~ 0.967pts.

Here are the current (Mar 12th 2017) Buchholz coefficients and remaining schedule strengths for the entire 30 times (and note how the Blues stand out with plenty of matchups vs Colorado and Arizona remaining).

+-----------------------+-----------+-------+-------+
| Team Name             | PPG       | Buch  | RStr  |
+-----------------------+-----------+-------+-------+
| Washington Capitals   | 1.4179105 | 1.119 | 1.133 |
| Pittsburgh Penguins   | 1.4029851 | 1.117 | 1.127 |
| Minnesota Wild        | 1.3939394 | 1.090 | 1.070 |
| Columbus Blue Jackets | 1.3731343 | 1.125 | 1.132 |
| Chicago Blackhawks    | 1.3283582 | 1.088 | 1.096 |
| San Jose Sharks       | 1.2985075 | 1.106 | 1.106 |
| New York Rangers      | 1.2941176 | 1.120 | 1.184 |
| Ottawa Senators       | 1.2537313 | 1.105 | 1.169 |
| Montreal Canadiens    | 1.2352941 | 1.122 | 1.097 |
| Edmonton Oilers       | 1.1791044 | 1.121 | 1.040 |
| Anaheim Ducks         | 1.1764706 | 1.102 | 1.150 |
| Calgary Flames        | 1.1764706 | 1.099 | 1.140 |
| Boston Bruins         | 1.1470588 | 1.115 | 1.151 |
| Toronto Maple Leafs   | 1.1343284 | 1.114 | 1.150 |
| Nashville Predators   | 1.1323529 | 1.105 | 1.116 |
| St. Louis Blues       | 1.1194030 | 1.144 | 0.943 |
| New York Islanders    | 1.1194030 | 1.142 | 1.103 |
| Tampa Bay Lightning   | 1.0895522 | 1.121 | 1.134 |
| Los Angeles Kings     | 1.0746269 | 1.118 | 1.104 |
| Philadelphia Flyers   | 1.0447761 | 1.122 | 1.179 |
| Florida Panthers      | 1.0298507 | 1.118 | 1.175 |
| Carolina Hurricanes   | 1.0000000 | 1.138 | 1.136 |
| Buffalo Sabres        | 0.9855072 | 1.127 | 1.158 |
| Winnipeg Jets         | 0.9565217 | 1.110 | 1.143 |
| Vancouver Canucks     | 0.9558824 | 1.115 | 1.152 |
| Dallas Stars          | 0.9552239 | 1.119 | 1.100 |
| Detroit Red Wings     | 0.9545455 | 1.151 | 1.059 |
| New Jersey Devils     | 0.9117647 | 1.148 | 1.132 |
| Arizona Coyotes       | 0.8358209 | 1.133 | 1.098 |
| Colorado Avalanche    | 0.6119403 | 1.128 | 1.164 |
+-----------------------+-----------+-------+-------+

In tne next installment we're going to talk about the application of the Sonneborn-Berger coefficient to the NHL regular season.