Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

[pokechecker]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

We're using our Pairwise, which is correct, instead of USCHO's, which is incorrect.

never use USCHO stats unless you have no other source for the stats you need
and don't bet your paycheck on them

I'm pretty confident that both Clarkson and Colgate are destined for the top four, unless something unexpected happens (like OSU beating Wisconsin three times in a row). Those two ECAC teams are comfortably ahead of the rest of the pack.

yes, and OSU seems pretty much a lock on #5

forgotten in all of this is the very real possibility of an upset in the conference tourneys
I could see 3 teams other than Wisco winning WCHA
I could see any of the also-rans upsetting BC
and there are 3-4 teams that could get hot at the right time other than Clarkson or Colgate in the ECAC

 

[TonyTheTiger20]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

So for a while I've wondered to myself if there was an easy way to rank teams in order of who gives you the biggest PWR boost for beating them. Not as a way of ranking teams -- but just to see how much it's worth to beat each team. It isn't just "beating #1 is best," because who is "best" involves how good your opponent's winning % is and how good your opponent's opponent's winning % is -- not your opponent's final ranking.

Anyway I should have figured this out sooner because it's actually a simple formula. Since RPI is calculated as 30% Win%, 24% OppWin%, and 46% OppOppWin%, you can calculate how much each team will affect your RPI by doing:

1 (since you would have won that game, so your winning percentage in that game is 1.000) times 0.3
plus that team's winning percentage times 0.24 (since that team's Win% is now your OppWin%)
plus that team's opponent's winning percentage times 0.46 (since that team's OppWin% is now your OppOppWin%

Do that for all 40 teams, and voila --

<img src="https://i.imgur.com/kaqRlgV.png">

The colored numbers at the right are each team's ranking relative to their PWR position.

 

[Quiet Riot]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

So for a while I've wondered to myself if there was an easy way to rank teams in order of who gives you the biggest PWR boost for beating them. Not as a way of ranking teams -- but just to see how much it's worth to beat each team. It isn't just "beating #1 is best," because who is "best" involves how good your opponent's winning % is and how good your opponent's opponent's winning % is -- not your opponent's final ranking.

Anyway I should have figured this out sooner because it's actually a simple formula. Since RPI is calculated as 30% Win%, 24% OppWin%, and 46% OppOppWin%, you can calculate how much each team will affect your RPI by doing:

1 (since you would have won that game, so your winning percentage in that game is 1.000) times 0.3
plus that team's winning percentage times 0.24 (since that team's Win% is now your OppWin%)
plus that team's opponent's winning percentage times 0.46 (since that team's OppWin% is now your OppOppWin%

Do that for all 40 teams, and voila --

<img src="https://i.imgur.com/kaqRlgV.png">

The colored numbers at the right are each team's ranking relative to their PWR position.

Why is anyone wasting any kind of spots on St A's. Because they beat Post 21-0? There is no way they are going to be placed in the NCAA D1 tournament.

 

[TonyTheTiger20]

Why is anyone wasting any kind of spots on St A's. Because they beat Post 21-0? There is no way they are going to be placed in the NCAA D1 tournament.

They are not -- but whether they're selected or not, we do know from the committee chair that games against them do count. That's all this list is! It's not a ranking, but a list of how much beating each team helps <I>your</I> ranking

 

[robertearle]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

Do that for all 40 teams, and voila --

So this afternoon, there is one and only one game (an interesting circumstance that let's you see how much or little a game can affect teams in isolation, and "third-party" teams): St Cloud and Wisconsin. If we assume Wisconsin wins, does that means we should see Wisconsin's RPI go up be something like the St Cloud value (.6237) divided by the number of games UW has played (28), or about 0.0223? Because that seems like a lot for a win over St Cloud (sorry, SCS fans).

 

[Eeyore]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

So this afternoon, there is one and only one game (an interesting circumstance that let's you see how much or little a game can affect teams in isolation, and "third-party" teams): St Cloud and Wisconsin. If we assume Wisconsin wins, does that means we should see Wisconsin's RPI go up be something like the St Cloud value (.6237) divided by the number of games UW has played (28), or about 0.0223? Because that seems like a lot for a win over St Cloud (sorry, SCS fans).

No. I'd have to play around with some numbers to be sure, and I won't have time for that until this evening, but I'm pretty sure that your RPI will increase by:

(Above calculated value for your opponent - Your RPI before the game) / Number of games you've played

So, if your RPI is higher than your opponent's value, that win would drop your RPI and it gets tossed out.

 

[robertearle]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

No. I'd have to play around with some numbers to be sure, and I won't have time for that until this evening, but I'm pretty sure that your RPI will increase by:

(Above calculated value for your opponent - Your RPI before the game) / Number of games you've played

So, if your RPI is higher than your opponent's value, that win would drop your RPI and it gets tossed out.

The RPI page shows both the 'unadjusted' and 'adjusted' values. So, we'll see the unadjusted go down a bit for UW, but the adjusted not change? This is going to be interesting...

 

[TonyTheTiger20]

No. I'd have to play around with some numbers to be sure, and I won't have time for that until this evening, but I'm pretty sure that your RPI will increase by:

(Above calculated value for your opponent - Your RPI before the game) / Number of games you've played

So, if your RPI is higher than your opponent's value, that win would drop your RPI and it gets tossed out.

The RPI page shows both the 'unadjusted' and 'adjusted' values. So, we'll see the unadjusted go down a bit for UW, but the adjusted not change? This is going to be interesting...

Yep, that's correct, (both of you). Also one other caveat is it's not exact, because "OppWin%" is actually your opponent's winning percentage in games not against you -- so for the UW vs. SCSU example, you'd subtract out the prior UW vs. SCSU games from the winning percentage before doing the calculation. But the difference at this point in the season is very small.

 

[TonyTheTiger20]

No. I'd have to play around with some numbers to be sure, and I won't have time for that until this evening, but I'm pretty sure that your RPI will increase by:

(Above calculated value for your opponent - Your RPI before the game) / Number of games you've played

So, if your RPI is higher than your opponent's value, that win would drop your RPI and it gets tossed out.

Put another way -- Wisconsin's new (unadjusted) RPI would be:

{[UW's unadjusted RPI times (games played minus 1)]
Plus the RPI above for beating SCSU}
All divided by number of games played.

I'm on my phone multitasking so if that's wrong then oops lol... But I'm pretty sure that's right.

 

[robertearle]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

Put another way -- Wisconsin's new (unadjusted) RPI would be:

{[UW's unadjusted RPI times (games played minus 1)]
Plus the RPI above for beating SCSU}
All divided by number of games played.

I'm on my phone multitasking so if that's wrong then oops lol... But I'm pretty sure that's right.

What about the other way? How much is losing to Wisconsin worth to St Cloud? Does the gain of 'opponents' and 'opponents-opponents' outweigh the hit to their own won-loss? Similar table?

 

[TonyTheTiger20]

What about the other way? How much is losing to Wisconsin worth to St Cloud? Does the gain of 'opponents' and 'opponents-opponents' outweigh the hit to their own won-loss? Similar table?

Same calculation but different RPI value --

So losing to Wisconsin would be worth

0 (winning percentage in a loss) times 30%
plus .9375 (UW Win% taking out the SCSU games) times 24%
plus .4761 (UW OppWin%) times 46%

So that game RPI would be .4440

St. Cloud's unadjusted RPI is .4601. So their new RPI would be:

.4601 times (26-1=) 25
plus .4440
= 11.9465

Divided by 26 = .4595

<B>Final answer: With a loss today, St. Cloud's RPI should be something like .4595, down from their current .4601.</b>

Can you tell I was a math major???

This won't be exact, because after playing this game, Wisconsin's OppWin% is going to change (and everyone's else's factors are going to change, which will change everyone else's game RPIs... etc). But close enough for back of the envelope guesstimation. That's why we need a big ol' calculator to do RPI lol

EDIT: Checking my work modifying results on the BCI Pairwise calculator ( <URL>https://www.bcinterruption.com/boston-college-bc-eagles-womens-hockey/2017/11/4/16605602/2017-2018-ncaa-womens-hockey-pairwise-rankings-calculator</URL> ) looks like it'll be exactly .4586, which gives you an idea of how much peripheral results can affect things. Most teams' RPI is going to change today even though only UW and SCSU are playing.

 

[Eeyore]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

This won't be exact, because after playing this game, Wisconsin's OppWin% is going to change (and everyone's else's factors are going to change, which will change everyone else's game RPIs... etc). But close enough.

Will it? I know that the games you play against a team are dropped from their Win% for the calculation of your RPI. Based upon that, I had assumed that the games your opponent has played against their opponents (as well as the games you have played against their opponents) would be dropped from the calculation of your OppOppWin% component. That would seem like the consistent, and logical, way to do it, but I've never checked to see if it is. Done right, no game should affect more than one component of a team's RPI rating.

Then again, no sentence that starts, "If RPI were done right . . ." ends in anything but tears.

 

[TonyTheTiger20]

Based upon that, I had assumed that the games your opponent has played against their opponents (as well as the games you have played against their opponents) would be dropped from the calculation of your OppOppWin% component. That would seem like the consistent, and logical, way to do it, but I've never checked to see if it is.

It's actually not -- games are dropped from OppWin%, but not OppOppWin%.

From the RPI wiki:

The OWP is calculated by taking the average of the WP's for each of the team's opponents with the requirement that all games against the team in question are removed from the calculation. Continuing from the example above, assume Syracuse has played one other game and lost, while Cincinnati has played two other teams and won. The team in question has played Syracuse twice and therefore Syracuse must be counted twice. Thus the OWP of the team is (0/1 + 0/1 + 2/2) / 3 (number of opponents – Syracuse, Syracuse, Cincinnati). OWP = 0.3333

The OOWP is calculated by taking the average of each Opponent's OWP. Note that the team in question is part of the team's OOWP. In fact, the most re-occurring opponent of your opponents is the team in question.

 

[Eeyore]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

It's actually not -- games are dropped from OppWin%, but not OppOppWin%.

facepalm

 

[TonyTheTiger20]

facepalm

Hahaha

 

[Eeyore]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

So, does OppOppWin% include the games that opponent has played against its opponents, or is it the average of your opponents' OppWin% as calculated for the second component of their own RPI?

 

[robertearle]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

The RPI page shows both the 'unadjusted' and 'adjusted' values. So, we'll see the unadjusted go down a bit for UW, but the adjusted not change? This is going to be interesting...

Wisconsin's unadjusted went from .6417 to .6421, adjusted went from .6655 to .6650. (I had written down their 'before' numbers this AM)

(St Cloud from .4601 to .4586)

 

[vicb]

Re: Fun With Numbers 2018: Pairwise, KRACH, GRaNT, and other mathematical excitement

.......
(St Cloud from .4601 to .4586)

.......
<B>Final answer: With a loss today, St. Cloud's RPI should be something like .4595, down from their current .4601.</b>

Can you tell I was a math major???

........

.4586 is almost something like .4595

 

[TonyTheTiger20]

So, does OppOppWin% include the games that opponent has played against its opponents, or is it the average of your opponents' OppWin% as calculated for the second component of their own RPI?

It's the latter -- this is great, I went through the exact mental exercise you're going through a few years ago when I first built my calculator lol

 

[TonyTheTiger20]

.4586 is almost something like .4595

Well my math check using the RPI calculator was spot on at least!