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!