Minimally Adjusted Reduced Score (MARS) Ranking Method

 

 

 

Nicholas F. Trombetta

September 21, 2009

 

 

 

Explanation:

 

The Minimally Adjusted Reduced Score Ranking Method, or MARS Rank, or MARS, is an accurate and impartial ranking system primarily used for college football (NCAA FBS teams) though it can be applied to any sport.  The ranking system begins with all teams being of equal rank, where teams exchange “rank points” based on the results of the actual games played.  Rank points can only be exchanged among teams based on game results and are not created or destroyed within a season.  The MARS Rank method is based on the following:

 

Rank Points – this is the value that determines the rank of the team where the team with the highest rank point value is considered to be in “first place”, and so on.

 

Game Value – Each game is worth a certain amount of rank points based on the teams playing in that game.  For exapmle, if team A has a total of 1.5 rank points and team B has a total of 2.4 rank points, the game between A and B would be worth a total of 1.95 rank points, or the midpoint of rank point values of the teams.  In this way, each team contributes half its rank points to every game.  The results of the game will determine how many of those points a team receives from that game.  This sytem is not winner take all, so a team will always regain some of their original rank point contribution after a game.

 

Game Results – The game value is determined by the rank points of the teams participating, and the results of the game determine the redistribution of the game value.  In the example above, if the final score is team A, 35, team B, 23, team A would receive 1.95*(35/58) = 1.176724  rank points, and team B would receive the balance, or (.773276) rank points.  The end result is team A now has 1.926724 rank points and team B now has 1.973276 rank points.  So while team A upset team B, team B will retain a better rank.

 

 

 

 

 

 

 

 

 

 

Reduced Score – Now we will consider two more possible scores as examples:

 

1.)    Team A, 45; Team B, 7

2.)    Team A, 45; Team B, 0

 

In the first example above, team A has dominated team B and the resulting apportionment of rank points available in this game would be (.865385) for team A with the remainder for Team B - is this too much?  In the second example, team A has gained 100% of the available game value, leaving zero for team B – is this winner-take-all?

 

MARS is a system designed to acknowledge close games as such (winning does not have a value of 1, with losing having a value of 0) while also limiting the effect of runaway scoring.  Therefore a cap is needed on the per-game apportionment – what is the cap and is it arbitrary?  The cap is (.75) and it is not arbitrary.  Consider football, where a team can score a touchdown for six points, a field goal for three points, or a safety for two points.  The “extra point” is an extra kick worth one point allowed after a touchdown.  It is only allowed after a touchdown; therefore a score of one point can only happen after a score of six points.  In football, therefore, the lowest naturally occuring winning score is 2 – 0.  In most other sports the lowest naturally occuring winning score is 1 – 0.

 

Since the MARS Rank System is not winner-take-all, it is necessary to assume that games begin at a tie score of 1 – 1, rather than a tie score of 0 – 0, this is the minimal adjustment.  Therefore the lowest winning score of a football game is minimally adjusted to 3 – 1; where the winning score corresponds to a take away of (.75) of the available game value for the winning team.  For most other sports, the lowest winning score is minimally adjusted to 2 – 1; where the winning score corresponds to a take away of (2/3) of the available game value for the winning team.  Runaway scoring is capped at the value of the minimally adjusted lowest shutout score.  In shutout games, or games where the winner would have gained more than (.75) based on the actual score, the winner’s take is reduced to (.75).  This is the reduction.  In games where the actual score results in an apportionment where the winner gains (.75) or less of the total, no adjustment is necessary.

 

 

 

 

 

 

 

 

 

Algorithm:

 

Begin with all teams having one rank point and, therefore, equal rank.

 

Games are worth the average of the rank points of the two teams in the game, so each team gives half its rank points towards the game value.

 

The score of the game determines the apportionment of the available rank points (the game value) with a max of (.75) to the winner for football, and a max of (2/3) to the winner for most other sports.

 

Add the corresponding apportioned value back to the rank points of each team.

 

At any given time, the team with the highest rank point value is considered to be in “first place”, the team with the next highest value is second and so on.

 

Continue through the season.

 

Only games amongst teams in the same league are considered (for the MARS rankings of NCAA FBS football teams, only games amongst two FBS opponents are considered.)

 

 

 

 

Conclusion:

 

MARS is a margin of victory system where the winner’s take is capped in such a way as to allow for the loser of a close game to be rewarded for playing a close game.  The cap is not arbitrary, but based on the lowest possible naturally occuring shutout score.  The adjustment is the minimum value added to the beginning score to avoid winner take all scenarios.  Only “runaway scores” will even be subject to the “cap” and “adjustment” as the intent is to disturb the actual and natural outcome of the game as little as possible.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Please see the MARS “top 25” for the first five weeks of the 2009 NCAA FBS season.

 

Week

Team

MARS Rank

Week

Team

MARS Rank

1

Arizona

1

2

Nebraska

1

1

California

2

2

Texas

2

1

Cincinnati

3

2

Penn State

3

1

Idaho

4

2

Auburn

4

1

Kentucky

5

2

Tulsa

5

1

Michigan

6

2

Boise St

6

1

Mississippi

7

2

Missouri

7

1

Missouri

8

2

Alabama

8

1

Nebraska

9

2

UCLA

9

1

Notre Dame

10

2

Michigan

10

1

Penn State

11

2

LSU

11

1

Southern Cal

12

2

Arizona

12

1

Stanford

13

2

Boston College

13

1

Tennessee

14

2

California

14

1

Texas A&M

15

2

Cincinnati

15

1

Texas

16

2

Iowa

16

1

Auburn

17

2

Kentucky

17

1

Tulsa

18

2

Mississippi

18

1

Clemson

19

2

Texas A&M

19

1

Oklahoma St

20

2

Southern Cal

20

1

Boise St

21

2

Kansas

21

1

UCLA

22

2

Notre Dame

22

1

South Carolina

23

2

Duke

23

1

Bowling Green

24

2

Texas Tech

24

1

Utah

25

2

Utah

25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Week

Team

MARS Rank

Week

Team

MARS Rank

3

Penn State

1

4

Iowa

1

3

Alabama

2

4

Alabama

2

3

Texas

3

4

Texas

3

3

LSU

4

4

Florida

4

3

Kansas

5

4

Virginia Tech

5

3

Clemson

6

4

Boise St

6

3

UCLA

7

4

Nebraska

7

3

Michigan

8

4

Stanford

8

3

Auburn

9

4

UCLA

9

3

Iowa

10

4

Kansas

10

3

Nebraska

11

4

LSU

11

3

Oklahoma

12

4

Oregon

12

3

California

13

4

Auburn

13

3

Boise St

14

4

South Florida

14

3

Pittsburgh

15

4

Ohio State

15

3

Florida

16

4

Oklahoma

16

3

Missouri

17

4

Texas A&M

17

3

Cincinnati

18

4

TCU

18

3

Virginia Tech

19

4

Michigan

19

3

Ohio State

20

4

South Carolina

20

3

Mississippi

21

4

Cincinnati

21

3

Washington

22

4

Southern Cal

22

3

Kentucky

23

4

Penn State

23

3

Miami FL

24

4

Clemson

24

3

Texas A&M

25

4

Missouri

25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Week

Team

MARS Rank

5

Alabama

1

5

Texas

2

5

Florida

3

5

Stanford

4

5

Ohio State

5

5

Boise St

6

5

Iowa

7

5

TCU

8

5

Southern Cal

9

5

Nebraska

10

5

Virginia Tech

11

5

LSU

12

5

Oregon

13

5

Kansas

14

5

South Florida

15

5

Auburn

16

5

Cincinnati

17

5

South Carolina

18

5

Pittsburgh

19

5

Penn State

20

5

UCLA

21

5

Georgia Tech

22

5

Missouri

23

5

Brigham Young

24

5

Oklahoma

25