Now to break down how all these values are compiled....
Why the Golden Ratio (φ)?
The Golden Ratio (φ) is the ratio between a and b if a>b>0 and (a+b)/a = a/b. If:
a = a team's winning percentage
b = a team's opponents' combined winning percentage
c = a team's opponents' opponents' combined winning percentage,
the original football BCS SOS formula was (2b+c)/3. To have a simple 2:1 ratio when compiling SOS always seemed too contrived and easy to me, so I thought, rhetorically, that a ratio of a team's record and its strength of schedule should be the same as that of its "b" to its "c."
The Golden Ratio = φ = (1+√5)/2 = 1.6180339887499
Wins, Losses, Ties, etc.
All wins and losses (even those to lower-division teams) are counted. A game that Team A plays against Team B does not factor into each other’s opponents’ record. That is, if Team A drops to 30-1 after a loss to Team B, Team B notches 30-0 into its SOS. This same exclusion applies to the opponents’ opponents’ total. Plainly: a team’s games are excluded from reflexively factoring into its SOS. When Team A plays a lower-division opponent, that opponent is deemed hypothetically to go winless vs. Team A’s schedule. If Team A is 10-0 against opponents who go a combined 54-46, an upcoming game against Lower Division U would account for 0-10 in their opponents’ combined record and 54-46 in their opponents’ opponents’ combined record. The higher Team A’s overall winning percentage is, the less a game against Lower Division U. hurts their SOS.
Most remember the BCS controversies in 2003 and 2006, when LSU/USC & Florida/Michigan were in respective conversations for spots in their title games. In each case, the former was 12-1, having beaten an FCS opponent and won a conference championship game, and the latter was 11-1, having played no FCS opponents and no “extra” conference championship game. If not every team plays the same number of games, this means that different opponents carry different weights in a team’s SOS. After literally years of deliberating on how to account for these differences (if at all), I decided to count “sitting at home” the same as beating a lower-division team. Therefore, 2003 USC and 2006 Michigan are given applied records of 12-1, each given a win vs. a hypothetical lower-division team.
As it applies today in the middle of a college basketball regular season, the maximum number of games played by a given team as of this morning (2/1/19) is 24, and any team who has played fewer games–that is to say, almost every team–is given the number of wins vs. a hypothetical lower-division team that equals the difference between its total and the running maximum. As of right now, my Golden Formula applies to all 353 teams' totals 24 games, 352 opponents’ games (24²-24), and 13,248 opponents’ opponents’ games (24³-24²). This has a considerable effect on all postseason games, giving them extra weight. As a team advances in its conference tournament or in a postseason tournament, beating increasingly good teams along the way, most other teams are, of course, “sitting at home.” Postseason wins, therefore, cause strengths of schedule and overall ratings to rise considerably more than do regular season games, while postseason losses show little or no negative effect to a team’s rating. On April 8, two teams will play for the national title, each adding to its SOS a strong opponent, while each of the other 351 teams effectively will be given a 0-40 hypothetical team to theirs.
All winning percentage totals are “graded on a curve” before applying φ to them. This addresses the relatively low variance in SOS, which tends to be only about ⅓ of the variance in winning percentage. As of this morning, the lowest applied winning percentage is Maryland-Eastern Shore’s .167, while the highest is .958, held by the five 1-loss teams. The highs and lows of opponents’- and opponents’ opponents’ winning percentages are .663/.313 and .588/.387. Each high is represented by 1.000, and each low, .000, with all other values represented proportionally between. All elements of the Golden Formula are now determined.
The Golden Formula (a × φ²) + (b × φ) + c = Rating
2/1/19 Rankings (Expanded)
I received some questions about Virginia's surprisingly low #20 ranking that I published Monday (and you might notice that they quickly have jumped to #8). Virginia's schedule before the new year was on the weak end of the Power 6's, and as of this past weekend, their inevitably strong ACC schedule accounted for less than 1/3 of their overall schedule; to that, add four "sitting at home" instances since they've played only 20 games, compared to the running high of 24. This will be more than mitigated by the time UVA finishes their regular season with 10 ACC opponents and likely 2-4 ACC tournament opponents; by selection Sunday, they will have played as many games as virtually anyone else, and their relatively strong conference slate will represent more than 2/3 of their overall schedule.
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