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Thread: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

  1. #1
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    ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    There are just five weekends left in the ECAC Regular Season. And everybody knows what that means! It's time to take an ill-advised, extraordinarily ridiculous approach to determining the final standings.

    Come and join me as we walk down this path towards the Cleary Cup together. Along the way, I will show you the wonders of TBRW's ECAC Page (currently down), Sioux Sports' What-If Calculator (which doesn't use tiebreakers), and the ECAC Tiebreakers page (which you will come to know and love). And, if you want to use Playoff Status, then I get to punch you in the face. For those that are curious, I've also included links to last year's and the year before's threads, too.

    Current Standings (by Points %):
    --- Bye Lock - 31+
    --- Home Lock - 25+
    Quinnipiac 23-43 [1-11]
    Yale 17-35 [1-12]
    Dartmouth 15-33 [1-12]
    Union 15-33 [1-12]
    Princeton 13-33 [1-12]
    St. Lawrence 11-31 [1-12]
    Colgate 11-31 [1-12]
    Cornell 10-30 [1-12]
    Clarkson 10-30 [1-12]
    Brown 10-28 [1-12]
    Rensselaer 9-29 [1-12]
    Harvard 6-22 [2-12]
    --- Bye Eligible - 18+
    --- Home Eligible - 13+

    Dartmouth defeated Union in Hanover in November and holds third place on a head-to-head tiebreaker. They play tomorrow night to finalize this tiebreaker.

    St. Lawrence went on the road to Hamilton in November and took home the win, so they also lead their tiebreaker for 6th place. The Saints and Raiders are also playing this weekend to finalize that tiebreaker.

    Cornell beat Clarkson at home that same night in November and holds the 10th place tiebreaker because of it. That tiebreaker will be finalized this Saturday in Potsdam.

    Brown is not tied with either Clarkson or Cornell because the Bears have played an extra game.

    Remaining League Schedules:
    Code:
    ----------------------------------------------------------------------------------
    |              |    ||F2/01|S2/02|F2/08|S2/09|F2/15|S2/16|F2/22|S2/23|F3/01|S3/02|
    ----------------------------------------------------------------------------------
    | Quinnipiac   | QN || @BN | @YA | @CR | @CG |  SL |  CK |  YA |  BN | @HA | @DA |
    | Yale         | YA ||  PN |  QN |     | @BN | @UC | @RP | @QN | @PN |  CG |  CR |
    | Dartmouth    | DA || @UC | @RP |     | @HA |  CR |  CG | @CK | @SL |  PN |  QN |
    | Union        | UC ||  DA |     | @CK | @SL |  YA |  BN | @CG | @CR |  SL |  CK |
    | Princeton    | PN || @YA | @BN | @CG | @CR |  CK |  SL |  BN |  YA | @DA | @HA |
    | St. Lawrence | SL ||  CR |  CG |  RP |  UC | @QN | @PN |  HA |  DA | @UC | @RP |
    | Colgate      | CG || @CK | @SL |  PN |  QN | @HA | @DA |  UC |  RP | @YA | @BN |
    | Cornell      | CR || @SL | @CK |  QN |  PN | @DA | @HA |  RP |  UC | @BN | @YA |
    | Clarkson     | CK ||  CG |  CR |  UC |  RP | @PN | @QN |  DA |  HA | @RP | @UC |
    | Brown        | BN ||  QN |  PN |     |  YA | @RP | @UC | @PN | @QN |  CR |  CG |
    | Rensselaer   | RP ||  HA |  DA | @SL | @CK |  BN |  YA | @CR | @CG |  CK |  SL |
    | Harvard      | HA || @RP |     |     |  DA |  CG |  CR | @SL | @CK |  QN |  PN |
    ----------------------------------------------------------------------------------
    Individual Team Limits:
    Teams Losing Out (Floors)
    Holy crap, I don't know how I did it, but I did. I found a way to put Quinnipiac all the way back to 11th place. The end result is three teams (Brown, RPI, and Union) each getting a share of the Cleary Cup at 24 points, Harvard finishing in 12th at 8 points, and everybody else finishing in a tie for 4th place at 23 points. Clarkson, Colgate, Cornell, Princeton, and St. Lawrence all get 15 points against the 7 other teams tied for 4th, leaving just Dartmouth, Yale, and Quinnipiac tied for 9th place after the first round of tiebreakers. The tiebreaking process starts over, with Yale winning the head-to-head having taken 6 points against the Big Green and the Bobcats. Then, it's down to just DC and QU fighting for 10th having (hypothetically) split the season series. I also had it so that Dartmouth only won or lost to get to 23 points, meaning that both teams have 11 wins still. And, with Brown, RPI, and Union all locked into the Top 4, Dartmouth can't lose the Points vs Top 4 tiebreaker, so Quinnipiac falls all the way to 11th place.

    Game-by-game results (if anyone's curious):
    Code:
    02/01 CG < CK | CR < SL | DA > UN | HA < RP | PN > YA | QN < BN
    02/02 CG < SL | CR > CK | DA < RP | PN < BN | QN < YA
    02/08 PN < CG | QN < CR | RP > SL | UN > CK
    02/09 DA > HA | PN = CR | QN < CG | RP > CK | UN > SL | YA < BN
    02/15 BN < RP | CK > PN | CG > HA | CR < DA | SL > QN | YA < UN
    02/16 BN > UN | CK > QN | CG > DA | CN > HA | SL = PN | YA > RP
    02/22 BN > PN | DA < CK | HA < SL | RP < CR | UN > CG | YA > QN
    02/23 BN > QN | DA < SL | HA < CK | RP > CG | UN < CR | YA < PN
    03/01 CK = RP | CG > YA | CR < BN | PN > DA | QN < HA | SL = UN
    03/02 CK > UN | CG > BN | CR > YA | PN > HA | QN < DA | SL < RP
    
    Notes:
    XX > YY indicates that XX wins
    XX = YY indicates a tie
    XX < YY indicates that YY wins
    XX is away; YY is home
    Games are listed in alphabetical order of the away teams that night
    PS I'm not saying that this is the only way to get Quinnipiac to fall to 11th. I'm sure there at least one or two other game combinations, haha.

    Teams Winning Out (Ceilings)
    Raising Harvard to 2nd wasn't nearly as difficult as I imagined. With the Crimson still capable of cresting 22 points and Quinnipiac getting to 41 at the same time (they couldn't hit 43, because that would require beating Harvard in Boston on the final weekend), the other ten teams have to finish a net 19 points below 0.500 (which is, coincidentally, 22 points). With only ten teams and nineteen points to spread around, you don't even need to start going into tiebreakers.

    "Other"
    Getting the other schools to the limits isn't all that difficult. Have them win out, then let teams that can't catch them win out, and as more and more games get played, the picture of how to boost them up becomes clearer and clearer.
    Last edited by burgie12; 01-31-2013 at 02:46 PM. Reason: Quinnipiac to 11th isn't formatting properly
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  2. #2
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Thresholds:
    The Bye Lock refers to the most number of points that can be reached by 5th place and correspondingly, how many points a team would need to earn in order to guarantee themselves a 4th place finish. How I reached that scenario this time involves Yale, Dartmouth, Union, and Princeton winning every remaining game they don't have against each other, Princeton defeating Yale, and then ties for the remaining four games. That leaves all four of them sitting at 31 points and Quinnipiac is still capable of getting to (or above) that total, so to be guaranteed a weekend off, you need to finish with at least 31 points and win the appropriate tiebreakers.

    Conversely, Bye Eligible refers to the least number of points that a team can earn and still have the first weekend of playoffs off. I was able to cause a large logjam at 18 points with Quinnipiac, Yale, and Dartmouth ahead of the pack and three teams just behind this logjam at 17 points. So, as long as you can get to 18 points, if you can win the right tiebreakers, you could potentially finish in 4th place.

    Home Lock is similar to Bye Lock, except that it refers to boosting the points of 9th place, trying to determine the maximum number of points that a team would need in order to secure home-ice in the first round. By minimizing the points that I gave to Brown, Rensselaer, and Harvard (and then Quinnipiac, since they were clearly above this maximum after having hypothetically swept Brown and Harvard), I was able to cause the other eight teams to conglomerate around 25 or 26 points. Two teams ended up tied for eighth place at 25 points apiece, indicating that 25 points should be enough to guarantee yourself two extra home games.

    That point merits some extra exploring, then. Let's say that Quinnipiac beats Brown on Friday night and somehow, that's the only game that gets played that night. Quinnipiac now has 25 points. Are they guaranteed an extra two home games after the conclusion of the regular season? Well, let's have the Bobcats lose out the rest of the regular season. And, we know that the Crimson can't catch them, so have Harvard lose out, too. Now, to have each of the ten remaining teams get to 25 points, 99 points would have to be distributed, but with both Quinnipiac's and Harvard's regular season finished, there are only eighty points remaining. So, clearly, not every team can catch and/or pass Quinnipiac. RPI, with nine hypothetical games remaining (and the least number of points accumulated so far), is next on the chopping block. We're still short. 67 points need to be accumulated, but there are only 31 games (and therefore 62 points) left. Good bye Brown. Finally, we're on the right side of the ledger. But, two teams will need to finish with 25 points while the remaining six can finish with 26. What two teams should finish with 25 points so that Quinnipiac would lose the tiebreakers and finish in 9th? Yale, who hasn't played Quinnipiac yet and would therefore have swept them in this hypothetical is an obvious choice. But, we need somebody else who could win the three-way tiebreaker to make sure that the head-to-head comes down to Yale and Quinnipiac. And, in the way that I generated this scenario, it falls to Colgate to become said third team. End result? Quinnipiac loses the relevant tiebreakers, finishes 9th, and is not guaranteed another weekend at home. So, don't forget, just because you've reached the relevant point plateau does not mean that you are guaranteed to have a bye or have earned home-ice. Tiebreakers matter, people.

    And, finally, Home Eligible. The least number of points that can be had by the 8th place team at the end of the season. I was able to get a three-way tie for 8th place at 13 points. So, for now, if you can get to 13 points (and have the requisite tiebreakers), you can have home-ice in the playoffs.

    I'm going to apologize in advance for the mathiness of the rest of this post (and the next one).

    You all know what KRACH is, right? It's the application of the Bradley-Terry ranking system, but falls into the fatal flaw that it doesn't treat ties properly. It treats them like half-wins and half-losses. Now, there's a way to fix this. P. Rao and L. Kupper wrote a paper in 1967 which found a better way to treat ties. PDF paper with the details of the ranking system can be found here (Section 7.3). The problem is, I still don't like it. Luckily, Dr. Michael Rutter of Penn State Erie is awesome. He took the Mease College Football Rankings and adapted them for college hockey (slides 10-12 are the keys there). And then he allowed me to steal his spreadsheet and adapt it. The Mease Rankings are based on the idea that each team's abilities can be represented by a normal distribution centered at some value (THETA_i) with a variance of 0.5. Then, whenever two teams play each other, the likelihood of outcomes when Team i plays Team j is represented by a normal distribution centered at the difference of their THETA values with a variance of 1.0 (supporting webpage from Wolfram). We then try and maximize the log of these likelihoods by changing the thetas of each team and phi. This makes a lot of sense to me. Teams have good nights and they have bad ones. No team has a fixed ranking. It's a distribution of rankings centered around some number. And, a normal distribution sounds good to me.

    Before we go any further, Mease suffers from many of the same flaws as B-T. It can only evaluate the games as they were played. It doesn't take into account referee screw-jobs, injuries, academic disqualifications, game misconducts or anything of the like. It doesn't matter if you win by one goal or fifteen, Mease treats them the same. It doesn't take into account hot streaks or home-ice advantage (at least, this version doesn't, it would be non-trivial to code, but it could be done). It is retrodictive, not predictive. It should not be used to predict games. With that out of the way, let's predict some games!

    PS If someone knows how to code CHODR, please let me know. It doesn't have a distribution, but it uses real scores instead of just game results and is actually a predictive model.
    Last edited by burgie12; 01-31-2013 at 02:49 PM.
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  3. #3
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Mease Rankings (THETA):
    Code:
    -------------------------
    |     Team     |  Theta |
    -------------------------
    | Quinnipiac   |  0.888 |
    | Yale         |  0.539 |
    | Dartmouth    |  0.405 |
    | Union        |  0.223 |
    | Princeton    | -0.086 |
    | St. Lawrence | -0.065 |
    | Colgate      |  0.234 |
    | Cornell      |  0.041 |
    | Clarkson     | -0.443 |
    | Brown        | -0.190 |
    | Rensselaer   |  0.021 |
    | Harvard      | -0.493 |
    -------------------------
    
    PHI = 0.196221
    The probability that the away team wins is equal to the probability that a standard normal distribution (NORMSDIST in Excel) will have a value less than [(their theta) - (home team's theta) - (phi)]. The probability that the home team wins is equal to 1 - {the probability that a standard normal distribution will have a value less than [(their theta) - (away team's theta) - (phi)]}. The probability of a tie is the leftover remainder. So, have Excel generate a random number between 0 and 1 and see where it falls on this distribution scale. Repeat it for each of the remaining 57 games, find the standings after said 57 games have been completed, and voila, that is one trial.

    And, here are the results, organized by expected final standings...
    Mease Simulation (51,500 trials; phi = 0.196221):
    Code:
    ------------------------------------------------------------------------------------------------------------
    |              |  1st |  2nd |  3rd |  4th |  5th |  6th |  7th |  8th |  9th | 10th | 11th | 12th | ExpPl |
    ------------------------------------------------------------------------------------------------------------
    | Quinnipiac   | 99.5 |  0.5 |  0.0 |  0.0 | XXXX | XXXX | XXXX | XXXX | XXXX | XXXX | XXXX | ---- |  1.01 |
    | Yale         |  0.5 | 54.3 | 25.7 | 12.0 |  4.8 |  1.9 |  0.7 |  0.2 |  0.0 |  0.0 | XXXX | XXXX |  2.76 |
    | Dartmouth    |  0.0 | 24.7 | 34.6 | 20.8 | 10.9 |  5.3 |  2.5 |  0.9 |  0.3 |  0.0 |  0.0 | XXXX |  3.50 |
    | Union        |  0.0 | 15.1 | 23.8 | 30.9 | 15.9 |  7.7 |  3.9 |  1.8 |  0.7 |  0.2 |  0.1 | XXXX |  4.01 |
    | Colgate      |  0.0 |  2.7 |  6.7 | 12.9 | 19.4 | 19.8 | 16.3 | 11.8 |  6.9 |  2.7 |  0.8 |  0.1 |  6.00 |
    | Princeton    | XXXX |  2.1 |  5.4 | 11.5 | 21.5 | 21.2 | 16.2 | 11.1 |  6.3 |  3.3 |  1.3 |  0.1 |  6.08 |
    | Cornell      | XXXX |  0.4 |  1.6 |  4.6 |  9.3 | 14.0 | 18.2 | 19.8 | 16.3 | 10.0 |  5.2 |  0.6 |  7.51 |
    | St. Lawrence | XXXX |  0.2 |  1.3 |  4.0 |  8.3 | 12.3 | 16.0 | 18.7 | 20.5 | 12.4 |  5.6 |  0.6 |  7.75 |
    | Rensselaer   | XXXX |  0.2 |  0.9 |  2.9 |  8.3 | 13.9 | 17.9 | 19.5 | 17.0 | 11.9 |  6.8 |  0.8 |  7.77 |
    | Brown        | XXXX | XXXX |  0.0 |  0.2 |  0.7 |  1.9 |  4.2 |  8.2 | 16.2 | 31.6 | 31.6 |  5.3 |  9.84 |
    | Clarkson     | XXXX |  0.0 |  0.0 |  0.2 |  0.9 |  1.9 |  4.0 |  7.8 | 14.9 | 25.2 | 37.8 |  7.2 |  9.96 |
    | Harvard      | ---- | XXXX | XXXX | XXXX |  0.0 |  0.0 |  0.1 |  0.2 |  0.7 |  2.8 | 10.9 | 85.3 | 11.80 |
    ------------------------------------------------------------------------------------------------------------
    "XXXX" indicates that the finish is mathematically possible, but did not occur in any of the 51,500 trails. "----" indicates that it is not possible. "0.0" means that it occurred 24 times or fewer.

    First-Round Match-ups:
    Code:
    -------------------------------------------------------------------------------------------------
    |TEAM|  QN  |  YA  |  DA  |  UC  |  CG  |  PN  |  CR  |  SL  |  RP  |  BN  |  CK  |  HA  | HOST |
    -------------------------------------------------------------------------------------------------
    | QN |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |
    | YA |  0.0 |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.1 |  0.4 |  0.2 |  1.1 |  1.4 |  4.3 |  7.5 |
    | DA |  0.0 |  0.0 |  XX  |  0.0 |  0.1 |  0.1 |  0.7 |  0.8 |  0.9 |  3.7 |  3.6 |  9.8 | 19.6 |
    | UC |  0.0 |  0.0 |  0.0 |  XX  |  0.3 |  0.4 |  1.1 |  1.1 |  1.8 |  4.9 |  5.2 | 14.6 | 29.3 |
    | CG |  0.0 |  0.0 |  0.1 |  0.1 |  XX  |  1.3 |  4.7 |  6.0 |  5.7 | 14.9 | 15.2 | 19.3 | 67.2 |
    | PN |  0.0 |  0.0 |  0.0 |  0.1 |  1.1 |  XX  |  4.8 |  5.6 |  6.6 | 15.4 | 15.4 | 21.0 | 70.1 |
    | CR |  0.0 |  0.0 |  0.1 |  0.2 |  2.7 |  2.0 |  XX  | 10.2 |  7.0 | 14.5 | 14.5 | 10.2 | 61.4 |
    | SL |  0.0 |  0.0 |  0.1 |  0.2 |  2.5 |  2.5 |  6.8 |  XX  |  8.3 | 13.6 | 12.1 |  9.2 | 55.3 |
    | RP |  0.0 |  0.0 |  0.0 |  0.1 |  2.2 |  1.7 |  8.1 |  9.7 |  XX  | 14.0 | 14.3 |  9.4 | 59.6 |
    | BN |  0.0 |  0.0 |  0.0 |  0.1 |  0.8 |  1.5 |  3.4 |  2.0 |  3.0 |  XX  |  3.4 |  0.9 | 15.1 |
    | CK |  0.0 |  0.0 |  0.0 |  0.1 |  0.7 |  1.4 |  2.3 |  3.4 |  3.0 |  2.6 |  XX  |  1.1 | 14.6 |
    | HA |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.1 |  0.0 |  XX  |  0.3 |
    -------------------------------------------------------------------------------------------------
    |ROAD|  0.0 |  0.0 |  0.3 |  1.0 | 10.5 | 10.9 | 32.1 | 39.2 | 36.5 | 84.7 | 85.1 | 99.7 |      |
    -------------------------------------------------------------------------------------------------
    Home teams are listed on the left while road teams are listed on top. So, for example, the Cornell hosting SLU matchup occurred in 5,274 of the 51,500 trials while SLU hosted Cornell another 3,491. A number of "0.0" indicates that the match-up did not occur (and I am not going to make any judgements on the mathematical possibility of said matchup). "XX" means a team can't play themselves in the playoffs (duh). The bottom row is a representation of the percentage of times that a team finished between 9th and 12th in the simulation. The far right column is an indicator of how often they finished between 5th and 8th. And, the difference between 100 and the sum of those two numbers is the indicator of how often they placed 1st, 2nd, 3rd, or 4th.

    Wow, I am tired. Well, that's all I have for today. I'll try and provide some more insight as it gets closer to March 2nd. As a reminder, Brian Sullivan is an awesome blogger and is much quicker about getting these things up and out there before I do, I just try and fill in the gaps.
    Last edited by burgie12; 01-31-2013 at 03:27 PM.
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  4. #4
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    I can identify the difference between Mease and B-T in one sheet of paper. These aren't world breaking ideas, folks.

    (Short hand, B-T logit, Mease probit, ... , B-T no penalty, Mease penalty, Mease frequentist and treatments, Rutter Bayesian and random effects).
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  5. #5
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    I've programmed CHODR, or something that is essentially the same thing that I programmed up a couple of years ago. PM me and we can discuss....

  6. #6
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Standings (by Points %):
    Quinnipiac 24-42 [1-10]
    Yale 19-35 [1-12]
    Dartmouth 16-32 [1-12]
    Union 16-32 [1-12]
    St. Lawrence 13-31 [1-12]
    Princeton 13-31 [1-12]
    Colgate 12-30 [1-12]
    Clarkson 11-29 [1-12]
    Rensselaer 11-29 [1-12]
    Brown 11-27 [1-12]
    Cornell 10-28 [1-12]
    Harvard 6-20 [3-12]

    Miscellaneous Links:
    TBRW's ECAC Page (currently down)
    Sioux Sports' What-If Calculator (which doesn't use tiebreakers)
    ECAC Tiebreakers page (which you will come to know and love)

    Dartmouth won the season series 3-1 and have clinched the two-way tiebreaker against Union.

    Princeton and St. Lawrence tied their game at Appleton Arena in November and they each have five wins in thirteen games, so the tiebreaker goes down to "Record vs Top X" values. Both the Saints and the Tigers have earned 0.5 points per game played against a Top 4 team, so it goes to Top 8. Both have earned 6 points, but SLU has played one less game against a Top 8 team, so they own the tiebreaker.

    Clarkson and RPI have not yet faced each other, have four wins in thirteen games, and have earned 0.5 points per game played against a Top 4 team, so this head-to-head tiebreaker also goes to Top 8. Both teams have played nine games against Top 8 teams and Clarkson has earned eight points while Rensselaer has only earned six. Therefore, Clarkson currently owns the tiebreaker, and with it, 8th place.

    Brown is not tied with either Clarkson or Rensselaer because the Bears have played an extra game

    Remaining League Schedules:
    Code:
    ----------------------------------------------------------------------------
    |              |    ||S2/02|F2/08|S2/09|F2/15|S2/16|F2/22|S2/23|F3/01|S3/02|
    ----------------------------------------------------------------------------
    | Quinnipiac   | QN || @YA | @CR | @CG |  SL |  CK |  YA |  BN | @HA | @DA |
    | Yale         | YA ||  QN |     | @BN | @UC | @RP | @QN | @PN |  CG |  CR |
    | Dartmouth    | DA || @RP |     | @HA |  CR |  CG | @CK | @SL |  PN |  QN |
    | Union        | UC ||     | @CK | @SL |  YA |  BN | @CG | @CR |  SL |  CK |
    | St. Lawrence | SL ||  CG |  RP |  UC | @QN | @PN |  HA |  DA | @UC | @RP |
    | Princeton    | PN || @BN | @CG | @CR |  CK |  SL |  BN |  YA | @DA | @HA |
    | Colgate      | CG || @SL |  PN |  QN | @HA | @DA |  UC |  RP | @YA | @BN |
    | Clarkson     | CK ||  CR |  UC |  RP | @PN | @QN |  DA |  HA | @RP | @UC |
    | Rensselaer   | RP ||  DA | @SL | @CK |  BN |  YA | @CR | @CG |  CK |  SL |
    | Brown        | BN ||  PN |     |  YA | @RP | @UC | @PN | @QN |  CR |  CG |
    | Cornell      | CR || @CK |  QN |  PN | @DA | @HA |  RP |  UC | @BN | @YA |
    | Harvard      | HA ||     |     |  DA |  CG |  CR | @SL | @CK |  QN |  PN |
    ----------------------------------------------------------------------------
    Individual Team Limits:
    Teams Losing Out (Floors)
    Quinnipiac can still find themselves down in 10th place if they fall hard enough. Having RPI, Qpac, and Harvard all lose out leaves enough points so that four teams (Cornell, Clarkson, Union, and Princeton... in that order for seeding purposes) can claim the Cleary Cup and the remaining five teams (plus the Bobcats) tie for 5th at 24 points. [Side note, I didn't have RPI actually lose out. They tied Yale on the 16th so that Yale could finish at 24, instead of 25.] Anyways, teams separated themselves from Quinnipiac one-by-one until it was just the Bears and Bobcats duking it out for 9th and Brown obviously won the (hypothetical) head-to-head tiebreaker 3-1.

    Yale can still theoretically finish in 12th place. Harvard can still reach 20, so they can be passed, even by the Crimson.

    Teams Winning Out (Ceilings)
    Harvard can no longer finish in 2nd. Even if they win out and Quinnipiac wins out their other games, it would only take 68 points to get every team up to 20 and there would be 36 games (or 72 points) remaining. So, let Yale win out and those numbers drop to 67 points with 60 remaining. With nine teams that still need to be pushed up to 20, that means two teams would need to tie with the Crimson. Cornell is an obvious choice, having been hypothetically swept, while anyone other than Union gets to play the victim of "that other team." So, giving that role to RPI allows Harvard to win the three-way tiebreaker and finish in 3rd place.

    I didn't redo the thresholds, so I did not include them in the standings.

    Current Mease Rankings (THETA):
    Code:
    -------------------------
    |     Team     |  Theta |
    -------------------------
    | Quinnipiac   |  0.842 |
    | Yale         |  0.592 |
    | Dartmouth    |  0.411 |
    | Union        |  0.241 |
    | St. Lawrence |  0.001 |
    | Princeton    | -0.119 |
    | Colgate      |  0.212 |
    | Clarkson     | -0.394 |
    | Rensselaer   |  0.042 |
    | Brown        | -0.109 |
    | Cornell      |  0.026 |
    | Harvard      | -0.540 |
    -------------------------
    
    PHI = 0.206102
    And, here are the results, organized by expected final standings...
    Mease Simulation (10,000 trials; phi = 0.206102):
    Code:
    ------------------------------------------------------------------------------------------------------------
    |              |  1st |  2nd |  3rd |  4th |  5th |  6th |  7th |  8th |  9th | 10th | 11th | 12th | ExpPl |
    ------------------------------------------------------------------------------------------------------------
    | Quinnipiac   | 98.4 |  1.5 |  0.1 | XXXX | XXXX | XXXX | XXXX | XXXX | XXXX | XXXX | ---- | ---- |  1.02 |
    | Yale         |  1.5 | 68.0 | 19.5 |  8.2 |  2.1 |  0.5 |  0.1 |  0.0 |  0.0 | XXXX | XXXX | XXXX |  2.44 |
    | Dartmouth    |  0.0 | 16.3 | 42.8 | 23.7 |  9.9 |  4.5 |  1.9 |  0.7 |  0.2 |  0.0 |  0.0 | XXXX |  3.53 |
    | Union        |  0.0 | 11.8 | 24.7 | 37.0 | 14.7 |  6.5 |  2.9 |  1.5 |  0.7 |  0.2 |  0.1 | XXXX |  3.98 |
    | Colgate      | XXXX |  1.1 |  4.4 |  9.2 | 18.6 | 19.1 | 18.1 | 14.3 |  9.2 |  4.5 |  1.6 |  0.0 |  6.46 |
    | Princeton    | XXXX |  0.7 |  2.9 |  7.6 | 19.8 | 19.8 | 17.5 | 12.9 |  9.4 |  5.8 |  3.5 |  0.1 |  6.67 |
    | St. Lawrence | XXXX |  0.4 |  4.0 |  9.0 | 16.0 | 18.4 | 17.5 | 15.2 | 10.7 |  6.4 |  2.5 |  0.1 |  6.71 |
    | Rensselaer   | XXXX |  0.2 |  1.2 |  3.4 | 11.8 | 16.7 | 17.5 | 16.7 | 15.4 | 10.4 |  6.6 |  0.1 |  7.49 |
    | Cornell      | XXXX |  0.0 |  0.4 |  1.2 |  4.2 |  7.6 | 12.0 | 16.6 | 19.9 | 20.4 | 16.5 |  1.1 |  8.68 |
    | Brown        | XXXX | XXXX |  0.1 |  0.5 |  1.8 |  3.8 |  7.1 | 12.5 | 19.6 | 27.2 | 26.5 |  1.1 |  9.35 |
    | Clarkson     | XXXX | XXXX |  0.0 |  0.4 |  1.1 |  3.1 |  5.5 |  9.5 | 14.8 | 24.5 | 38.6 |  2.6 |  9.73 |
    | Harvard      | ---- | ---- | XXXX | XXXX | XXXX | XXXX | XXXX |  0.0 |  0.2 |  0.6 |  4.2 | 95.0 | 11.94 |
    ------------------------------------------------------------------------------------------------------------
    "XXXX" indicates that the finish is mathematically possible, but did not occur in any of the 10,000 trails. "----" indicates that it is not possible. "0.0" means that it occurred 9 times or less.

    First-Round Match-ups:
    Code:
    -------------------------------------------------------------------------------------------------
    |TEAM|  QN  |  YA  |  DA  |  UC  |  CG  |  PN  |  SL  |  RP  |  CR  |  BN  |  CK  |  HA  | HOST |
    -------------------------------------------------------------------------------------------------
    | QN |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |
    | YA |  0.0 |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.2 |  0.2 |  0.3 |  2.0 |  2.7 |
    | DA |  0.0 |  0.0 |  XX  |  0.0 |  0.1 |  0.2 |  0.2 |  0.6 |  1.3 |  2.4 |  2.5 |  9.6 | 17.0 |
    | UC |  0.0 |  0.0 |  0.0 |  XX  |  0.3 |  0.7 |  0.3 |  1.4 |  2.0 |  3.1 |  3.4 | 14.4 | 25.5 |
    | CG |  0.0 |  0.0 |  0.1 |  0.0 |  XX  |  3.0 |  3.3 |  5.1 | 10.4 | 14.3 | 15.2 | 18.6 | 70.1 |
    | PN |  0.0 |  0.0 |  0.0 |  0.1 |  2.4 |  XX  |  2.8 |  6.0 | 11.1 | 13.8 | 14.3 | 19.7 | 70.0 |
    | SL |  0.0 |  0.0 |  0.1 |  0.1 |  2.8 |  3.2 |  XX  |  5.8 | 10.5 | 13.9 | 14.6 | 16.2 | 67.1 |
    | RP |  0.0 |  0.0 |  0.1 |  0.1 |  3.0 |  2.8 |  3.8 |  XX  | 12.5 | 13.8 | 14.8 | 12.0 | 62.7 |
    | CR |  0.0 |  0.0 |  0.1 |  0.2 |  3.8 |  3.3 |  4.4 |  5.8 |  XX  |  9.0 |  9.5 |  4.4 | 40.4 |
    | BN |  0.0 |  0.0 |  0.0 |  0.1 |  1.9 |  3.3 |  2.0 |  4.2 |  5.9 |  XX  |  5.9 |  2.0 | 25.2 |
    | CK |  0.0 |  0.0 |  0.0 |  0.2 |  1.1 |  2.5 |  2.8 |  3.5 |  4.1 |  3.9 |  XX  |  1.1 | 19.2 |
    | HA |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  XX  |  0.0 |
    -------------------------------------------------------------------------------------------------
    |ROAD|  0.0 |  0.0 |  0.3 |  0.9 | 15.3 | 18.8 | 19.6 | 32.5 | 58.0 | 74.2 | 80.4 |100.0 |      |
    -------------------------------------------------------------------------------------------------
    Go Red!!

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  7. #7
    RPI '11 Go Red!
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Standings (by Points %):
    --- Bye Lock - 29
    Quinnipiac 26-42 [1-7]
    --- Home Lock - 25+
    Yale 19-33 [1-12]
    Union 16-32 [1-12]
    Dartmouth 16-30 [1-12]
    St. Lawrence 14-30 [1-12]
    Princeton 13-29 [1-12]
    Rensselaer 13-29 [1-12]
    Colgate 13-29 [1-12]
    Clarkson 13-29 [1-12]
    Brown 13-27 [1-12]
    Cornell 10-26 [1-12]
    Harvard 6-20 [3-12]
    --- Bye Eligible - 18+
    --- Home Eligible - 14+

    Union has a game-in-hand against Dartmouth (which will be resolved on Friday), so the Dutchmen and Big Green are not tied.

    My understanding of the tiebreaker rules says that head-to-head results can still be used as long as everyone has played somebody that they're tied against at least once, even if each team vs team matchup has not occurred. So, despite the fact that RPI has not played Clarkson, head-to-head results (on a point per game basis) can still be used.

    So, the tie for 6th is broken as follows: Princeton has earned 5 points in 4 games while Clarkson and RPI each have 3 points in 3 games and Colgate has 3 points in 4 games, so the Tigers "lock up" 6th.

    It then falls to 7th place and restarts. RPI beat Colgate on Jan 18th and Clarkson didn't sweep Colgate, so the Engineers have the highest point-per game result of the bunch and sit in 7th.

    Colgate and Clarkson then battle it out for 8th. Colgate won the season series 3-1 and have therefore locked up the 2-way tiebreaker.

    Brown has played one more game than the other teams with 13 points, so they are not tied for 6th place.

    Miscellaneous Links:
    TBRW's ECAC Page (currently down)
    Sioux Sports' What-If Calculator (which doesn't use tiebreakers)
    ECAC Tiebreakers page (which you will come to know and love)

    Remaining League Schedules:
    Code:
    ----------------------------------------------------------------------
    |              |    ||F2/08|S2/09|F2/15|S2/16|F2/22|S2/23|F3/01|S3/02|
    ----------------------------------------------------------------------
    | Quinnipiac   | QN || @CR | @CG |  SL |  CK |  YA |  BN | @HA | @DA |
    | Yale         | YA ||     | @BN | @UC | @RP | @QN | @PN |  CG |  CR |
    | Union        | UC || @CK | @SL |  YA |  BN | @CG | @CR |  SL |  CK |
    | Dartmouth    | DA ||     | @HA |  CR |  CG | @CK | @SL |  PN |  QN |
    | St. Lawrence | SL ||  RP |  UC | @QN | @PN |  HA |  DA | @UC | @RP |
    | Princeton    | PN || @CG | @CR |  CK |  SL |  BN |  YA | @DA | @HA |
    | Rensselaer   | RP || @SL | @CK |  BN |  YA | @CR | @CG |  CK |  SL |
    | Colgate      | CG ||  PN |  QN | @HA | @DA |  UC |  RP | @YA | @BN |
    | Clarkson     | CK ||  UC |  RP | @PN | @QN |  DA |  HA | @RP | @UC |
    | Brown        | BN ||     |  YA | @RP | @UC | @PN | @QN |  CR |  CG |
    | Cornell      | CR ||  QN |  PN | @DA | @HA |  RP |  UC | @BN | @YA |
    | Harvard      | HA ||     |  DA |  CG |  CR | @SL | @CK |  QN |  PN |
    ----------------------------------------------------------------------
    Individual Team Limits:
    Teams Losing Out (Floors)
    Quinnipiac has clinched themselves home-ice. If they lose out, 124 points would have to be distributed to have every team finish at 26 points with just 76 points remaining. Harvard can't catch the Bobcats. With there still too few points to distribute, Cornell, despite the fact that they can catch Quinnipiac individually, is next on the chopping block. We're still way short (70 points to go, 52 points left). RPI, with the least number of accumulated points, is next out. Down to a 45 / 38 split, but that's still too far. It could be either Princeton or Union next, but I'll go for Union because only one game on their remaining schedule hasn been accounted for so far. Yale is now past Quinnipiac, so they can lose out, too. Now, there are enough points to distribute where every team can reach 26. But, with Quinnipiac having already swept Princeton and hypothetically splitting with nearly everyone else, it's too tough to make them lose the tiebreakers. So, let's leave Princeton losing out. Now, there are enough points remaining so that everyone can reach 27 except for one team. And, we'll let Brown take that hit to tie them with Quinnipiac and win the head-to-head tiebreaker having "won" the "season series" 3-1. And, voila, Quinnipiac's floor is 7th place. Congratulations to the Bobcats on having clinched home-ice.

    Since Yale lost last night and Harvard didn't have the chance to, the Elis can still drop to 12th.

    Teams Winning Out (Ceilings)
    It's the same scenario for Harvard as yesterday, it's just that the numbers have changed.

    Cornell can still get to 26 points, so they can tie Quinnipiac for 1st place. Since the two teams would have split (with Cornell winning this upcoming Friday's matchup) and both teams having 12-8-2 records, it comes down to Records vs Top 4 / 8 to determine the #1 seed. Cornell would want Brown, Colgate, or Dartmouth up there while Quinnipiac would want Princeton or RPI. If the right teams win, then Cornell can still finish in 1st.

    Thresholds:
    Bye Lock - Since there's no clear-cut 5th place team and Union still needs to face SLU twice this season, I figured it would be better to let Yale be the 5th team whose points get boosted. Using Dartmouth, Princeton, Quinnipiac, Union, and Yale, I got every team to have at least 28 points, with Princeton and Yale tied for 3rd at 29, so 29 points is enough for a guaranteed weekend off.

    Bye Eligible - The lack of a clear-cut 5th place team helped the Bye Eligible scenario. Letting Quinnipiac, Union, and Yale win out allowed everyone else to conglomerate around 18 points. In fact, everyone finished at 18 except for one team, who finished at 17. So, 18 and the requisite tiebreakers can still get you into the "weekend off" crowd.

    Home Lock - The obvious first play is to have the teams way behind (Cornell, Harvard) and way ahead (Quinnipiac) of the pack lose out. Next is to have the team that now has the least amount of points (RPI) lose out. So, that determines the makeup of 10th-12th. My spreadsheet then tells me that there are 38 points still to distribute and it would only take 37 points to get every team up to 25 points at the end of the season. Now, it's just a matter of distribution. Turns out, it is possible. I ended up with Quinnipiac and Dartmouth at 26 points each and the seven remaining teams tied at 25 points, with St. Lawrence drawing the short stick and having to go on the road in the first round. So, 25 points and tiebreakers still guarantees you an extra pair of home games.

    Home Eligible - And, this is where the conglomeration around 6th place hurts. Having the top 5 win out is all well and good, but to truly reduce the points necessary for 8th, we need two more teams to win out. Since Princeton and RPI had the least number of games consumed by the previous process, letting them win out is the answer. So, we're now left with Brown, Clarkson, Colgate, Cornell, and Harvard fighting for home-ice. Harvard can only get to 12 points now (with Brown, Colgate, and Clarkson already at 13), so the Crimson can win out and put even more damage to this threshold. Cornell's ceiling is also 12 points now, leaving Brown and Colgate tied at 13 points with their season-ending game remaining. So, assuming they tie, Colgate gets to play Brown again the weekend of March 8th-9th, but this time in Hamilton. That makes the threshold 14 plus tiebreakers. That's right, it is possible for Colgate to earn one tie over the course of the rest of the season and still finish in 8th place.
    Last edited by burgie12; 02-03-2013 at 11:45 AM. Reason: speeling
    Go Red!!

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  8. #8
    RPI '11 Go Red!
    Join Date
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Mease Rankings (THETA):
    Code:
    -------------------------
    |     Team     |  Theta |
    -------------------------
    | Quinnipiac   |  0.891 |
    | Yale         |  0.526 |
    | Union        |  0.239 |
    | Dartmouth    |  0.323 |
    | Rensselaer   |  0.103 |
    | St. Lawrence |  0.018 |
    | Colgate      |  0.214 |
    | Princeton    | -0.176 |
    | Brown        | -0.053 |
    | Clarkson     | -0.320 |
    | Cornell      | -0.104 |
    | Harvard      | -0.535 |
    -------------------------
    
    PHI = 0.207019
    And, here are the results, organized by expected final standings...
    Mease Simulation (12,100 trials; phi = 0.207019):
    Code:
    ------------------------------------------------------------------------------------------------------------
    |              |  1st |  2nd |  3rd |  4th |  5th |  6th |  7th |  8th |  9th | 10th | 11th | 12th | ExpPl |
    ------------------------------------------------------------------------------------------------------------
    | Quinnipiac   |100.0 |  0.0 | XXXX | XXXX | XXXX | XXXX | XXXX | ---- | ---- | ---- | ---- | ---- |  1.00 |
    | Yale         |  0.0 | 66.3 | 20.3 |  8.7 |  3.0 |  1.2 |  0.3 |  0.1 | XXXX |  0.0 | XXXX | XXXX |  2.54 |
    | Union        | XXXX | 20.3 | 30.4 | 25.3 | 12.1 |  6.4 |  2.9 |  1.5 |  0.8 |  0.2 |  0.0 | XXXX |  3.75 |
    | Dartmouth    | XXXX |  9.3 | 30.7 | 27.6 | 15.3 |  9.1 |  4.5 |  2.4 |  0.7 |  0.2 |  0.1 | XXXX |  4.13 |
    | Rensselaer   | XXXX |  1.5 |  4.8 | 10.6 | 20.4 | 18.4 | 15.1 | 11.9 |  8.8 |  6.0 |  2.6 |  0.0 |  6.40 |
    | St. Lawrence | XXXX |  0.9 |  6.1 | 10.3 | 15.9 | 17.5 | 17.1 | 14.4 |  9.4 |  5.9 |  2.4 |  0.0 |  6.51 |
    | Colgate      | XXXX |  1.4 |  5.0 | 10.0 | 15.6 | 18.4 | 17.2 | 14.5 |  9.9 |  5.5 |  2.5 |  0.0 |  6.55 |
    | Princeton    | XXXX |  0.3 |  1.9 |  4.7 | 10.0 | 13.3 | 16.3 | 16.0 | 15.2 | 14.0 |  8.3 |  0.2 |  7.67 |
    | Brown        | XXXX |  0.0 |  0.2 |  1.1 |  3.3 |  7.1 | 11.5 | 17.4 | 22.4 | 20.6 | 16.1 |  0.2 |  8.72 |
    | Clarkson     | XXXX |  0.0 |  0.6 |  1.4 |  3.5 |  6.3 | 10.8 | 14.1 | 19.3 | 24.1 | 19.3 |  0.4 |  8.85 |
    | Cornell      | XXXX |  0.0 |  0.1 |  0.3 |  0.9 |  2.2 |  4.3 |  7.7 | 13.3 | 23.0 | 44.6 |  3.7 |  9.95 |
    | Harvard      | ---- | ---- | XXXX | XXXX | XXXX | XXXX | XXXX |  0.0 |  0.1 |  0.5 |  4.1 | 95.3 | 11.95 |
    ------------------------------------------------------------------------------------------------------------
    "XXXX" indicates that the finish is mathematically possible, but did not occur in any of the 12,100 trails. "----" indicates that it is not possible. "0.0" means that it occurred 6 times or less.

    First-Round Match-ups:
    Code:
    ------------------------------------------------------------------------------------------
    |TEAM|  YA  |  UC  |  DA  |  RP  |  SL  |  CG  |  PN  |  BN  |  CK  |  CR  |  HA  | HOST |
    ------------------------------------------------------------------------------------------
    | QN |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |
    | YA |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.2 |  0.2 |  0.5 |  0.7 |  2.9 |  4.6 |
    | UC |  0.0 |  XX  |  0.0 |  0.7 |  0.3 |  0.4 |  1.6 |  2.0 |  1.8 |  4.1 | 12.0 | 22.9 |
    | DA |  0.0 |  0.0 |  XX  |  0.7 |  0.6 |  0.4 |  1.3 |  3.9 |  3.4 |  6.2 | 14.9 | 31.4 |
    | RP |  0.0 |  0.0 |  0.2 |  XX  |  2.8 |  2.6 |  6.0 |  9.2 | 10.5 | 14.6 | 20.0 | 65.8 |
    | SL |  0.0 |  0.1 |  0.1 |  2.1 |  XX  |  3.8 |  5.7 | 11.8 | 11.0 | 14.1 | 16.2 | 64.9 |
    | CG |  0.0 |  0.1 |  0.1 |  2.0 |  2.3 |  XX  |  6.1 | 12.0 | 12.4 | 15.0 | 15.7 | 65.7 |
    | PN |  0.0 |  0.1 |  0.2 |  3.8 |  3.4 |  3.8 |  XX  | 10.5 | 10.8 | 12.8 | 10.1 | 55.5 |
    | BN |  0.0 |  0.3 |  0.1 |  3.5 |  2.5 |  3.0 |  7.3 |  XX  |  9.7 |  9.3 |  3.7 | 39.3 |
    | CK |  0.0 |  0.3 |  0.2 |  3.2 |  4.2 |  2.2 |  6.8 |  6.5 |  XX  |  7.7 |  3.6 | 34.7 |
    | CR |  0.0 |  0.1 |  0.2 |  1.3 |  1.6 |  1.8 |  2.7 |  3.3 |  3.1 |  XX  |  0.9 | 15.0 |
    | HA |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  XX  |  0.0 |
    ------------------------------------------------------------------------------------------
    |ROAD|  0.0 |  1.1 |  1.0 | 17.4 | 17.7 | 18.0 | 37.7 | 59.4 | 63.2 | 84.5 |100.0 |      |
    ------------------------------------------------------------------------------------------
    Go Red!!

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    Houston Field House, Cheel Arena, Agganis Arena, Magness Arena, Ritter Arena, Messa Rink, Matthews Arena, Von Braun Center, Lynah Rink, Starr Rink, Appleton Arena, Dwyer Arena, Buffalo State Ice Arena, Kelley Rink (also Verizon Center (DC), Herb Brooks Arena, Fenway Park (Frozen Fenway I), Times Union Center, DCU Center, Blue Cross Arena)

  9. #9
    Let's Go 'Tute! Ralph Baer's Avatar
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Burgie,

    How much of an effort would it take to run similar simulations on the women's ECAC side? There has been some discussion on the RPI women's thread http://board.uscho.com/showthread.ph...81#post5622781 about how safe RPI is currently for making the playoffs (8th or better).

  10. #10
    RPI '11 Go Red!
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Standings (by Points %):
    Quinnipiac 30-42 [1-2]
    --- Bye Lock - 28+
    --- Home Lock - 24+
    Yale 19-33 [1-11]
    Princeton 17-29 [2-12]
    Rensselaer 17-29 [2-12]
    Dartmouth 17-29 [2-12]
    St. Lawrence 16-28 [2-12]
    Union 16-28 [2-12]
    Clarkson 15-27 [2-12]
    Brown 13-27 [2-12]
    Colgate 13-25 [2-12]
    Cornell 10-22 [2-12]
    Harvard 7-19 [4-12]
    --- Bye Eligible - 19+
    --- Home Eligible - 15

    Princeton wins the head-to-head matchup for 3rd place with a 2-0-1 record against Rensselaer and Dartmouth.

    RPI wins the tiebreaker for 4th based on Record vs Top 4. They've split their season series and both have identical 7-6-3 records, so it goes to the third tiebreaker. RPI has earned 4 points in 5 games, which bests Dartmouth's 2 points in 4 games. So, the Engineers currently hold the last bye position.

    St. Lawrence beat Union this past Saturday, so they're currently sitting in 6th place. The two teams will meet again on the final weekend to firm up this tiebreaker.

    Brown has a game in hand oveer Colgate (which will be resolved tomorrow), so the Bears sit in 9th place by themselves.

    Miscellaneous Links:
    TBRW's ECAC Page (currently down)
    Sioux Sports' What-If Calculator (which doesn't use tiebreakers)
    ECAC Tiebreakers page (which you will come to know and love)

    Remaining League Schedules:
    Code:
    ----------------------------------------------------------------
    |              |    ||T2/12|F2/15|S2/16|F2/22|S2/23|F3/01|S3/02|
    ----------------------------------------------------------------
    | Quinnipiac   | QN ||     |  SL |  CK |  YA |  BN | @HA | @DA |
    | Yale         | YA || @BN | @UC | @RP | @QN | @PN |  CG |  CR |
    | Union        | UC ||     |  YA |  BN | @CG | @CR |  SL |  CK |
    | Dartmouth    | DA ||     |  CR |  CG | @CK | @SL |  PN |  QN |
    | St. Lawrence | SL ||     | @QN | @PN |  HA |  DA | @UC | @RP |
    | Princeton    | PN ||     |  CK |  SL |  BN |  YA | @DA | @HA |
    | Rensselaer   | RP ||     |  BN |  YA | @CR | @CG |  CK |  SL |
    | Colgate      | CG ||     | @HA | @DA |  UC |  RP | @YA | @BN |
    | Clarkson     | CK ||     | @PN | @QN |  DA |  HA | @RP | @UC |
    | Brown        | BN ||  YA | @RP | @UC | @PN | @QN |  CR |  CG |
    | Cornell      | CR ||     | @DA | @HA |  RP |  UC | @BN | @YA |
    | Harvard      | HA ||     |  CG |  CR | @SL | @CK |  QN |  PN |
    ----------------------------------------------------------------
    Individual Team Limits:
    Teams Losing Out (Floors)
    Quinnipiac has has guaranteed themselves a Top 2 finish. Yale is the only team that can still catch the Bobcats, and even that is a looooooooooooooooooooooooooooong shot.

    With Yale having swept Harvard and the Crimson no longer being able to outright pass the Elis, I can no longer find a way for Yale to finish in 12th place. Harvard just hasn't won enough season series to find a "beneficial third team" to win the head-to-head tiebreaker and force Yale to face some other team in a two-way tiebreaker. I just can't find a way to make it work. The only teams against whom Harvard could end up with a better series record than Yale are Quinnipiac (who can't finish with 19 points), RPI, Colgate and Cornell. And if any or all of the other three teams finish in a tie at 19, then it's either Yale or the teams that I bring in that always work themselves out of that tie more quickly than the Crimson. So, congratulations Coach Allain, you can't finish in 12th this season.

    Any of the remaining ten teams can finish in 12th place by themselves.

    Teams Winning Out (Ceilings)
    If Harvard wins out, they'll reach 19 points. Like I said above, I can't find a way for Harvard to win a tiebreaker that involves Yale, so let Yale and Quinnipiac win out, too. Cornell now has a maximum of 18 points, so we can let the Big Red win out (and we'll turn the loss against Yale into a tie because Harvard swept them, so we want them to end up tied at 19 to boost Harvard's head-to-head scores). Now, there are eight teams that don't have a finished schedule and it would take only 28 points to get everyone to 19 with 32 left to distribute. So, at least one team still has to win out. Princeton, RPI, and Dartmouth are the viable candidates, but since Dartmouth has already won the season series against Harvard, they're the team I'm going to push past everyone else. And, with that, we're now on the correct side of the ledger at 26 points to get everyone up to 19 points and 24 points to distribute. That means that two teams get stuck at 18 and everyone else can move into a tie with Harvard at 19. Union, having swept Harvard is an obvious choice to leave at 18. If RPI is the other team that we leave at 18, then I got Harvard to finish at the heap of the pack, taking 14 points in their twelve games against Brown, Clarkson, Colgate, Cornell, Princeton, and St. Lawrence, more than any other team in that bunch, and enough to finish in 4th place. The Crimson are still bye eligible.

    Yale is the only team that can catch Quinnipiac (as mentioned above).

    The nine remaining teams can all finish by themselves in 2nd place.

    Thresholds:
    Bye Lock - This is a pretty simple scenario, really. Let Dartmouth, Princeton, RPI, and Yale (the Top 5 who actually still need to win), win their games against everyone not just listed. That leaves RPI and Dartmouth at 27 points with one game remaining, Yale at 29 points with two, and Princeton at 25 with two. If we let Yale lose their two remaining games, then Dartmouth and Princeton are tied at 27 points apiece, they tie each other, and the last bye spot can be claimed with 28 points. So, 28 points and the requisite tiebreakers (which Dartmouth clearly do not own at this time) are enough to guarantee yourself a weekend off.

    Bye Eligible - This was pretty well laid out in the Harvard ceiling section. 19 points plus tiebreakers can potentially get you into a bye spot.

    Home Lock - With Colgate, Cornell, Harvard, and Quinnipiac losing out, it would take 34 points to get the eight remaining teams to 25 points and there are 16 games (aka 32 points) left to distribute, so two teams end up shafted at 24 points. One of them gets an extra pair of home games while the other ends up in the 9 seed.

    Home Eligible - If the Bottom 5 (Brown, Clarkson, Colgate, Cornell, and Harvard) lose all of their remaining games against the Top 7, then they all have the same number of points as they do currently, but significantly less games remaining. Clarkson has just their one against Harvard remaining, while the Colgate / Cornell travel pair has to face both Brown and Harvard next weekend and then the final weekend of the season. Since Harvard has seven points and three games remaining, their ceiling is 13 points (less than Clarkson's current 15), so we let them win out. Now, Cornell is still at 10 points with just one game remaining. Let them win their game against Brown and you're left with Colgate and Brown tied for 9th place at 13 points each with just the one game remaining on the composite schedule. If the Bears and Raiders tie, then Clarkson earns exactly zero points over the remainder of the season and still gets to host a playoff series. So, if you can get to 15 points, you are still eligible for a home series.
    Last edited by burgie12; 02-11-2013 at 02:45 PM. Reason: colors
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  11. #11
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Mease Rankings (THETA):
    Code:
    -------------------------
    |     Team     |  Theta |
    -------------------------
    | Quinnipiac   |  0.931 |
    | Yale         |  0.527 |
    | Princeton    | -0.035 |
    | Rensselaer   |  0.192 |
    | Dartmouth    |  0.272 |
    | St. Lawrence |  0.031 |
    | Union        |  0.141 |
    | Clarkson     | -0.289 |
    | Brown        | -0.036 |
    | Colgate      |  0.124 |
    | Cornell      | -0.186 |
    | Harvard      | -0.494 |
    -------------------------
    
    PHI = 0.201928
    Please remember, this is just a simulation. There were only 45,000 trials run with no guarantee of uniqueness between trials and there are over 450 * 10^15 different possibilities.

    And, here are the results, organized by expected final standings...
    Mease Simulation (45,000 trials):
    Code:
    ------------------------------------------------------------------------------------------------------------
    |              |  1st |  2nd |  3rd |  4th |  5th |  6th |  7th |  8th |  9th | 10th | 11th | 12th | ExpPl |
    ------------------------------------------------------------------------------------------------------------
    | Quinnipiac   |100.0 |  0.0 | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- |  1.00 |
    | Yale         |  0.0 | 77.9 | 12.8 |  5.5 |  2.4 |  1.0 |  0.3 |  0.1 |  0.0 |  0.0 | XXXX | ---- |  2.37 |
    | Dartmouth    | ---- |  5.4 | 27.2 | 23.0 | 17.5 | 12.6 |  7.8 |  4.1 |  1.8 |  0.6 |  0.1 | XXXX |  4.58 |
    | Rensselaer   | ---- |  8.9 | 22.0 | 20.7 | 17.7 | 14.1 |  8.5 |  4.7 |  2.4 |  0.9 |  0.1 | XXXX |  4.68 |
    | Princeton    | ---- |  3.7 | 16.4 | 17.1 | 18.1 | 17.1 | 13.1 |  8.2 |  4.6 |  1.7 |  0.1 | XXXX |  5.34 |
    | Union        | ---- |  3.3 | 11.8 | 17.1 | 18.7 | 17.3 | 13.3 |  9.1 |  5.6 |  3.2 |  0.5 | XXXX |  5.63 |
    | St. Lawrence | ---- |  0.6 |  7.6 | 10.7 | 14.2 | 17.4 | 21.0 | 14.6 |  9.6 |  3.8 |  0.5 |  0.0 |  6.36 |
    | Colgate      | ---- |  0.1 |  0.7 |  2.0 |  3.8 |  7.3 | 13.3 | 22.7 | 23.3 | 20.2 |  6.3 |  0.3 |  8.32 |
    | Clarkson     | ---- |  0.1 |  1.0 |  2.2 |  4.3 |  7.1 | 12.0 | 17.7 | 24.7 | 26.3 |  4.8 |  0.1 |  8.39 |
    | Brown        | ---- |  0.1 |  0.6 |  1.7 |  3.3 |  5.9 |  9.9 | 16.7 | 22.7 | 27.5 | 11.2 |  0.3 |  8.71 |
    | Cornell      | ---- | XXXX | XXXX |  0.0 |  0.0 |  0.2 |  0.7 |  2.1 |  5.2 | 14.9 | 64.4 | 12.4 | 10.77 |
    | Harvard      | ---- | ---- | ---- | XXXX | XXXX | XXXX | XXXX |  0.0 |  0.1 |  0.8 | 12.1 | 87.0 | 11.86 |
    ------------------------------------------------------------------------------------------------------------
    "XXXX" indicates that the finish is mathematically possible, but did not occur in any of the 45,000 trails. "----" indicates that it is not possible. "0.0" means that it occurred 22 times or less.

    First-Round Match-ups:
    Code:
    ------------------------------------------------------------------------------------------
    |TEAM|  YA  |  DA  |  RP  |  PN  |  UC  |  SL  |  CG  |  CK  |  BN  |  CR  |  HA  | HOST |
    ------------------------------------------------------------------------------------------
    | YA |  XX  |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.1 |  0.2 |  0.2 |  1.0 |  2.2 |  3.8 |
    | DA |  0.0 |  XX  |  0.2 |  0.2 |  0.7 |  0.5 |  2.9 |  3.8 |  5.3 | 11.8 | 16.7 | 42.0 |
    | RP |  0.0 |  0.2 |  XX  |  0.3 |  0.5 |  0.9 |  4.1 |  4.3 |  5.0 | 12.9 | 16.9 | 45.0 |
    | PN |  0.0 |  0.2 |  0.4 |  XX  |  0.7 |  1.2 |  7.4 |  6.1 |  6.9 | 15.3 | 18.3 | 56.5 |
    | UC |  0.0 |  0.2 |  0.5 |  0.9 |  XX  |  1.3 |  6.5 |  6.9 |  7.8 | 16.0 | 18.3 | 58.4 |
    | SL |  0.0 |  0.3 |  0.2 |  0.8 |  1.2 |  XX  | 10.7 | 10.3 | 12.0 | 16.9 | 14.8 | 67.2 |
    | CG |  0.0 |  0.8 |  0.5 |  0.8 |  1.7 |  2.2 |  XX  | 13.1 | 15.4 |  8.3 |  4.3 | 47.2 |
    | CK |  0.0 |  0.5 |  0.6 |  2.0 |  2.5 |  5.5 |  8.7 |  XX  |  8.5 |  8.1 |  4.6 | 41.0 |
    | BN |  0.0 |  0.3 |  0.8 |  1.4 |  1.8 |  1.9 |  8.8 | 10.5 |  XX  |  6.6 |  3.8 | 35.9 |
    | CR |  0.0 |  0.1 |  0.1 |  0.0 |  0.3 |  0.4 |  1.0 |  0.6 |  0.5 |  XX  |  0.1 |  3.0 |
    | HA |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  XX  |  0.0 |
    ------------------------------------------------------------------------------------------
    |ROAD|  0.0 |  2.5 |  3.4 |  6.4 |  9.4 | 13.9 | 50.0 | 55.7 | 61.8 | 97.0 |100.0 |      |
    ------------------------------------------------------------------------------------------
    The home team is listed on the left and the away team is listed on top. These matchups adds up the number of occurrences of the 5v12, 6v11, 7v10, and 8v9 matchups and lists it all in one block.

    HOST (the far right column) indicates how often a team finished in Spots 5-8. ROAD (bottom row) indicates how often a team finished in Spots 9-12. The difference of 100% and the sum of HOST and ROAD indicates how often a team earned a bye and finished in the Top 4.

    And, here's something that I experimented with in the RPI women's thread... the breakdown of where a team will finish based on how many points they earned over the remainder of the season. This time around, I'm only going to do it for the three teams tied for third. Later in the season, I may expand it.

    Princeton:
    Code:
    ---------------------------------------------------------------------------------------------
    |Pl\Pts| 17  | 18  | 19  | 20  | 21  | 22  | 23  | 24  | 25  | 26  | 27  | 28  | 29  |      |
    ---------------------------------------------------------------------------------------------
    |  2nd |   0 |   0 |   0 |   0 |   0 |   0 |   2 |  21 | 214 | 392 | 599 | 278 | 140 | 1646 |
    |  3rd |   0 |   0 |   0 |   0 |   0 |  16 | 401 |1266 |2759 |1806 | 965 | 136 |  24 | 7313 |
    |  4th |   0 |   0 |   0 |   0 |  21 | 300 |2090 |2711 |2042 | 466 |  69 |   2 |   0 | 7701 |
    |  5th |   0 |   0 |   0 |   6 | 321 |1628 |3690 |1909 | 512 |  60 |   1 |   0 |   0 | 8127 |
    |  6th |   0 |   0 |  11 | 177 |1739 |3050 |2249 | 434 |  49 |   1 |   0 |   0 |   0 | 7710 |
    |  7th |   0 |   6 | 155 |1005 |2830 |1553 | 319 |  24 |   1 |   0 |   0 |   0 |   0 | 5893 |
    |  8th |   2 |  64 | 704 |1387 |1294 | 215 |   7 |   0 |   0 |   0 |   0 |   0 |   0 | 3673 |
    |  9th |  52 | 256 | 933 | 661 | 162 |   9 |   0 |   0 |   0 |   0 |   0 |   0 |   0 | 2073 |
    | 10th | 173 | 211 | 310 |  73 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  771 |
    | 11th |  19 |  13 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   33 |
    | 12th |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |    0 |
    ---------------------------------------------------------------------------------------------
    |      | 246 | 550 |2114 |3309 |6371 |6771 |8758 |6365 |5577 |2725 |1634 | 416 | 164 |      |
    ---------------------------------------------------------------------------------------------
    Rensselaer:
    Code:
    ---------------------------------------------------------------------------------------------
    |Pl\Pts| 17  | 18  | 19  | 20  | 21  | 22  | 23  | 24  | 25  | 26  | 27  | 28  | 29  |      |
    ---------------------------------------------------------------------------------------------
    |  2nd |   0 |   0 |   0 |   0 |   0 |   0 |   1 |  52 | 402 | 890 |1462 | 735 | 475 | 4017 |
    |  3rd |   0 |   0 |   0 |   0 |   0 |  11 | 323 |1459 |3425 |2631 |1753 | 271 |  20 | 9893 |
    |  4th |   0 |   0 |   0 |   0 |  19 | 309 |1882 |3065 |3005 | 872 | 162 |   4 |   0 | 9318 |
    |  5th |   0 |   0 |   0 |   3 | 199 |1295 |3396 |2092 | 885 |  92 |   5 |   0 |   0 | 7967 |
    |  6th |   0 |   0 |   3 | 108 |1096 |2366 |2122 | 548 |  85 |   2 |   0 |   0 |   0 | 6330 |
    |  7th |   0 |   0 |  60 | 512 |1736 |1131 | 377 |  17 |   0 |   0 |   0 |   0 |   0 | 3833 |
    |  8th |   0 |  13 | 269 | 758 | 913 | 167 |  12 |   0 |   0 |   0 |   0 |   0 |   0 | 2132 |
    |  9th |   6 |  87 | 463 | 366 | 146 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   0 | 1072 |
    | 10th |  54 | 106 | 178 |  63 |   1 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  402 |
    | 11th |  14 |  17 |   5 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   36 |
    | 12th |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |    0 |
    ---------------------------------------------------------------------------------------------
    |      |  74 | 223 | 978 |1810 |4110 |5283 |8113 |7233 |7802 |4487 |3382 |1010 | 495 |      |
    ---------------------------------------------------------------------------------------------
    Dartmouth:
    Code:
    ---------------------------------------------------------------------------------------------
    |Pl\Pts| 17  | 18  | 19  | 20  | 21  | 22  | 23  | 24  | 25  | 26  | 27  | 28  | 29  |      |
    ---------------------------------------------------------------------------------------------
    |  2nd |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  25 | 275 | 545 | 885 | 375 | 305 | 2410 |
    |  3rd |   0 |   0 |   0 |   0 |   0 |  19 | 416 |1581 |4083 |3156 |2366 | 477 | 143 |12241 |
    |  4th |   0 |   0 |   0 |   0 |  21 | 284 |2221 |3426 |3245 | 925 | 200 |  15 |   0 |10337 |
    |  5th |   0 |   0 |   0 |   2 | 229 |1384 |3472 |2046 | 691 |  72 |   1 |   0 |   0 | 7897 |
    |  6th |   0 |   0 |   2 |  92 |1108 |2218 |1835 | 348 |  44 |   1 |   0 |   0 |   0 | 5648 |
    |  7th |   0 |   2 |  46 | 481 |1643 |1074 | 243 |  11 |   0 |   0 |   0 |   0 |   0 | 3500 |
    |  8th |   0 |  21 | 267 | 678 | 756 | 129 |  11 |   0 |   0 |   0 |   0 |   0 |   0 | 1862 |
    |  9th |  11 |  77 | 348 | 283 |  83 |   2 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  804 |
    | 10th |  30 |  83 | 117 |  35 |   4 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  269 |
    | 11th |  19 |  10 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   32 |
    | 12th |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |    0 |
    ---------------------------------------------------------------------------------------------
    |      |  60 | 193 | 783 |1571 |3844 |5110 |8198 |7437 |8338 |4699 |3452 | 867 | 448 |      |
    ---------------------------------------------------------------------------------------------
    Last edited by burgie12; 02-11-2013 at 02:32 PM. Reason: re-jiggering the Place \ Points matrices
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  12. #12
    Let's Go 'Tute! Ralph Baer's Avatar
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Any reason that you didn't include 28 or 29 points for the three schools?

  13. #13
    RPI '11 Go Red!
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Quote Originally Posted by Ralph Baer View Post
    Any reason that you didn't include 28 or 29 points for the three schools?
    Because I was still using the RPI women's set of formulas as a basis and that was from 15 to 27 points. I remembered to make it start at 17, but forgot to include 28 and 29. It should be fixed now.

  14. #14
    Let's Go 'Tute! Ralph Baer's Avatar
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Quote Originally Posted by burgie12 View Post
    Because I was still using the RPI women's set of formulas as a basis and that was from 15 to 27 points. I remembered to make it start at 17, but forgot to include 28 and 29. It should be fixed now.
    Looks good. Thanks.

    That was an explanation that I didn't think of. So it was my fault for asking you about the women.

  15. #15
    RPI '11 Go Red!
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Standings (by Points %):
    Quinnipiac 30-42 [1-2]
    --- Bye Lock - 28+
    --- Home Lock - 24+
    Yale 19-31 [1-11]
    Princeton 17-29 [2-12]
    Rensselaer 17-29 [2-12]
    Dartmouth 17-29 [2-12]
    St. Lawrence 16-28 [2-12]
    Union 16-28 [2-12]
    Brown 15-27 [2-12]
    Clarkson 15-27 [2-12]
    Colgate 13-25 [2-12]
    Cornell 10-22 [2-12]
    --- Bye Eligible - 19+
    Harvard 7-19 [5-12]
    --- Home Eligible - 15+

    Princeton wins the head-to-head matchup for 3rd place with a 2-0-1 record against Rensselaer and Dartmouth.

    RPI wins the tiebreaker for 4th based on Record vs Top 4. They've split their season series and both have identical 7-6-3 records, so it goes to the third tiebreaker. RPI has earned 4 points in 5 games, which bests Dartmouth's 2 points in 4 games. So, the Engineers currently hold the last bye position.

    St. Lawrence beat Union this past Saturday, so they're currently sitting in 6th place. The two teams will meet again on the final weekend to firm up this tiebreaker.

    Brown won the season series against Clarkson, so they have moved themselves into 8th place.

    Miscellaneous Links:
    TBRW's ECAC Page (currently down)
    Sioux Sports' What-If Calculator (which doesn't use tiebreakers)
    ECAC Tiebreakers page (which you will come to know and love)

    Remaining League Schedules:
    Code:
    ----------------------------------------------------------
    |              |    ||F2/15|S2/16|F2/22|S2/23|F3/01|S3/02|
    ----------------------------------------------------------
    | Quinnipiac   | QN ||  SL |  CK |  YA |  BN | @HA | @DA |
    | Yale         | YA || @UC | @RP | @QN | @PN |  CG |  CR |
    | Union        | UC ||  YA |  BN | @CG | @CR |  SL |  CK |
    | Dartmouth    | DA ||  CR |  CG | @CK | @SL |  PN |  QN |
    | St. Lawrence | SL || @QN | @PN |  HA |  DA | @UC | @RP |
    | Princeton    | PN ||  CK |  SL |  BN |  YA | @DA | @HA |
    | Rensselaer   | RP ||  BN |  YA | @CR | @CG |  CK |  SL |
    | Colgate      | CG || @HA | @DA |  UC |  RP | @YA | @BN |
    | Clarkson     | CK || @PN | @QN |  DA |  HA | @RP | @UC |
    | Brown        | BN || @RP | @UC | @PN | @QN |  CR |  CG |
    | Cornell      | CR || @DA | @HA |  RP |  UC | @BN | @YA |
    | Harvard      | HA ||  CG |  CR | @SL | @CK |  QN |  PN |
    ----------------------------------------------------------
    Individual Team Limits:
    Teams Losing Out (Floors)
    Quinnipiac has has guaranteed themselves a Top 2 finish. They only need one more point to guarantee themselves a share of the Cleary Cup and just two more to lock themselves into the #1 seed. A magic number of two with three weekends to play. Wow.

    Yale still can't be passed or lose a tiebreaker to Harvard. They still have a floor of 11th.

    Any of the remaining ten teams can finish in 12th place by themselves.

    Teams Winning Out (Ceilings)
    If Harvard wins out, they'll reach 19 points. Let Yale and Quinnipiac win out, too, since Harvard can't beat / catch them. Cornell now has a maximum of 18 points, so we can let the Big Red win out (and we'll turn the loss against Yale into a tie because Harvard swept the Big Red, so we want them to end up tied at 19 to boost Harvard's head-to-head scores). Now, there are eight teams that don't have a finished schedule and it would take only 26 points to get everyone to 19 with 32 left to distribute. So, at least one team still has to win out. Princeton, RPI, and Dartmouth are the viable candidates, but since Dartmouth has already won the season series against Harvard, they're the team I'm going to push past everyone else. We're now dead even. Since Brown beat Yale last night, it would take only 24 points to get everyone to a tie at 19 points and there are 24 points left to distribute. If everyone ends up in a bunch at 19 (with Quinnipiac, Yale, and Dartmouth above the fray), then Harvard comes out in the middle of the pack somewhere (8th, to be specific). (Incidentally, if we had let Cornell lose to Yale instead of tying, then Harvard would come out of the massive tiebreaker in 5th place instead). It's better (from Harvard's perspective), to let one more team (Rensselaer) win out and then try again to keep everyone below 19 points. Now, we can do it with just one team tying Harvard. And, we have a wide variety of teams to choose for this task. I chose SLU because of their large number of ties, allowing Harvard to finish in 5th on the basis of ECAC wins. TL;DR? Harvard can finish no higher than 5th place.

    Yale is the only team that can catch Quinnipiac (as mentioned above).

    The nine remaining teams can all finish by themselves in 2nd place.

    Thresholds:
    Bye Lock - This is a pretty simple scenario, really. Let Dartmouth, Princeton, RPI, and Yale (the Top 5 who actually still need to win), win their games against everyone not just listed. Then, let Princeton beat Yale and have the remaining two games (Yale-RPI and Princeton-Dartmouth) end in ties. Quinnipiac wins the Cleary Cup and the 2nd through 5th seeds finish with 28 points each. So, 28 plus tiebreakers is the scenario here.

    Bye Eligible - I was able to get a 7-way tie for 4th place at 19 points (Princeton, Quinnipiac, and Yale above and Cornell and Harvard below). So, if your favorite team can still get to 19 points and win some tiebreakers, they can still earn themselves a weekend off.

    Home Lock - With Colgate, Cornell, Harvard, and Quinnipiac losing out, it would take 32 points to get the eight remaining teams to 25 points and there are 15 games (aka 30 points) left to distribute, so two teams end up shafted at 24 points. One of them gets an extra pair of home games while the other ends up in the 9 seed.

    Home Eligible - If the Bottom 5 (Brown, Clarkson, Colgate, Cornell, and Harvard) lose all of their remaining games against the Top 7, then Harvard can then win out without fear of overtaking 8th place. Then, it's just Cornell and Colgate at Brown to round out the schedule. If the Bears lose both of their games at home that final weekend, then it becomes a three-way tie for 8th place between Brown, Clarkson, and Colgate. The Raiders would have swept Brown and beaten Clarkson 3 points to 1 on the season series, so they would get the extra pair of home games.
    Go Red!!

    National Champions: 1954, 1985, 201x

    Houston Field House, Cheel Arena, Agganis Arena, Magness Arena, Ritter Arena, Messa Rink, Matthews Arena, Von Braun Center, Lynah Rink, Starr Rink, Appleton Arena, Dwyer Arena, Buffalo State Ice Arena, Kelley Rink (also Verizon Center (DC), Herb Brooks Arena, Fenway Park (Frozen Fenway I), Times Union Center, DCU Center, Blue Cross Arena)

  16. #16
    RPI '11 Go Red!
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Current Mease Rankings (THETA):
    Code:
    -------------------------
    |     Team     |  Theta |
    -------------------------
    | Quinnipiac   |  0.939 |
    | Yale         |  0.459 |
    | Princeton    | -0.023 |
    | Rensselaer   |  0.202 |
    | Dartmouth    |  0.284 |
    | St. Lawrence |  0.037 |
    | Union        |  0.153 |
    | Brown        |  0.056 |
    | Clarkson     | -0.279 |
    | Colgate      |  0.130 |
    | Cornell      | -0.175 |
    | Harvard      | -0.389 |
    -------------------------
    
    PHI = 0.200059
    And, here are the results, organized by expected final standings...
    Mease Simulation (35,000 trials):
    Code:
    ------------------------------------------------------------------------------------------------------------
    |              |  1st |  2nd |  3rd |  4th |  5th |  6th |  7th |  8th |  9th | 10th | 11th | 12th | ExpPl |
    ------------------------------------------------------------------------------------------------------------
    | Quinnipiac   |100.0 | XXXX | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- |  1.00 |
    | Yale         | XXXX | 58.3 | 19.8 | 10.5 |  6.1 |  3.2 |  1.3 |  0.5 |  0.1 |  0.0 | XXXX | ---- |  2.83 |
    | Dartmouth    | ---- | 11.5 | 24.8 | 21.2 | 16.1 | 11.7 |  7.2 |  4.4 |  2.1 |  0.9 |  0.1 | XXXX |  4.47 |
    | Rensselaer   | ---- | 14.9 | 19.2 | 18.3 | 15.9 | 12.9 |  9.0 |  5.6 |  2.8 |  1.2 |  0.2 | XXXX |  4.64 |
    | Princeton    | ---- |  6.6 | 13.9 | 15.5 | 16.5 | 16.0 | 13.3 |  9.5 |  6.3 |  2.4 |  0.0 | XXXX |  5.45 |
    | Union        | ---- |  6.3 | 11.4 | 15.6 | 16.9 | 16.0 | 13.0 |  9.7 |  6.5 |  3.9 |  0.6 | XXXX |  5.63 |
    | St. Lawrence | ---- |  1.5 |  6.7 |  9.8 | 13.0 | 16.4 | 18.6 | 16.4 | 11.3 |  5.8 |  0.6 |  0.0 |  6.53 |
    | Brown        | ---- |  0.5 |  2.5 |  5.2 |  8.4 | 11.6 | 16.0 | 19.8 | 19.1 | 13.6 |  3.2 |  0.0 |  7.55 |
    | Colgate      | ---- |  0.2 |  0.9 |  1.9 |  3.7 |  6.1 | 11.2 | 17.6 | 24.0 | 26.1 |  7.7 |  0.5 |  8.54 |
    | Clarkson     | ---- |  0.1 |  0.8 |  2.0 |  3.4 |  5.9 |  9.8 | 15.1 | 23.3 | 33.2 |  6.2 |  0.1 |  8.64 |
    | Cornell      | ---- | XXXX | XXXX |  0.0 |  0.1 |  0.1 |  0.5 |  1.4 |  4.2 | 11.8 | 65.6 | 16.3 | 10.89 |
    | Harvard      | ---- | ---- | ---- | ---- | XXXX | XXXX | XXXX |  0.0 |  0.2 |  1.1 | 15.7 | 83.0 | 11.82 |
    ------------------------------------------------------------------------------------------------------------
    "XXXX" indicates that the finish is mathematically possible, but did not occur in any of the 35,000 trails. "----" indicates that it is not possible. "0.0" means that it occurred 17 times or less.

    First-Round Match-ups:
    Code:
    ------------------------------------------------------------------------------------------
    |TEAM|  YA  |  DA  |  RP  |  PN  |  UC  |  SL  |  BN  |  CG  |  CK  |  CR  |  HA  | HOST |
    ------------------------------------------------------------------------------------------
    | YA |  XX  |  0.1 |  0.0 |  0.0 |  0.0 |  0.2 |  0.4 |  0.6 |  0.9 |  3.4 |  5.5 | 11.2 |
    | DA |  0.0 |  XX  |  0.3 |  0.3 |  0.9 |  0.6 |  3.0 |  3.7 |  4.2 | 11.4 | 15.0 | 39.4 |
    | RP |  0.0 |  0.2 |  XX  |  0.5 |  0.6 |  1.3 |  2.4 |  5.3 |  5.4 | 12.4 | 15.3 | 43.4 |
    | PN |  0.0 |  0.2 |  0.5 |  XX  |  1.0 |  1.7 |  3.3 |  8.9 |  8.0 | 14.7 | 17.0 | 55.3 |
    | UC |  0.0 |  0.2 |  0.6 |  1.4 |  XX  |  1.7 |  4.1 |  7.7 |  8.0 | 15.4 | 16.5 | 55.6 |
    | SL |  0.0 |  0.5 |  0.4 |  1.1 |  1.6 |  XX  |  7.1 | 12.1 | 11.8 | 16.0 | 13.7 | 64.3 |
    | BN |  0.0 |  0.4 |  0.9 |  1.8 |  2.2 |  3.0 |  XX  | 12.7 | 14.0 | 11.8 |  8.9 | 55.8 |
    | CG |  0.1 |  0.9 |  0.7 |  1.0 |  1.9 |  2.7 | 10.8 |  XX  | 10.1 |  6.5 |  4.1 | 38.6 |
    | CK |  0.0 |  0.6 |  0.7 |  2.6 |  2.6 |  6.1 |  4.5 |  6.9 |  XX  |  6.4 |  3.9 | 34.3 |
    | CR |  0.0 |  0.1 |  0.0 |  0.0 |  0.2 |  0.4 |  0.4 |  0.5 |  0.4 |  XX  |  0.1 |  2.1 |
    | HA |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  0.0 |  XX  |  0.0 |
    ------------------------------------------------------------------------------------------
    |ROAD|  0.2 |  3.1 |  4.2 |  8.7 | 11.1 | 17.7 | 36.0 | 58.4 | 62.9 | 97.9 |100.0 |      |
    ------------------------------------------------------------------------------------------
    The home team is listed on the left and the away team is listed on top. These matchups adds up the number of occurrences of the 5v12, 6v11, 7v10, and 8v9 matchups and lists it all in one block.

    HOST (the far right column) indicates how often a team finished in Spots 5-8. ROAD (bottom row) indicates how often a team finished in Spots 9-12. The difference of 100 and the sum of HOST and ROAD indicates how often a team earned a bye and finished in the Top 4.
    Go Red!!

    National Champions: 1954, 1985, 201x

    Houston Field House, Cheel Arena, Agganis Arena, Magness Arena, Ritter Arena, Messa Rink, Matthews Arena, Von Braun Center, Lynah Rink, Starr Rink, Appleton Arena, Dwyer Arena, Buffalo State Ice Arena, Kelley Rink (also Verizon Center (DC), Herb Brooks Arena, Fenway Park (Frozen Fenway I), Times Union Center, DCU Center, Blue Cross Arena)

  17. #17
    Registered User
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    I hope you derive a lot of enjoyment and personal satisfaction from performing these calculations....reminds me of quantum mechanics class in college. you calculate all these probabilities, and then "boom" they all crystallize into one and only one actuality.
    "Hope is a good thing; maybe the best of things."

    "While I thought it was the light at the end of the tunnel, in reality it was the headlight of an oncoming train."

    "Beer is a sign that God loves us and wants us to be happy." -- Benjamin Franklin

  18. #18
    I Still bleed RPI Red
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    My head is spinning. I think i will just wait and see how it all turns out after 6 more games.
    Take the shortest distance to the puck and arrive in ill humor

  19. #19
    The Force is with Brown hockey.
    Join Date
    Feb 2010
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    265

    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Quote Originally Posted by DrDemento View Post
    My head is spinning. I think i will just wait and see how it all turns out after 6 more games.

    I know, right? I am suffering from GEEK OVERLOAD. But I respect the enthusiasm and effort.

  20. #20
    I Still bleed RPI Red
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    Re: ECAC Byes and Home-Ice - A Mathematical Journey 2012-13

    Quote Originally Posted by DarthBruno View Post
    I know, right? I am suffering from GEEK OVERLOAD. But I respect the enthusiasm and effort.
    If you follow most of the postings on any of the RPI threads you will discover that we have an incredibly intelligent group of posters who have entirely too much time on their hands.
    Take the shortest distance to the puck and arrive in ill humor

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