ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

[StayPuftMMcu]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

Yes, but:

a) I didn't have time to write the code to cycle through all the possibilities methodically - much easier just to assign random results to the remaining games and let 'er rip

and

b) I wanted to keep it general so that I might be able to use it in future seasons earlier in the season, when the # of permutations would be prohibitive

For ties, here's what I did so far (I emphasize that I am NOT happy with this):

I looked across all games played this year, and about 13% have been ties. Say that KRACH tells me that the probability of A beating B is 70%. I choose a random number between 0 and 1 for the result. Normally, you if the number is less than .7, you would call that a W for team A, and if it's above .7, it's an L. I added a "gray" zone centered on 0.7 that is .13 wide, and if the random number ends up in that range, it's a tie. So, for this example, a random number between .635 and .765 would be called a tie. If the raw probability is within 6.5% of 0 or 100, I slide the "tie zone" so that it ends halfway between the probability and the end of the scale. For example, if the probability of a win is 95%, then the "tie zone" would go from 84.5% to 97.5% (still 13% wide). Therefore, each game has a 13% chance of being a tie, so the total number of ties will be 13% (for a large sample).

Of course, the obvious problem is that every game should NOT have the same probability of having a tie. A pair of teams with a large discrepancy in KRACH ratings should clearly have less probability of tying than two teams with nearly identical ratings. So I need to figure out how to scale the "tie" zone based on the discrepancy in the pair under consideration, while making sure that the total number of ties comes out approximately correct, and I just didn't have time to think about that any farther yet. So, the results I have given are probably a little bit biased in favor of the lower teams - it gives them a higher probability of tying the really good teams than they probably really should get credit for.

Wouldn't you be able to create a sliding scale of probability that would result in a total of ~13% ties, but with a wider range as the game is rated at about 50-50 and a narrower range as the game approaches a 100-0 chance of victory/loss for one team? Say at 50-50 there's a 20% chance of tying, at 100-0 there's 0% chance of tying, and a sliding scale in between that allows for the total probability to equal out to 13% when all games are played?

I haven't done the math at all, but hopefully I've explained it well enough.

[Ralph Baer]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

I threw together a little Monte Carlo simulation of the last 12 games for a thread over on eLynah. Here are the probabilities I came up with, where the results of the remaining games are based on the current KRACH ratings (including all games, not just ECAC games) of the teams. My method does allow for ties (though this could use some improvement) AND my code is not up to date with the latest tiebreaking procedure (it's from 2003 or so), but with those caveats, here's approximately the situation (after 1 million trial runs):

	1	2	3	4	5	6	7	8	9	10	11	12
Union	81.0%	19.0%										
Yale	19.0%	81.0%										
Cornell			83.1%	14.7%	2.2%							
Dartmouth		9.8%	36.2%	42.7%	11.3%						
Rensselaer		2.5%	29.5%	25.3%	42.7%						
Princeton		4.6%	19.6%	29.8%	46.0%						
Quinnipiac						68.0%	30.6%	1.3%	0.1%		
Clarkson						32.0%	68.0%				
St. Lawrence							0.8%	54.9%	44.2%		
Brown								0.6%	43.8%	45.9%	9.7%	
Colgate										9.7%	73.2%	17.0%
Harvard											17.0%	83.0%

At least for me, the columns aren't lining up correctly. They don't on eLynah either. For example, all of Princeton's numbers should be one column to the right.

[LynahFan]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

Wouldn't you be able to create a sliding scale of probability that would result in a total of ~13% ties, but with a wider range as the game is rated at about 50-50 and a narrower range as the game approaches a 100-0 chance of victory/loss for one team? Say at 50-50 there's a 20% chance of tying, at 100-0 there's 0% chance of tying, and a sliding scale in between that allows for the total probability to equal out to 13% when all games are played?

I haven't done the math at all, but hopefully I've explained it well enough.

Yes - and that is exactly what I plan to do. The trick is in picking the sliding scale. There are a whole lot more 50-50 games played than 90-10s, because most teams' KRACH ratings aren't all that far apart - they tend to cluster together, with only a few high and low outliers. So the trick is to slide the scale so that it "feels" right (in that a lousy team should have some small, but finite, chance to tie a great team) while still making the overall proportion of ties come out correct.

The first (obvious, but also obviously incorrect) idea is to make the likelihood of a tie go from 0% (when there's 0% chance of a win) up to 26% when the game is exactly 50-50. This seems to make sense initially, since the "average" likelihood of a tie would then seem to be the average of 0 and 26, which is the desired 13%. However, given that there are so many 50-50 games played, the actual percentage of ties that you would get out of the simulation would be far too high - probably approaching 26%, in fact.

I'm working on some other ideas in the back of my mind, but haven't had a chance to work them out mathematically (or more likely, given my temperament, just code the suckers up and see if they work!).

[LynahFan]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

At least for me, the columns aren't lining up correctly. They don't on eLynah either. For example, all of Princeton's numbers should be one column to the right.

Hmmm - apologies! My html is far weaker than my math... Princeton should have a 4.6% chance of finishing 3rd, which is what it looks like to me (using Chrome 9.0.597.98) - even when you requote the message back to me. Interestingly, I copied and pasted that from Excel, and I did have to delete some of the tabs to get it to look correct in my browser. Here's what it looks like if I just paste in the raw excel and don't fix it so it looks right to me:

	1	2	3	4	5	6	7	8	9	10	11	12
Union	81.0%	19.0%										
Yale	19.0%	81.0%										
Cornell			83.1%	14.7%	2.2%							
Dartmouth			9.8%	36.2%	42.7%	11.3%						
Rensselaer			2.5%	29.5%	25.3%	42.7%						
Princeton			4.6%	19.6%	29.8%	46.0%						
Quinnipiac							68.0%	30.6%	1.3%	0.1%		
Clarkson							32.0%	68.0%				
St. Lawrence								0.8%	54.9%	44.2%		
Brown								0.6%	43.8%	45.9%	9.7%	
Colgate										9.7%	73.2%	17.0%
Harvard											17.0%	83.0%

I'm guessing that this has to do with the length of the team names. Maybe it would help if I shorten them:

	1	2	3	4	5	6	7	8	9	10	11	12
UC	81.0%	19.0%										
YU	19.0%	81.0%										
Corn			83.1%	14.7%	2.2%							
DC			9.8%	36.2%	42.7%	11.3%						
RPI			2.5%	29.5%	25.3%	42.7%						
PU			4.6%	19.6%	29.8%	46.0%						
QU							68.0%	30.6%	1.3%	0.1%		
CCT							32.0%	68.0%				
SLU								0.8%	54.9%	44.2%		
BU								0.6%	43.8%	45.9%	9.7%	
Gate										9.7%	73.2%	17.0%
HU											17.0%	83.0%

Yep - the first version above still looks wrong to me, but the 2nd one looks right.

[Ralph Baer]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

The new one is fine. The simplest way around it is to use spaces instead of tabs.

[goblue78]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

Thanks LynahFan. I was asking about ties because I'm in the process of creating a new "tie-enabled" Krach right now. I'm still checking the method, and I'm probably not going to release it until the end of the season and start a discussion about it, but essentially it uses (for the math-enabled) a mutinomial logit with independent team tie intercepts. I'm collecting academic references now to make sure I implement it correctly. Preliminary results are interesting, though....

As to enumeration vs. Monte Carlo, I guess it's really just a question of what sort of programming you're used to. I'm a Monte Carlo fan as well, but enumeration always seems much tidier to me. Once the season's over and ties are no longer an issue, an enumerative program I have will easily do both the ECAC and NCAA tournaments.

[licwinko]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

However, given that there are so many 50-50 games played, the actual percentage of ties that you would get out of the simulation would be far too high - probably approaching 26%, in fact.

Just lower the upper threshold of a tie to 18% or something. There's got to be a number that ensures the average will be 13% I havent had enough coffee to figure it out.

[alslammerz]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

I wouldn't call it a mathematical look but I tried going through every possible scenario and figuring out where that would leave teams. I think I got all of them but many other eyes would be helpful in case I missed anything or botched something. Will happily explain anything that I put if I can remember tomorrow why I thought that was the case. So here's a look at where every team can end up (and an explanation of how they can get there).

Help much appreciated.

PS- Here is Brian Sullivan's column as well.

[burgie12]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

I know that Brian Sullivan and Brendan Roche have already made significant posts on this subject, but I'm going to throw my hat in the ring anyways.

UC 34 - 36 [1-2]
Yale 33 - 35 [1-2]
DC 26 - 28 [3-4]
--- Bye Lock - 26+
Cornell 24 - 26 [3-6]
PU 24 - 26 [4-6]
RPI 23 - 25 [4-6]
--- Bye Eligible - 24+
--- Home Lock - 18+
QU 18 - 20 [7-9]
CCT 17 - 19 [7-9]
Brown 16 - 18 [7-9]
--- Home Eligible - 17+
Harvard 13 - 15 [10-11]
SLU 13 - 15 [10-11]
Colgate 11 - 13 [12-12]

Remaining League Schedules:
UC - PU
Yale - Cornell
DC - CCT
Cornell - @Yale
PU - @UC
RPI - QU
QU - @RPI
CCT - @DC
Brown - Colgate
Harvard - SLU
SLU - @Harvard
Colgate - @Brown

slack.net's ECAC Playoff Possibilities Script is really pretty awesome.

Cornell holds 4th place over Princeton based on Record vs Top 4 teams (3 v 0). Harvard holds 10th place over St. Lawrnece based on head-to-head points (2 v 0).

Possible Tiebreakers (winning team(s) listed first):
Yale / Union - A Yale win and Union tie gives the Elis the tiebreaker win based on ECAC wins (17 v 16). A Yale tie and Union loss lets Yale win the tiebreaker based on Record vs Top 4 teams (10 v 5).

Dartmouth / Princeton - Season series (4 v 0)

Cornell / Dartmouth - Season series (3 v 1)

Cornell / Princeton - Record vs Top 4 (3 v 0 or 4 v 1 or 5 v 2)

Cornell / Rensselaer - Season series (4 v 0)

Rensselaer / Princeton - An RPI win and Princeton tie gives the Engineers the tiebreaker win based on ECAC wins (12 v 11). An RPI tie and Princeton loss gives the Engineers the tiebreaker win on Record vs Top 4 teams (8 v 2).

Brown / Quinnipiac - ECAC wins (8 v 6)

Clarkson / Quinnipiac - ECAC wins (8 v 6 or 9 v 6)

Clarkson / Brown - Season series (4 v 0)

Harvard / St. Lawrence - Season series (3 v 1)

Harvard / Colgate - ECAC wins (6 v 5)

St. Lawrence / Colgate - Season series (4 v 0)

Cornell / Dartmouth / Princeton - head-to-head record (5 v 5 v 2) drops out Princeton. Cornell beats Dartmouth in the head-to-head tiebreaker

Cornell / Rensselaer / Princeton - head-to-head record (6 (Cornell) v 4 (Princeton) v 2 (Rensselaer)) gives the Big Red the 4th place position. Rensselaer beats Princeton in the head-to-head tiebreaker

Clarkson / Brown / Quinnipiac - head-to-head record (6 (Clarkson) v 4 (Quinnipiac) v 2 (Brown)) gives the Knights 7th place. The Bears take 8th based on the head-to-head tiebreaker

Best-case scenarios:
Union - 1st place
1) Win OR
2) Tie AND Yale Tie or Loss OR
3) Loss AND Yale Loss

Yale - 1st place
1) Win AND Union Tie or Loss OR
2) Tie AND Union Loss

Dartmouth - 3rd place
1) Win OR
2) Tie OR
3) Loss AND Cornell Tie or Loss (Princeton result doesn't matter)

Cornell - 3rd place
1) Win AND Dartmouth Loss (Princeton result doesn't matter)

Princeton - 4th place
1) Win AND Cornell Tie or Loss (Dartmouth result doesn't matter) OR
2) Tie AND Cornell Loss AND Rensselaer Tie or Loss

Rensselaer - 4th place
1) Win AND Cornell Loss AND Princeton Tie or Loss

Quinnipiac - 7th place
1) Win OR
2) Tie AND Clarkson Tie or Loss OR
3) Loss AND Clarkson Loss AND Brown Tie or Loss

Clarkson - 7th place
1) Win AND Quinnipiac Tie or Loss OR
2) Tie AND Quinnipiac Loss (Brown result doesn't matter)

Brown - 7th place
1) Win AND Quinnipiac Loss AND Clarkson Loss

Harvard - 10th place
1) Win OR
2) Tie

St. Lawrence - 10th place
1) Win

Other important scenarios:
Cornell - bye position
1) Win OR
2) Tie AND Princeton Tie or Loss OR
3) Loss AND Princeton Loss AND Rensselaer Tie or Loss

Quinnipiac - home-ice position
1) Win OR
2) Tie OR
3) Loss AND Clarkson Loss OR
4) Loss AND Brown Tie or Loss

Clarkson - home-ice position
1) Win OR
2) Tie OR
3) Loss AND Brown Tie or Loss

Brown - home-ice position
1) Win AND Quinnipiac Loss OR
2) Win AND Clarkson Loss

[alslammerz]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

Burgie, just wanted add, thanks for doing this. My post would have taken 5x as long without the ECAC playoff script you posted. And the tiebreaking scenarios you posted were definitely a plus.

[burgie12]

Re: ECAC Home Stretch - A mathematical approach to Byes and Home-ice (2010-2011)

Burgie, just wanted add, thanks for doing this. My post would have taken 5x as long without the ECAC playoff script you posted. And the tiebreaking scenarios you posted were definitely a plus.

Don't thank me. Your post was much more extensive and enlightening. Besides, I didn't make the playoff script. I'm just using the resources available, just like any good reporter / blogger / poster.

I hope that everyone enjoyed reading these past few weeks. I'd also like to thank LynahFan for the awesome KRACH Monte Carlo work that he's done.