Tuesday Football: New Victory Weighting definitions

Yes, the last third of April is an odd time to bring up my Victory Weighting scheme, but there's the little matter of the NFL Draft, which, for 2011, starts next Thursday.  That event provides the perfect pretext to introduce definitions for a couple of terms that occasionally occur in tiebreaking contexts.  [If you haven't done so already, please click on the Victory Weighting line in the top bar for an overview of the system.]

Strength of Victory

This figure provides an estimate of the quality of the opponents a team has defeated.  As a tiebreaker, it applied only when all comparisions of sub-records (e.g. records vs. common opponents) have failed.

The NFL calculates strength of victory as the combined won-lost-tied percentages of those opponents a team has beaten.  For Victory Weighting, it has a more formal definition.

For a given season, a team's strength of victory is the sum of the weighted current Strengths of its opponents, taken over all completed games.   For each game the team has played, the Strength it earned from that game weights the opponent's current Strength.  At the beginning of the season, strength of victory is 0 for all teams.

Written as a formula,

where

• W_i is the Strength the team earned from game i.  It ranges from 0 to 4, per the standard Victory Weighting rules.  Once established, it never changes.
• S_i(N) is the current strength of opponent i; that is, its Strength as of week N during the season.  This value usually increases over the course of the season.

For example, consider two teams who meet in the first week of the season.  While the game was in progress, N=0, and, by definition, both teams have zero strength of victory.  Upon completion of the game, N increments to 1.  If the game ended in overtime, the winner's strength of victory increases by 0.750 (W_i = 3, S_i(1) = 1, 3*1/4 = 0.75).   After eight weeks (N = 8), if the loser has earned 16 Strength, that contribution increases from 0.75 to 3*16/4=12.00.

Only completed games count toward strength of victory.  Games remaining on the schedule do not count, since their game weights cannot be determined until they have been played.

A few easy colloraries follow:

• After the first week, every team's strength of victory is either 0, 3 or 4.
• A team that has not managed even an overtime loss has a strength of victory of zero.  The  Detroit Lions accomplished this dubious feat in 2008.
• If a team has not played any overtime games, its strength of victory is equal to the combined Strength of all the teams it has defeated.  If, that team also finishes undefeated, its strength of victory is also equal to its strength of schedule.  The 2007 New England Patriots are the only team to achieve this distinction since the NFL adopted overtime.^*

Strength of Schedule

For the 32-team NFL, this statistic isn't as crititcal as it is for the NCAA, which has to evaluate 120 to 350 teams at once for its tournaments.  Accordingly, the NFL can use a less rigorous formula for each team: the combined won-loss-tied percentage of all its opponents.  Typically, strength of schedule is evaluated only at the end of a season, after all games have been played.

Victory Weighting provides a more rigorous definition:

For a given season, a team's strength of schedule is the sum of all its opponents' current Strengths, taken over the entire season.

Its formula is

where

• N_max is the length of the season in games, currently 16.
• S_i(N) is the current strength of opponent i; that is, its Strength as of week N during the season.  This value usually increases over the course of the season.

At any point in the season, the strength-of-schedule calculation includes games yet to be played, as well as games already played.

In both the official and VW schemes, strength of schedule is the first tiebreaker used to determine draft order.  If that fails, then any division and conference tiebreakers that were used for playoff seeding apply.

With both terms defined for Victory Weighting, two more corollaries emerge:

• After the first week, all winners have a strength of schedule equal to 0 or 1; all losers, 3 or 4.  If the first game ends in a tie, both teams have a strength of schedule equal to 2.
• If a team has not played any overtime games, the ratio between its strength of victory and its strength of schedule is equal to its won-loss-tied percentage.

Some examples follow the fold.

Example:  Denver vs. Cincinnati vs. Buffalo, 2010

The Bengals and Broncos finished 2010 tied for the second-worst Victory Weighted record, each with 16 Strength and each at 4-12.  The Bills also finished 4-12, with 19 Strength.  We'll use these teams to illustrate how strength of victory and strength of schedule work.

Strengths of victory

Both the Bengals and Broncos went through the 2010-1 campaign without playing an overtime period.  For each team all twelve losses were in regulation (W_i = 0), so they contribute nothing to either one's strength of victory.  In each case, it's necessary to look at only the four wins.

The Bills present a slightly more complicated case.  Three of their losses came in overtime(W_i = 1), so those actually add to the Bills' strength of victory.  We'll need to look at those games as well as the Bills' four wins.

Cincinnati:  Defeated Baltimore (47), Carolina (8), Cleveland (21) and San Diego (36).  The Bengals' strength of victory is (4*47 + 4*8 + 4*21 + 4*36)/4 = 47+8+21+36 = 112.00.
Denver:  Defeated Seattle (28), Tennessee (25), Kansas City (40) and Houston (24).  The Broncos' strength of victory is (4*28 + 4*25 + 4*40 + 4*24)/4 = 28+25+40+24 = 117.00.
Buffalo:  Defeated Detroit (24), Cincinnati (16), Cleveland (21) and Miami (27).   Lost in overtime to Baltimore (47), Kansas City (40) and Pittsburgh (46). The Bills' strength of victory is (4*24 + 4*16 + 4*21 + 4*27 + 1*47 + 1*40 + 1*46)/4 = (4*(24+16+21+27) + (47+40+46))/4 = (4*88 + 133)/4 = 121.25.

Strengths of schedule

Traditionally, the NFL makes it even easier to compute strengths of schedule by giving teams in the same division nearly identical schedules.  Under the current alignment, a team plays six games against its divisional rivals (two against each), four games against another division in its conference and four more against a division in the other conference.  Rivals' schedules thus differ by only two games.  In that light, it makes sense to calculate Strengths for divisions.  For any division, just add the Strengths of its members.

Under this scheme, you can look up each game individually, or you can add up division Strengths as appropriate.

Cincinnati:  The AFC North divisional strength was 130.  Subtracting the Bengals' Strength of 16, and noting that they played their rivals twice each means that the AFC North contributed 2*(130-16) = 228 to their strength of schedule.  The AFC East added 143; the NFC South; 146.  Finally the Bengals' "free" opponents, San Diego and Indianapolis, contributed 36 and 41 Strength, respectively.  Therefore, the Bengals' strength of schedule is 228+143+146+36+41= 594.
Denver:  AFC West divisional strength = 123, AFC South = 123, NFC West = 101, Baltimore = 47, New York Jets = 40.  Broncos' strength of schedule = (2*(123-16))+123+101+47+40 = 562.
Buffalo:  AFC East divisional strength = 143, AFC North = 130, NFC North = 133, Kansas City = 40,  Jacksonville = 33.  Bills' strength of schedule = (2*(143-19))+130+133+40+33 = 584.

For the 2011 Draft, Denver would draft second with a lower Victory Weighted strength of schedule than Cincinnati.  The Bengals would draft third, while the Bills, with Strength 19, would go fourth.  Officially, the Bills are going third, because they were tied with the other two teams, and the traditional strength-of-schedule calculations left them between the Broncos (still second) and the Bengals.

* The 1972 Miami Dolphins also finished perfectly, two years before the NFL adopted overtime.

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