On Buchholz and Sonneborn-Berger coefficients - Part II

Part I

2. The Sonneborn-Berger coefficient.
This stranger beast is a metric extensively used for tie-breaks in chess-round robins and as an auxiliary tie-break tool to the Buchholz coefficient in non-round robin. Let's start with the definition.

$$SB = Σ↙{n=1}↖N f(R_n,P_n)$$

where R_n is the result against the n-th opponent, and P_n is the opponent's points score.
The function  f(R_n, P_n) is defined as:

f(Win, P_n)  = P_n
f(Tie, P_n)  = P_n/2
f(Loss, P_n) = 0

The result value evaluates whether the participant performed better against stronger and weaker opposition. Actually, I do have a problem with this criteria as a tie-breaker, in my opinion ALL points are created equal, and it doesn't matter if they came from a contender or a bottom feeder. However, this metric does answer the notorious statements like "This team only shows up for big games" and "This team is only good against garbage opposition."

So, first of all, for the NHL application, we will modify the function f(R_n, P_n) to:

f(Win, P_n) = P_n
f(OW, P_n)  = 2*P_n/3
f(OL, P_n)  = P_n/3
f(L, P_n)   = 0

to account for the overtime point.

Then, we can calculate the minimal possible SB_min value for a team with the given schedule so far this season, by assigning Wins to be against the weakest teams played, and the OW/OL against the weakest remainder until the sum of W, OW and OL points add up to the number of points the team currently has.

Similarly we shall calculate the maximal possible SB_max value by assigning Wins to be against the strongest teams played, and the OW/OL against the strongest of the remainder, assuming OT wins are about 1/4 of the whole.

Then the closer the actual SB is to the SB_min or SB_max we may be able to say whether the team is successful more against the bottom feeders, the top guns, or whether it achieves its points from the whole spectrum available.

Here is the table describing how this season's teams have their SB positioned between SB_min and SB_max.

Team	Points	SBmin	SBopt	SB	SBmax
Pittsburgh Penguins	1.40	44.28	46.48	46.24	53.06
Washington Capitals	1.40	44.70	46.74	47.77	52.89
Minnesota Wild	1.37	42.25	44.36	46.63	50.66
Columbus Blue Jackets	1.37	43.10	45.36	46.44	52.15
Chicago Blackhawks	1.34	41.61	43.90	43.79	50.80
San Jose Sharks	1.31	40.68	42.97	44.16	49.84
New York Rangers	1.30	41.25	43.67	45.55	50.92
Ottawa Senators	1.25	37.84	40.07	41.79	46.78
Montreal Canadiens	1.25	39.37	41.74	41.05	48.87
Anaheim Ducks	1.19	36.86	39.43	40.12	47.15
Calgary Flames	1.18	35.97	38.49	38.20	46.05
Edmonton Oilers	1.16	35.86	38.32	37.43	45.70
Boston Bruins	1.15	34.73	37.23	37.74	44.72
Nashville Predators	1.13	33.28	36.14	38.04	44.72
Toronto Maple Leafs	1.13	34.64	36.99	35.66	44.02
St. Louis Blues	1.12	34.69	37.14	38.52	44.50
New York Islanders	1.12	34.36	36.94	37.94	44.71
Tampa Bay Lightning	1.09	32.62	34.98	35.41	42.06
Los Angeles Kings	1.07	32.10	34.66	33.56	42.34
Philadelphia Flyers	1.04	31.26	33.56	32.01	40.48
Florida Panthers	1.03	30.89	33.12	30.95	39.82
Carolina Hurricanes	1.00	29.43	31.78	32.41	38.85
Buffalo Sabres	0.99	30.09	32.49	33.43	39.68
Winnipeg Jets	0.96	27.55	30.35	31.48	38.75
Vancouver Canucks	0.96	28.48	30.91	29.02	38.21
Dallas Stars	0.94	28.05	30.62	31.16	38.34
Detroit Red Wings	0.94	29.12	31.12	30.02	37.13
New Jersey Devils	0.91	27.78	30.15	28.63	37.27
Arizona Coyotes	0.84	25.13	27.24	25.86	33.56
Colorado Avalanche	0.61	17.90	19.74	19.98	25.25

Once again, we use Point Per Game values because the teams and their opponents have a different number of games played at most of the moments within a season.

We would dare to make one more step forward and claim that the team that performs closer to SB_max seem to have a coach problem (notable differences highlighted in green in the table above). The roster is there to compete against the best, but the points aren't trickling in at a pace good enough against the fodder. Similarly, if the SB value is closer to SB_min is more likely to have a GM problem (notable differences highlighted in blue in the table above), that its roster is not good enough to compete, but the coach is able to squeeze close to the maximum out of it. However, it is natural to win more games against the weaker teams, so we set the balance point at SB_opt = (SB_max + 3*SB_min) / 4;

Wrapping up the talk about the Buchholz and the Sonneborn-Berger coefficients we would like to state that these values have an almost entirely descriptive value and without any predictive capability, with a small exception of the Buchholz-based remaining schedule strength metric. And even then, it's sort of a 'descriptive prediction'.

Please see more Buchholz and Berger-Sonneborn data on the website!

On schedule breaks - some crosstables

I took a look at how the teams performed one against another depending on the break length. All lengths longer than five were truncated to five, and the back-to-backs are designated as 0-length.

Here's the crosstable for the 2015 season:

#days	0	1	2	3	4	5
0	X	84-90-28	35-38-8	7-6-1	3-7-0	4-2-0
1	118-65-19	X	118-82-27	28-18-9	6-4-1	7-5-0
2	46-29-6	109-95-23	X	15-11-3	5-1-0	2-0-0
3	7-3-4	27-23-5	14-11-4	X	5-4-0	0-1-0
4	7-3-0	5-3-3	1-4-1	4-4-1	X	2-0-0
5	2-2-2	5-4-3	0-2-0	1-0-0	0-2-0	X

And here's the one for the ongoing, 2016 season, in the midst of bye weeks:

#days	0	1	2	3	4	5
0	X	59-48-23	21-23-7	7-3-3	3-0-0	6-3-2
1	71-48-11	X	61-64-21	16-7-3	3-4-0	11-4-0
2	30-13-8	85-44-17	X	5-3-5	0-0-0	1-1-0
3	6-5-2	10-12-4	8-4-1	X	3-1-0	0-0-0
4	0-2-1	4-1-2	0-0-0	1-1-2	X	2-0-0
5	5-5-1	4-9-2	1-0-1	0-0-0	0-2-0	X

It's obvious that the bye weeks are no good for the teams, and NHL should convince the NHLPA to rescind it for 2017/18.

Just for fun, here's the aggregate since 2005, when the ties were abolished:

#days	0	1	2	3	4	5
0	X	1013-1030-285	352-380-104	117-104-26	37-34-10	45-34-11
1	1315-760-253	X	1128-917-284	305-231-67	119-86-24	61-42-7
2	484-267-85	1201-852-276	X	149-103-34	44-24-8	15-10-2
3	130-87-30	298-232-73	137-114-35	X	20-19-1	3-3-1
4	44-32-5	110-87-32	32-34-10	20-9-11	X	6-4-1
5	45-30-15	49-44-17	12-13-2	4-2-1	5-6-0	X

Soon to become part of the website!

Another rule change suggestion

Better less, but better

V.I. Lenin

I've got another rule change suggestion, this one even simpler:

Allow teams to decline penalty shot awards in favor of a regular power-play.

I think it adds more tactical variety to the game and discourages penalties on breakaways that are worse in penalty shooting.

As a side matter, I think: a player who is charged with the offense after which the penalty shot is awarded should still be added a minor penalty (2 minutes) in the statistics.

On Streaks and Breaks

3 articles in two days... What's gotten into me.

So after remembering the Botwinnik's quote, and after publishing the stats how the teams actually play after different breaks, a new idea came to me - check whether the teams on streaks are affected positively or negatively by breaks.

For the sake of the analysis, I assumed the following:

• A break is a period of three days at least between games.
• A streak is a sequence of at least three wins in a row, or at least seven points in four games.

So we check for the last thirty years (as far as NHL.com would let us in) if the streaking team was able to keep the streak alive, or whether the streak was broken:

SEASON	ALIVE	BROKEN
1987/1988	5	11
1988/1989	12	7
1989/1990	8	14
1990/1991	13	11
1991/1992	17	13
1992/1993	20	16
1993/1994	19	20
1994/1995	2	7
1995/1996	15	11
1996/1997	15	11
1997/1998	12	20
1998/1999	12	9
1999/2000	18	12
2000/2001	21	11
2001/2002	17	6
2002/2003	13	10
2003/2004	12	14
2005/2006	31	15
2006/2007	16	16
2007/2008	23	24
2008/2009	15	20
2009/2010	14	17
2010/2011	19	11
2011/2012	22	11
2012/2013	6	3
2013/2014	15	15
2014/2015	16	16
2015/2016	16	14
2016/2017	8	11
TOTAL	432	376

Actually, it looks like the streaks weren't affected by the break either way. 53.4% of the times the streak continued, 46.6% of the time it went dead. There is a very large discrepancy between the seasons, although I'd attribute it to lesser parity between the teams overall in these years. For the last 5 years, the probability for the streak to stay alive has been 50.8% (61 cases of extended streaks out of 120).

Now, what would change, if we define a break a little bit longer, by a single day:

SEASON	ALIVE	BROKEN
1987/1988	2	2
1988/1989	4	1
1989/1990	3	1
1990/1991	4	4
1991/1992	7	8
1992/1993	8	2
1993/1994	7	7
1994/1995	1	1
1995/1996	6	7
1996/1997	6	2
1997/1998	5	5
1998/1999	6	2
1999/2000	9	3
2000/2001	7	4
2001/2002	6	4
2002/2003	5	1
2003/2004	3	6
2005/2006	16	4
2006/2007	8	5
2007/2008	10	6
2008/2009	8	8
2009/2010	6	6
2010/2011	9	3
2011/2012	8	4
2012/2013	2	1
2013/2014	3	9
2014/2015	5	9
2015/2016	7	6
2016/2017	3	8
TOTAL	174	129

The changes are rather interesting. Now, overall, the chances of streak to continue are up to 57.4%, and only in 42.6% of the cases it came to a stop. But in the last five years - since the last lockout - and with the schedule changes so that there are at least two games between every team (increasing travel), the ratio drops from 50.8% to the humble 37.7% (20 out of 53!)

Extending the breaks to five days provides too little data to draw any conclusions.

So I am inclined to agree with Dr. Botwinnik, that extended breaks of more than three days throw teams off their pace and should be reduced to minimum. Three days are borderline alright.

A rule change suggestion

There's no irreplaceable people.

I.V. Stalin

Rushing this one up, because this idea already came to my mind before, but I forgot about it. The age is taking its toll.

Anyways. Everyone is talking these days about rule changes. I've already expressed a few thoughts on the scoring systems, but I am not original there. Now, however, I want to make a suggestion I haven't seen mentioned yet.

Allow soccer (baseball, too)-like substitutions in hockey. Allow the coaches to replace players in the original lineup at the start of the game with one of the "healthy scratches", as submitted in the roster sheet, like the one Peter DeBoer recently messed up in the game against Edmonton.

The substitution goes ONE-WAY. That means that the player that was substituted cannot return to the game. The substitutions may occur:

• During the intermissions
• During the commercial breaks
• During a time-out

First and foremost this will allow teams to handle early injuries much better. Your D-man got injured at the 7:04 mark of the 1st period? Around 10:00 there will be a commercial break, you can substitute him with one of the scratches!

Second, it may allow coaches to send stronger messages to players they deem slacking. Rather than shorten the roster by benching that guy, you can send an eager healthy scratch in. Of course, then the "slacking" player is benched for the whole remainder of the game.

Third (oh, I did military service, so I have a natural obsession of providing three reasons for each thing), it may give the coaches some extra flexibility if a designated roster player gets slightly injured in the warm-ups. Then a scratch takes his place as usual, but if the original player is fixed by the 1st intermission, he can substitute the starting scratch.

The substitutes will have to come from the "scratch" list with the exception of the emergency goaltending contracts.

Oh, and I am sure the NHL website will make a mess out of it in their game reports.

On Intangibles (a small addendum)

One "intangible" being tossed around is "motivation" of the players. Which brings memories of an episode I was witness to.

In 2003/04, in the Israeli Top Tier Chess League (which is indeed no slouch) our club managed to assemble an outstanding team, featuring, among others, a former Champion of Russia and a former Champion of Europe. I was part of the management team, and orchestrated bringing the first of the two, who also happened to be my childhood friend back in Leningrad, Soviet Union.

And so, in round III we were to face our main rival for the title, and the club's GM (also a pedestrian chess player) gathered the team and carried out a pronounced motivational speech, how we have to beat the team we're facing, and so on, and so on.

We lost 1½-4½ without winning a single game and lost any chance for the championship we could have.