On the NHL Scoring System - Part III

Part I
Part II


Once again, driven by idea that if you want to encourage goal scoring, you need to reward the goal scoring in standings directly, not indirectly through winning. Then, based on the idea of a fellow hockey fan and blogger, a new suggestion was born in my mind.

Not so long ago I was involved in another discussion on the subject on Twitter, where an interesting alternative, 2-1-0-0 was described. The idea is that you still get two points for a win in regulation, just one point for a win in OT, but nothing if you lose, and, the key, both teams get nothing if the game is tied at the end of regulation (shootouts are abolished). This is a very sharp idea, but for me something felt very wrong, and then it crystallized:

It's not fair to reward a hard fought 5-5 tie with zero points, just like a lazy-skated 1-1. We still want to encourage goal scoring, and the simple 2-1-0-0 just unbalances the game. And so it dawned on me. We should reward goals with extra standings points!

The formula that first came to mind, and which seemed fair: give each goal a 0.1 point in the standings, while the win-scoring system shall be 2-1-0-0. If you or your database have an aversion against decimals, assign 20 points for a win, 10 points for OT loss, and 1 extra point for each goal scored. This will encourage goal scoring in any situation, and for both sides, including the games that go into garbage time pretty quickly. So, a 7-2 win will give the winner 2.7 points, and the loser 0.2 points. A 2-0 win will give the winner 2.2 points, the loser 0. A 4-3 OT win will give the winner 1.4 points, the loser 0.3 points. A 5-5 OT tie will give each side 0.5 points.

Wait, there's a caveat.

Imagine a situation where a team needs just 0.1 point to pass another one in the standings for the playoff spot. They are playing an opponent whose number of points in the standings does not have any effect on them. In such a situation, the team would play without a goaltender at all, because they don't care how much they lose, they just need that goal. Now, this is not really hockey, so to prevent this kind of play a restriction needs to be introduced:

Any goal scored without a goaltender on the ice, when not on a delayed penalty, and when trailing by more than two goals shall not yield any standings points.

Here is an example what the today's standings would look like under the suggested system:

Team                           W  OW T  L  GF  GA  P
Boston Bruins                  34 04 04 34 216 201 93.6
Montreal Canadiens             31 09 05 31 205 186 91.5
Ottawa Senators                32 04 08 31 191 191 87.1
--------------------------------------------------------
Washington Capitals            41 08 07 20 246 165 114.6
Columbus Blue Jackets          38 09 04 24 233 170 108.3
Pittsburgh Penguins            37 06 08 25 256 211 105.6
--------------------------------------------------------
New York Rangers               38 05 06 28 242 203 105.2
Toronto Maple Leafs            29 06 09 31 229 213 86.9
--------------------------------------------------------
New York Islanders             28 05 06 36 217 224 82.7
Tampa Bay Lightning            27 06 07 35 206 207 80.6
Carolina Hurricanes            28 04 07 36 198 208 79.8
Buffalo Sabres                 24 06 08 39 191 215 73.1
Philadelphia Flyers            22 07 11 36 193 218 70.3
Florida Panthers               21 07 11 37 192 210 68.2
New Jersey Devils              18 06 06 46 171 221 59.1
Detroit Red Wings              16 07 08 45 181 224 57.1
--------------------------------------------------------
Chicago Blackhawks             36 09 05 27 230 197 104.0
Minnesota Wild                 37 04 05 30 241 193 102.1
St. Louis Blues                35 06 02 33 213 200 97.3
--------------------------------------------------------
San Jose Sharks                35 06 03 32 204 185 96.4
Anaheim Ducks                  37 02 06 31 200 183 96.0
Edmonton Oilers                33 05 09 29 221 191 93.1
--------------------------------------------------------
Nashville Predators            33 04 06 33 224 206 92.4
Calgary Flames                 30 09 06 32 208 206 89.8
--------------------------------------------------------
Winnipeg Jets                  29 03 04 41 226 243 83.6
Dallas Stars                   27 04 02 43 207 240 78.7
Los Angeles Kings              23 11 06 36 183 185 75.3
Vancouver Canucks              19 07 06 44 169 221 61.9
Arizona Coyotes                17 04 08 48 176 245 55.6
Colorado Avalanche             14 06 01 55 150 257 49.0

Naturally, they would not be the same standings if the system were indeed implemented, but why not to take a look. And once again, try it in the AHL first, it won't hurt anyone.

On Players Evaluation - Part II (Elo)

Part I.

The Elo rating system is the system used for evaluation and comparison of competitors. Up until today it's been mostly applied in the domain of board games, most well-known in chess, but also in disciplines such as draughts or go. The Elo system, named after its inventor, Prof. Arpad Elo, who first published it in the 1950s in the US, is capable to produce a reliable score expectation for an encounter between two competitors who oppose each other.

For those who are not familiar with chess or draughts, let's take a look on how the Elo ratings work:

1) In an encounter between two competitors, A and B, assume they have ratings R_a and R_b.

2) There is a function that maps the expected result for each player given the opponent:

E_a = F(R_a, R_b)

E_b = F(R_b, R_a)

where F is a monotonic non-decreasing function bounded between minimum and maximum possible scores, such as 0 and 1 in chess. An example for such a function would be arctan(x)/π + 0.5 .

E_a+E_b should be equal to maximum possible score.

In practice a non-analytical table-defined function is used that relates only on the difference between Ra and Rb, and not their actual values. The function can be reliably approximated by the following expression:

E = 1 / [ 1 + 10^(R_b-R_a) / 400 ]

which works well with ratings in low 4-digit numbers and rating changes per game in 0-20 range.

3) After the encounter, when real scores S_a and S_b have been registered, the ratings are adjusted:

R_a1 = R_a + K*(S_a-E_a)

R_b1 = R_b + K*(S_b-E_b)

Where K is a volatility coefficient, which is usually higher for participants with shorter history, but ideally it should be equal for both participants. The new ratings are used to produce the new expected results and so on.

The Elo rating has several highly important properties:

1) It gravitates to the center. As rating R of a participant climbs higher, so does the expected result E, which becomes difficult to maintain, and a failure to maintain it usually results in a bigger drop in the rating.

2) It's approximately distributive. If we gather N performances and average the opponents as R_av, the expected average performance as E_av = F(R_a, R_av), and the actual performance as S_av, then the new rating R_aN' = R_a + N*K*(S_av-E_av) will be relatively close to R_aN obtained via direct R_a reciprocal update after each of the N games.

3) It reflects tendencies, but overall performance still trumps it. Given the three players with ten encounters against other players with the same rating, when the performances are (W - win, L - loss):

For player 1: L,L,L,L,L,W,W,W,W,W

For player 2: L,W,L,W,L,W,L,W,L,W

For player 3: W,W,W,W,W,L,L,L,L,L

player 1 will end up with the highest rating of the three, player 2 will be in the middle, and player 3 will have the lowest one - but not by a very big margin. Only when the streaks become really long the Elo of a lower performance may overcome the Elo of a higher one.

And how does Elo stack against the four Brits?

* Goodhart's Law: pass. It measures the same thing it indicates.
* Granger's Causality: pass. It is a consequence of a performance by definition, and a prediction of future peformance, by definition.
* Occam's Razor: pass. The ratings revolve around the same parameter they measure.
* Popper's Falsifiability: partial pass. The predictions of Elo sometimes fail, because they are probabilistic. However, the test of time and the wide acceptance indicate that the confidence level holds. Elo was even used for "paleostatistics" when the ratings were calculated backwards until middle XIX century, and the resulting calculations are well-received by the chess historians' community.

The only well-known drawback of Elo is the avoidance by top chess players of competition against much weaker oppositions, especially when facing White, as such a game can be drawn relatively easily by the opponent, and the Elo rating of the top player could take a significant hit resulting in a drop of several places in the rating list.

Now, to the question of the chicken and the egg - where do the initial Elo ratings come from? Well, they can be set to an arbitrary value of low 4-digit number. Currently a FIDE beginner starts with the rating of 1300. If the newcomer is recognized as being more skilled than a beginner, then a higher rating is assigned based on rating grades for each skill level, sort of an historical average of the newcomer's peers.

And... What does all this have to do with hockey?

To be continued...

On Player Evaluation Systems - Part I

In the previous post we mentioned the Goodhart's Law and how it threatens any evaluation of an object. We said that it traps the Corsi/Fenwick approach because it substitutes the complex function of evaluation of a hockey player by a remarkably simple stat - shots.

Goodhart's law is not alone. In any research it is preceded by the two pillars: Popper's law of falsifiability and the Occam's razor. A theory willing to bear any scientific value must comply with both, i.e. produce hypotheses that can be overthrown by experiment or observation (and then relegated to the trashcan), and must avoid introduction of new parameters beyond the already existing ones. Add Granger causality into the mix and we see that the four Brits presented the hockey analytics society with pretty tough questions that the society - at least the public one - seems to be trying to avoid.

The avoidance will not help. Any evaluation system will not be able to claim credibility unless it complies with the four postulates above, and within that compliance issues measurable projections.

To be continued...

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.