a suggestion for the nfl's head-to-head...

A suggestion for the NFL’s head-to-head tiebreaker

James A. Swenson Dan SwensonUW-Platteville Black Hills State University

[email protected] [email protected]

23 April 2016

Swenson/Swenson (UWP/BHSU) NFL Tiebreakers 23 April 2016 1 / 19

Welcome!

Thanks for coming!

I hope you’ll enjoy the talk; please feel free to get involved!


Outline

1 What the NFL does now

2 What the NFL should do instead


The NFL playoffs

What are we doing?

There are 32 teams in the National Football League.

There are two conferences (AFC/NFC) of 16 teams each.

Each conference is divided into four divisions(North/South/East/West) of 4 teams each.

Each team plays a regular season of 16 games, including two gamesagainst each team in its division.After this, six teams from each conference compete in a playoff, endingwith the Super Bowl.In each conference, the four division winners and two “wild card” teamsqualify for the playoffs.

Images: http://www.nationalchamps.net/Helmet_Project/nfl.htm


http://www.nationalchamps.net/Helmet_Project/nfl.htm

The NFL playoffs

What are we doing?




Each team plays a regular season of 16 games, including two gamesagainst each team in its division.

After this, six teams from each conference compete in a playoff, endingwith the Super Bowl.In each conference, the four division winners and two “wild card” teamsqualify for the playoffs.




The NFL playoffs

What are we doing?




Each team plays a regular season of 16 games, including two gamesagainst each team in its division.After this, six teams from each conference compete in a playoff, endingwith the Super Bowl.

In each conference, the four division winners and two “wild card” teamsqualify for the playoffs.




The NFL playoffs

What are we doing?




Each team plays a regular season of 16 games, including two gamesagainst each team in its division.After this, six teams from each conference compete in a playoff, endingwith the Super Bowl.In each conference, the four division winners and two “wild card” teamsqualify for the playoffs.




The NFL Playoffs

Image: http://nittanysportshuddle.com/2016/01/smittys-picks-nfl-playoff-week-1/


http://nittanysportshuddle.com/2016/01/smittys-picks-nfl-playoff-week-1/

NFL procedures

Breaking ties between 2 teams

Teams in same division:1 Head-to-head (best won-lost-tied percentage

in games between the clubs).

2 Best won-lost-tied percentage in gamesplayed within the division.

3 Best won-lost-tied percentage in commongames.

4 Best won-lost-tied percentage in gamesplayed within the conference.

5 Strength of victory.

6 Strength of schedule.

7 Best combined ranking among conferenceteams in points scored and points allowed.

8 Best combined ranking among all teams inpoints scored and points allowed.

9 Best net points in common games.

10 Best net points in all games.

11 Best net touchdowns in all games.

12 Coin toss

Teams in different divisions:1 Head-to-head, if applicable.


3 Best won-lost-tied percentage in commongames, minimum of four.





8 Best net points in conference games.



11 Coin toss.

“NFL Tie-Breaking Procedures.” http://www.nfl.com/standings/tiebreakingprocedures


http://www.nfl.com/standings/tiebreakingprocedures

NFL procedures

Breaking ties among 3+ teams

Teams in same division:1 Head-to-head (best won-lost-tied percentage

in games among the clubs).

2 Best won-lost-tied percentage in gamesplayed within the division.

3 Best won-lost-tied percentage in commongames.






9 Best net points in common games.



12 Coin toss

Teams in different divisions:1 Apply division tie breaker to eliminate all but

the highest ranked club in each division.Then...

2 Head-to-head sweep. (Applicable only if oneclub has defeated each of the others or if oneclub has lost to each of the others.)


4 Best won-lost-tied percentage in commongames, minimum of four.





9 Best net points in conference games.



12 Coin toss

(After finding a rule that eliminates a team, go back to the beginning.)


Example: NFC, 2014

Who’s #1?

Week 1: Seattle 36, Green Bay 16.

Week 6: Dallas 30, Seattle 23.

Dallas vs. Green Bay: no regular-season game.

Image: http://espn.go.com/nfl/standings/_/season/2014/


http://espn.go.com/nfl/standings/_/season/2014/

Example: NFC, 2014

Who’s #1?

Week 1: SEA 36, GB 16.

Week 6: DAL 30, SEA 23.

DAL vs. GB: no game.

Breaking ties among 3+teams



2 Head-to-head sweep. (Applicable only if oneclub has defeated each of the others or if oneclub has lost to each of the others.)


4 . . .

#1: SEA. #2: GB. #3: DAL.


2014 NFC Playoffs: What happened next?

First weekend:

CAR won at home against ARI,whose top 2 QBs were injured.

DAL had to play DET, while GBand SEA got a week off.

Second weekend:

SEA won a home game againstCAR, a team with a losing record.

GB got a controversial win athome against DAL.

NFC Championship:

SEA scored two TDs in the last2:09, then beat GB in OT.

Image: CBS Sports, via http://www.interbasket.net/news/16746/2014/12/print-nfl-bracket-2015-15-wildcards/


http://www.interbasket.net/news/16746/2014/12/print-nfl-bracket-2015-15-wildcards/

Outline

1 What the NFL does now

2 What the NFL should do instead


Modeling the regular season

Definition

Let x and y be teams. We say x ⊳ y ⇐⇒ x beat or tied y at least once.

Definition

The transitive closure of a relation R is the smallest transitive relationthat contains R.

Definition

We write ▸ for the transitive closure of ⊳. Thus x ▸ y when there is asequence of teams x = v0 ⊳ v1 ⊳ . . . ⊳ vn = y .


Team ordering

Definition

Let x and y be teams. We say x > y ⇐⇒ (x ▸ y) and not (y ▸ x).

Lemma

The relation > is antisymmetric: If x > y , then y /> x .

Lemma

The relation > is transitive: If x > y > z , then x > z .

Proof.

Suppose x > y and y > z . Then x ▸ y ▸ z , so x ▸ z .

Sftsoc that z ▸ x . Also, x ▸ y , so z ▸ y . But then y /> z . ⇒⇐

Theorem

> is a partial order.


Team ordering

Definition

Let x and y be teams. We say x > y ⇐⇒ (x ▸ y) and not (y ▸ x).

Lemma

The relation > is antisymmetric: If x > y , then y /> x .

Lemma

The relation > is transitive: If x > y > z , then x > z .

Proof.

Suppose x > y and y > z . Then x ▸ y ▸ z , so x ▸ z .Sftsoc that z ▸ x . Also, x ▸ y , so z ▸ y . But then y /> z . ⇒⇐

Theorem

> is a partial order.


Tiebreaker proposal

Idea

Let X be a set of teams. Suppose X =W ∪ L, where:

W ≠ ∅;

W ∩ L = ∅;

∀x ∈W ,∀y ∈ L, x > y .

Then the league should rank teams in W above teams in L.

What if there’s a choice?

For example, suppose x > y , x > z , and y > z . We could take W = {x} orW = {x , y}. We should pick W = {x}, eliminating both y and z . Ingeneral, we ought to take the smallest possible W .


“The smallest possible W ” is well-defined!

Theorem

Let X be a set, and let > be a binary relation on X such that[x > y ⇒ y /> x] for all x , y ∈ X .Suppose {W1,L1} and {W2,L2} are partitions of X such that fori ∈ {1,2}:

Wi is non-empty;

If x ∈Wi and y ∈ Li , then x > y .

Then W1 ⊆W2 or W2 ⊆W1.

Proof.

If W1 =W2, we’re done, so assume W1 ≠W2.

Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.

Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2.

We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1.

Thenw > z . By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z .

By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w .

Hence w /∈ L2. Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w . Hence w /∈ L2.

Thus w ∈W2.



Theorem


Wi is non-empty;



Proof.

If W1 =W2, we’re done, so assume W1 ≠W2. Wlog, let z ∈W2 ∖W1.Now z ∈ L1 ∩W2. We claim W1 ⊆W2. To see this, pick w ∈W1. Thenw > z . By hypothesis, z /> w . Hence w /∈ L2. Thus w ∈W2.


2014 NFC Playoffs: What would have happened?

Who’s #1?

Week 1: SEA 36, GB 16.

Week 6: DAL 30, SEA 23.

DAL vs. GB: no game.

DAL > SEA > GB.W = {DAL},L = {SEA,GB}.




2 Apply the partition rule.


4 . . .

#1: DAL. #2: SEA. #3: GB.



Who’s #1?

SEA ⊳ GB

DAL ⊳ SEA







4 . . .

#1: DAL. #2: SEA. #3: GB.



Who’s #1?

SEA ⊳ GB

DAL ⊳ SEA

DAL ▸ SEA ▸GB.







4 . . .

#1: DAL. #2: SEA. #3: GB.



Who’s #1?

SEA ⊳ GB

DAL ⊳ SEA

DAL ▸ SEA ▸GB.

DAL > SEA > GB.

W = {DAL},L = {SEA,GB}.






4 . . .

#1: DAL. #2: SEA. #3: GB.



Who’s #1?

SEA ⊳ GB

DAL ⊳ SEA

DAL ▸ SEA ▸GB.







4 . . .

#1: DAL.

#2: SEA. #3: GB.



Who’s #1?

SEA ⊳ GB

SEA ▸GB.

SEA > GB.W = {SEA},L = {GB}.






4 . . .

#1: DAL. #2: SEA. #3: GB.



First weekend:

CAR won at home against ARI,whose top 2 QBs were injured.

GB beats DET, while DAL andSEA get a week off.

Second weekend:

DAL gets an easy win over CAR,the worst division champion ever.

SEA, at home, beats GB, as theyreally did twice in 2014.

NFC Championship:

DAL, at home, beats SEA, thenbeats NE in Super Bowl XLIX.

AP Photo/Brandon Wade, via http://www.bostonherald.com/sports/patriots_nfl/nfl_coverage/2014/09/

Charlie Riedel/AP, via http://www.si.com/nfl/2014/09/29/


http://www.bostonherald.com/sports/patriots_nfl/nfl_coverage/2014/09/

http://www.si.com/nfl/2014/09/29/


http://www.nj.com/super-bowl/index.ssf/2015/02/


http://www.nj.com/super-bowl/index.ssf/2015/02/

Thanks!

http://www.sportslogos.net/logos/list_by_team/172/


http://www.sportslogos.net/logos/list_by_team/172/

a suggestion for the nfl's head-to-head...

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