One way to correct for the unbalanced schedule would be to make games
between teams who only play twice worth 4 points, like the CHA is
doing this year, and I believe the WCHA did some time in the past.
However, that still benefits weak teams who only play strong teams
twice, since they could get lucky and pull off a win or tie, which
they would get double-credit for, when they would have been very
unlikely to repeat the feat.
Another correction would be to project how each team would have fared
if they'd played a full balanced schedule. One way to do this is to
calculate a Bradley-Terry, or KRACH, rating based only on games within
the conference. The result of this is a KRACH for each team which is
proportional to its "odds" of winning any game; if you add up the
resulting probabilities of winning each of the games on a team's
schedule, you'll get the number they actually won. Ranking teams by
KRACH is equivalent to ranking them by Round-Robin Winning Percentage,
which is the average of their probabilities of winning a game with
each other team in the league, or by RRP, the number of points KRACH
says they would get if they played each other team a specified number
of times. If a league plays a balanced schedule, ranking by
KRACH/RRWP/RRP is guaranteed to be the same as ranking by winning
percentage. For example, here are the results of last season's Hockey
East race; each team played each other team three times, and the
actual number of points they received in those games are listed below:
Tm PF/GP Pct RRWP NH Me BC Pv BU MA ML Mr NE
NH 39/24 .812 .812 --- 4/3 3/3 4/3 4/3 6/3 6/3 6/3 6/3
Me 36/24 .750 .750 2/3 --- 4/3 6/3 4/3 3/3 6/3 6/3 5/3
BC 32/24 .667 .667 3/3 2/3 --- 3/3 4/3 6/3 6/3 4/3 4/3
Pv 25/24 .521 .521 2/3 0/3 3/3 --- 6/3 4/3 4/3 4/3 2/3
BU 19/24 .396 .396 2/3 2/3 2/3 0/3 --- 4/3 4/3 3/3 2/3
MA 18/24 .375 .375 0/3 3/3 0/3 2/3 2/3 --- 2/3 6/3 3/3
ML 18/24 .375 .375 0/3 0/3 0/3 2/3 2/3 4/3 --- 4/3 6/3
Mr 15/24 .312 .312 0/3 0/3 2/3 2/3 3/3 0/3 2/3 --- 6/3
NE 14/24 .292 .292 0/3 1/3 2/3 4/3 4/3 3/3 0/3 0/3 ---
Notice that the RRWP is exactly equal to the actual winning percentage
for each team. That's because the teams really did play a round-robin
schedule. Here are the actual KRACH ratings and the probabilities
they predict of winning against each opponent:
Tm KRACH RRP/TGP NH Me BC Pv BU MA ML Mr NE
NH 428.5 39.0/24 .--- .584 .677 .797 .868 .878 .878 .904 .912
Me 304.7 36.0/24 .416 .--- .599 .737 .824 .837 .837 .871 .881
BC 204.1 32.0/24 .323 .401 .--- .652 .759 .774 .774 .818 .832
Pv 109.0 25.0/24 .203 .263 .348 .--- .627 .647 .647 .706 .725
BU 64.95 19.0/24 .132 .176 .241 .373 .--- .522 .522 .589 .612
MA 59.48 18.0/24 .122 .163 .226 .353 .478 .--- .500 .568 .591
ML 59.48 18.0/24 .122 .163 .226 .353 .478 .500 .--- .568 .591
Mr 45.32 15.0/24 .096 .129 .182 .294 .411 .432 .432 .--- .524
NE 41.24 14.0/24 .088 .119 .168 .275 .388 .409 .409 .476 .---
For example, UNH's conference KRACH is 428.5, about four times
Providence's rating of 109.0, so the predicted odds that they'll win
each game against PC are around four-to-one; more precisely, the
probability is .797=428.5/(428.5+109.0). Their RRP are found by
multiplying their head-to-head probability against each team by 3
(since the round-robin in question is three games between each pair of
teams), so .584 times 3 plus .677 times 3, etc. That adds up to 39,
which is exactly the number of points they actually got.
Now, apply this to a schedule that wasn't balanced, like last year's
WCHA:
Tm PF/GP Pct RRWP ND CC DU Wi Mn AA SC MT MD
ND 50/28 .893 .898 --- 6/4 8/4 6/4 7/4 4/2 8/4 4/2 7/4
CC 40/28 .714 .728 2/4 --- 4/4 2/2 8/4 8/4 4/4 8/4 4/2
DU 32/28 .571 .593 0/4 4/4 --- 5/4 3/2 4/4 6/4 8/4 2/2
Wi 29/28 .518 .511 2/4 2/2 3/4 --- 3/4 3/4 4/2 4/4 8/4
Mn 26/28 .464 .472 1/4 0/4 1/2 5/4 --- 4/4 4/4 4/2 7/4
AA 25/28 .446 .432 0/2 0/4 4/4 5/4 4/4 --- 2/2 4/4 6/4
SC 20/28 .357 .360 0/4 4/4 2/4 0/2 4/4 2/2 --- 0/4 8/4
MT 18/28 .321 .304 0/2 0/4 0/4 4/4 0/2 4/4 8/4 --- 2/4
MD 12/28 .214 .202 1/4 0/2 2/2 0/4 1/4 2/4 0/4 6/4 ---
Tm KRACH RRP/TGP ND CC DU Wi Mn AA SC MT MD
ND 863.1 57.5/32 .--- .751 .852 .892 .907 .920 .940 .953 .972
CC 286.8 46.6/32 .249 .--- .657 .732 .764 .793 .840 .871 .919
DU 149.8 37.9/32 .148 .343 .--- .588 .628 .667 .732 .779 .856
Wi 104.8 32.7/32 .108 .268 .412 .--- .542 .584 .657 .712 .806
Mn 88.61 30.2/32 .093 .236 .372 .458 .--- .543 .618 .676 .778
AA 74.65 27.6/32 .080 .207 .333 .416 .457 .--- .577 .637 .747
SC 54.76 23.1/32 .060 .160 .268 .343 .382 .423 .--- .563 .684
MT 42.47 19.5/32 .047 .129 .221 .288 .324 .363 .437 .--- .627
MD 25.29 12.9/32 .028 .081 .144 .194 .222 .253 .316 .373 .---
Everyone played 4 games against seven opponents and 2 against the
other two. The KRACH ratings are as always defined so that they
predict the correct number of points against whatever schedule was
actually played. For instance, UMD played 2 games each against
Colorado College and Denver, so if you add 4 (number of games against
North Dakota) times .028 (predicted probability of beating UND), 2
times .081 (for CC), 2 times .144 (for DU), 4 times .194 (for
Wisconsin), etc, you get 12, the number of points UMD actually got
against that schedule. However, to obtain the round-robin points or
winning percentage, we need to include an equal number of games
against each team, for instance by adding two hypothetical games each
against CC and DU. That would give UMD an additional 2 times .081
plus 2 times .144, or 0.9 points for a total of 12.9 against a 32-game
balanced schedule. Since the four "unplayed games" for UMD were
against two of the strongest teams in the league, their RRWP of .202
is lower than their winning percentage of .214.
As it turns out, correcting for the unbalanced schedule wouldn't have
changed the ranking of the teams last season, partly because it was
only slightly unbalanced. Here is the same analysis for the WCHA so
far this season:
Tm PF/GP Pct RRWP Wi ND SC Mn CC Mk AA MD DU MT
Wi 36/22 .818 .811 --- 4/2 6/4 4/2 4/2 2/2 2/2 2/2 4/2 8/4
ND 33/24 .688 .700 0/2 --- 5/4 5/4 3/2 2/2 8/4 4/2 2/2 4/2
SC 26/22 .591 .622 2/4 3/4 --- 3/2 0/0 4/2 4/2 4/4 2/2 4/2
Mn 24/22 .545 .587 0/2 3/4 1/2 --- 4/4 2/2 2/2 8/4 4/2 0/0
CC 26/22 .591 .555 0/2 1/2 0/0 4/4 --- 2/2 3/2 4/4 8/4 4/2
Mk 25/22 .568 .525 2/2 2/2 0/2 2/2 2/2 --- 3/4 0/0 6/4 8/4
AA 23/24 .479 .459 2/2 0/4 0/2 2/2 1/2 5/4 --- 0/0 5/4 8/4
MD 16/22 .364 .341 2/2 0/2 4/4 0/4 4/4 0/0 0/0 --- 2/2 4/4
DU 15/24 .312 .319 0/2 2/2 2/2 0/2 0/4 2/4 3/4 2/2 --- 4/2
MT 4/24 .083 .080 0/4 0/2 0/2 0/0 0/2 0/4 0/4 4/4 0/2 ---
Tm KRACH RRP/TGP Wi ND SC Mn CC Mk AA MD DU MT
Wi 433.4 58.4/36 .--- .642 .718 .748 .774 .795 .838 .898 .908 .980
ND 242.0 50.4/36 .358 .--- .587 .624 .656 .684 .743 .832 .847 .965
SC 170.1 44.8/36 .282 .413 .--- .539 .573 .604 .670 .776 .795 .951
Mn 145.7 42.3/36 .252 .376 .461 .--- .534 .566 .635 .748 .769 .943
CC 126.9 40.0/36 .226 .344 .427 .466 .--- .532 .602 .721 .743 .935
Mk 111.6 37.8/36 .205 .316 .396 .434 .468 .--- .571 .695 .718 .927
AA 83.81 33.1/36 .162 .257 .330 .365 .398 .429 .--- .631 .657 .905
MD 48.99 24.6/36 .102 .168 .224 .252 .279 .305 .369 .--- .528 .847
DU 43.82 22.9/36 .092 .153 .205 .231 .257 .282 .343 .472 .--- .832
MT 8.830 5.7/36 .020 .035 .049 .057 .065 .073 .095 .153 .168 .---
Micheal Neal is right that Minnesota is punished by their stronger
schedule; they are the 6th place team according to winning percentage
(or total points) and 4th place according to conference
KRACH/RRWP/RRP. Of course, that's working with a partially-completed
schedule, so it's even more unbalanced (Minnesota hasn't played
Michigan Tech at all, for instance). I tried inventing hypothetical
results for the last few weeks of the season which looked more or less
reasonable, and got the following as one possibility:
Tm PF/GP Pct RRWP Wi ND SC Mn CC Mk AA DU MD MT
Wi 46/28 .821 .821 --- 4/2 6/4 8/4 7/4 2/2 5/4 4/2 2/2 8/4
ND 40/28 .714 .715 0/2 --- 5/4 5/4 3/2 5/4 8/4 2/2 8/4 4/2
SC 34/28 .607 .627 2/4 3/4 --- 5/4 2/2 4/2 4/2 6/4 4/4 4/2
Mn 30/28 .536 .575 0/4 3/4 3/4 --- 4/4 2/2 2/2 4/2 8/4 4/2
CC 31/28 .554 .558 1/4 1/2 2/2 4/4 --- 4/4 3/2 8/4 4/4 4/2
Mk 32/28 .571 .547 2/2 3/4 0/2 2/2 4/4 --- 3/4 6/4 4/2 8/4
AA 27/28 .482 .478 3/4 0/4 0/2 2/2 1/2 5/4 --- 5/4 3/2 8/4
DU 19/28 .339 .319 0/2 2/2 2/4 0/2 0/4 2/4 3/4 --- 2/2 8/4
MD 17/28 .304 .296 2/2 0/4 4/4 0/4 4/4 0/2 1/2 2/2 --- 4/4
MT 4/28 .071 .064 0/4 0/2 0/2 0/2 0/2 0/4 0/4 0/4 4/4 ---
Tm KRACH RRP/TGP Wi ND SC Mn CC Mk AA DU MD MT
Wi 473.1 59.1/36 .--- .640 .729 .772 .786 .793 .839 .917 .926 .986
ND 266.5 51.5/36 .360 .--- .602 .656 .674 .684 .746 .862 .876 .975
SC 176.1 45.2/36 .271 .398 .--- .558 .577 .588 .660 .805 .823 .963
Mn 139.7 41.4/36 .228 .344 .442 .--- .520 .531 .607 .766 .787 .954
CC 129.1 40.1/36 .214 .326 .423 .480 .--- .511 .588 .752 .774 .951
Mk 123.3 39.4/36 .207 .316 .412 .469 .489 .--- .577 .743 .766 .948
AA 90.61 34.4/36 .161 .254 .340 .393 .412 .423 .--- .680 .706 .931
DU 42.58 22.9/36 .083 .138 .195 .234 .248 .257 .320 .--- .530 .864
MD 37.76 21.3/36 .074 .124 .177 .213 .226 .234 .294 .470 .--- .849
MT 6.713 4.6/36 .014 .025 .037 .046 .049 .052 .069 .136 .151 .---
Note that because of their varying strengths of schedule, Minnesota,
Colorado College, and Mankato State are ranked in opposite order
according to RRWP and (unbalanced) winning percentage.
John Whelan, Cornell '91
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http://www.amurgsval.org/joe/
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