The Website - A Page A Day, Part I - Home Page

The main page of the website shows the summary of all its features.

The site menu features the main statistical sections:

• League - league-wide statistics
• Teams - team-based statistics
• Players - personal statistics
• Coaches - statistics for the NHL coaches
• Drafts - statistics for currently drafted players and the historical performance by draft
• Fantasy - tools to help the fantasy player

On the top we have a ticker of the scores, as predicted by the model. The ticker is always scrolled to the current date, however you can navigate it back and forth using the two arrows at the edges. The away team is on the top, and the home team is on the bottom. The predicted score is in the Prdct column. The actual score is in the Act column. The projected winner is displayed in bold. If the prediction failed, the displayed teams will be painted red. We never adjust our predictions backwards. Only the scores for the current season are featured.

Then, on the top right we have three very important links that would help you with understanding the pages:

• Learn More - about the methodology of the Website
• Glossary - about the terms used in the pages
• Blog - link to this blog which, as you see, also takes time to elaborate on the site.

We also display our latest addition to the website and the latest blog entry.

Then we have three random snippets in the columns. The snippets represent excerpts from the tables published elsewhere on the site and may change after each publication, which happens overnight. The snippets currently (hopefully they would become more diverse) are:

• Team data snippets:
• Buchholz/Berger-Sonneborn coefficients
• Empty Net Statistics (probably our most popular page)
• Teams Elo Ratings and predicted standings, per division
• Strikebacks
• Player data snippets:
• Goaltender elo ratings
• Deadly pairs of the NHL (goals+primary assist tandems)
• Clutch players excelling scoring lead-changing goals
• Reverse stats (players being hit, drawing penalty, etc.)
• Coach data snippets:
• Face to face coaching records
• Coach challenge statistics

Only current season data is displayed in the snippets.

Below the random snippets we feature a permanent snippet that shows the best projected picks for the daily fantasy competitions. We provide a model-based evaluation of the expected score for players in the three most popular Daily Fantasy websites - Yahoo, FanDuel and DraftKings.

Below each snippet there is a link to the page with the full data.

At the bottom we have a collection of information links. Make sure you visit the Glossary and the About pages. The Links page has an ever-growing collection of hockey-related links. The Forum link leads to hockeyforums.net, the site we're partnering with - this blog is broadcast there as well. The Data section shows the software and the data sources this website is built with.

Hopefully this small introduction proves helpful and entices you to pay another visit to MoreHockeyStats, and to share this site with your friends. Follow us on Twitter, through the site or through this blog!

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.

On The NHL Scoring System (Part II)

Part I.

Goodhart's law is the bane, the safeguard and the watchdog of everyone who tries to make conclusions from sample data. The "Schroedinger Cat of Social Sciences" practically says, if you want people to do X, but you reward them for doing Y, they will be doing Y rather than X. We start seeing that in the "possession analytics", based on shots taken, that the players begin to shoot from everywhere to get their possession ratings up. But we digress - the topic is the scoring system, we'll save that note for another blog entry.

We want the NHL hockey to be spectacular. That's the main objective (beyond being fair and competitive, otherwise look for Harlem Globetrotters). In the past the spectacular was fighting as much scoring and winning games, but that taste of the public changed, and the fighting went away. It was not directly related to scoring and winning, it was just an extra free show provided.

Now we're left with scoring and winning. These two are closely tied, and not necessarily as a positive feedback, since not allowing your opponent to score also helps winning. A 2-1 win is practically just as valuable as a 7-2. So in the mid-2000s, the winning objective, the points objective took over the scoring objective. And from the previous post we see that the existing 2-2-1-0 points system encourages low-intensity game preferably slipping into the OT. On the other hand we also noted that the 3-2-1-0 points system would encourage teams to clamp down and protect their minuscule leads. Looks like a circle to break...

Well, here comes Goodhart's law. You want teams to score, or at least try to score, but you reward them for achieving points. So what they do is concentrate on getting points. Therefore, if the NHL want to see score-oriented hockey, the NHL needs to reward scoring, and not points. Still, the points have been used to determine playoff spots, so something has to give.

First, let's take a wild ride by suggesting that we rank teams by the amount of goals scored. That would lead to a pretty drastic change and the end of hockey as we know it. This will lead to situations where a team might play for a period without a goaltender in a playoff race. In general, the goaltending position will deteriorate, and aren't we loving the spectacular saves just as we love slick goals? Probably, that would be too much.

Thus, we can mitigate to allow the goal differential to be the ranking criteria. At the end of the season, the teams with the higher goal differentials will be ranked at the top, and the wins-ties-losses, well, they get relegated to tie-breaks. The incentive to score rather than to hold the opponent increases, because while now the competitors in their games cannot score more than two points, they still can score a bigger goal differential! All the lazy skating to finish the game after it's 4-0 or 5-1? Gone.

This idea is actually not novel. It's been used for a long while in team chess tournaments. Such tournaments consist of matches, where each player of one team plays against an opponent of the opposing team at the same time. Each player's individual score (win, draw or loss) is accumulated into the total score. So a match of 8-player teams, where one team had 5 wins, 1 draw and two losses ends up with 5.5-2.5 score, essentially "the goal differential". At the end of the tournament, the scores accumulate, and the teams are ranked according to them. You can see the crosstables of historical chess tournaments at the wonderful Olimpbase website.

And if you feel that the fact of winning or losing the game should be have more weight than just a tie-break (by the way, there will be less tie-breaks on goal differential), that is easy to factor in, just add a bonus "goal" to the winner, like it is done in the shootout now. Better, add two bonus "goals" for winning in regulation, one bonus "goal" for winning in the OT, abolish shootouts.

On The NHL Scoring System (Part I)

There was nothing wrong with ties. The 2-1-0 point system works fine in various sports around the world. It's just ... not fitting into the mind of a North American sports fan. "Who won?" - "It was a tie." - "Who won on a tiebreak?" Basketball and baseball do not have ties, and American Football has them at a rate of 1-2 times per whole season. So more than ten years ago NHL went with the flow and abolished ties, introducing the shootout, and with a twist, where the team making it past the regulation would still get the point, and a 2-2-1-0 point system came to life.

Since then the argument rages, whether the ties should come back, or whether the consolation point should be taken away, or whether the much more energetic 3-2-1-0 point system, adopted across the ocean and by the IIHF should make its way into the NHL as well. The feeling that there is something unhealthy when a team loses and still gets something, while the winner is not penalized is nagging.

The argument from the NHL leadership claims the system creates denser standings and thus more interest and drama throughout the season is a valid one. However, this system, as we show below, creates a wrong incentive.

The standings in the NHL are defined by a points total, and the seeding in the playoffs are first and foremost the divisional standings. The relative standings across conferences have a rather minor effect of the potential home advantage in the Stanley Cup Finals, the same standings within the same conference but across divisions have an impact on the seedings in the whole playoffs, but also to a limited effect. Therefore, at least with the exception of intradivisional games, but possibly including these games too (especially against the competition that has fallen out of the playoff picture), the only thing that matters are the points accrued by the team itself, and not the points the opposition gathers. Let's wield the statistic that says that 25% of the games go to the overtime and the

So what are the point expectations in a 2-2-1-0 system? Let's compare a few situations when teams A and B play.

1. Team A has 75% chance of winning the game (that's a huge, possibly maximum imaginable favorite odds)
2. Team A has 67% chance of winning.
3. Team A has 60% chance of winning.
4. Team A has 50% chance of winning.

Let's wield the statistic that says that 25% of all games go to the overtime and the shootout occurs in 40% of these games. Let's also assume that the 3-vs-3 overtime is more random and reduces by half the advantage of the better team (i.e. 75-25 becomes 62.5-37.5), and that the shootout is completely random, so the chances of winning it are 50/50. Then, the probabilities of the outcome become:

	Chance	PwReg	PwOT	PwSO	xPoints
Team A	75%	0.5625	0.09375	0.05	1.51875
Team B	25%	0.1875	0.05625	0.05	0.73125
Team A	67%	0.5025	0.08775	0.05	1.39275
Team B	33%	0.2475	0.06225	0.05	0.85725
Team A	60%	0.45	0.0825	0.05	1.2825
Team B	40%	0.3	0.0675	0.05	0.9675
Team A	50%	0.375	0.075	0.05	1.125
Team B	50%	0.375	0.075	0.05	1.125

Now let's consider than the stronger team A plays intentionally for overtime and manages to force it in 75% of the cases.

	Chance	PwReg	PwOT	PwSO	xPoints
Team A	75%	0.1875	0.28125	0.15	1.55625
Team B	25%	0.0625	0.16875	0.15	1.19375
Team A	67%	0.1675	0.26325	0.15	1.49825
Team B	33%	0.0825	0.18675	0.15	1.25175
Team A	60%	0.15	0.2475	0.15	1.4475
Team B	40%	0.1	0.2025	0.15	1.3025
Team A	50%	0.125	0.225	0.15	1.375
Team B	50%	0.125	0.225	0.05	1.375

In ALL cases it's worth for both teams to steer the game into OT. For the even odds case, the expectation gain is a whopping 0.25 points! Even in the case of super, uber favorite, it's still worth for that team to head to overtime, as it projects a gain of 0.04 points. And the gains for the underdogs are so big that there is no reason for the underdog to disturb the force of the overtime, so they will happily comply! Meaning: we'll see more fun overtime, we'll see more dumb shootouts, but more importantly the 60 minutes of hockey will lose a lot of their significance. The only quantative incentive to finish the game in regulation becomes denying extra points for your opponents - hardly a significant matter in what, fifty out of the eighty-two season games!

Now, let's repeat these calculations with 3-2-1-0 point system and combine them into another table:

		2-2-1-0			3-2-1-0
	Chance	Exp25%OT	Exp75%OT	Δexp	Exp25%OT	Exp75%OT	Δexp
Team A	75%	1.51875	1.55625	+0.0375	2.08125	1.74375	-0.3375
Team B	25%	0.73125	1.19375	+0.4625	0.91875	1.25625	+0.3375
Team A	67%	1.39275	1.49825	+0.1055	1.89525	1.66575	-0.2295
Team B	33%	0.85725	1.25175	+0.3945	1.10475	1.33425	+0.2295
Team A	60%	1.2825	1.4475	+0.165	1.7325	1.5975	-0.135
Team B	40%	0.9675	1.3025	+0.335	1.2675	1.4025	+0.135
Team A	50%	1.125	1.375	+0.25	1.5	1.5	0
Team B	50%	1.125	1.375	+0.25	1.5	1.5	0

Now there is no incentive for the stronger team to push for overtime, and even the gain for the weaker team decreased. 3-2-1-0 definitely encourages a regulation decision!

Reasons where brought up against the 3-2-1-0 system. One states that the spread over the standings will be too thin, and more teams will be eliminated from the playoff race early. This argument has had no statistical support, and the element of drama when a team pulls a goalie in a tied score trying to force a 3-0 point win may actually more than make up for it. Another argument refers to soccer studies that claim the 3-1-0 point system there encourages teams to sit on their early leads trying to stifle the game, which decreases the attractiveness of the game. This argument is more valid, although it's notably harder to preserve a lead in hockey than in soccer. But beyond that this argument prompts for another, a truly revolutionary suggestion...

To be continued.