Off-day Math Exercise: Bradley-Terry Baseball Rankings (by JonathanF)

So I woke up this morning and decided that what baseball needs is another set of power rankings. Not really. Actually, I have done a fair amount of work in ranking methodologies and was surprised to discover that nobody regularly produces (AFAIK) for baseball the most common ranking procedure around: Bradley-Terry ranking. No, not Milton Bradley and Bill Terry. Bradley-Terry ranking involves the creation of a single number per team Rt which yields a simple formula for the probability that team A will beat team B the next time they meet:

Bradley-Terry ratings, or minor variations of them, are used in Chess rankings, some college football ranking schemes, and, of course, NCAA hockey rankings, where they are known as KRACH rankings.

Bradley-Terry ratings, or minor variations of them, are used in Chess rankings, some college football ranking schemes, and, of course, NCAA hockey rankings, where they are known as KRACH rankings.

The key difference between Bradley-Terry rankings and most other baseball rankings is that they use wins and losses alone, not runs. The vast majority of sabermetric ranking schemes use some variant of aggregate run differences to create rankings, but Bradley-Terry doesn’t care whether you win by 1 or 10. On average, that’s throwing away a lot of information, but on the other hand winning is what the game is supposed to be about, and lots of scores in lopsided games are the result of teams abandoning defense or offense to better prepare for the next game. College football ranking schemes in the BCS forbid the use of scores, so Bradley-Terry –like schemes are common.

Figuring out the simplest version of Bradley-Terry rankings for baseball is pretty simple. Since

We can sum up both sides and take expectations to get the new equation:

Rearranging,

Now this equation doesn’t quite get you there, because RA is on both sides, but it turns out that so long as teams play each other enough and nobody has an undefeated record, you can keep updating the left hand side by an assumed value of the right hand side and eventually it all converges. So the only data you need are the wins by each team and the matrix of games played between each team to generate the rankings, which are conventionally normalized so that the best team gets a rank of 100.

Now this is the simple version. The more complicated version (with more complicated formulas) take into account home field advantage (and require data on the home-road games for all pairs of teams.) The easy version simply makes home field advantage common to all teams and then adjusts the rankings to reflect this common rating. But the more interesting (and even more complicated) method treats every team as two different teams, a home team and a road team, and creates a separate rating (and ranking) for each. The following table consolidates all three of these analyses:

Team
(by average rank)Basic RatingConstant Home FieldSeparate Home/Road Ratings
   HomeRoad
Boston100.0 (1)100.0 (1)86.5 (2)70.6 (1)
Detroit93.8 (2)94.9 (2)86.3 (3)61.4 (5)
Tampa Bay90.7 (3)91.4 (3)81.8 (5)61.7 (4)
Pittsburgh89.3 (4)89.9 (4)83.0 (4)57.2 (9)
Atlanta84.5 (5)87.0 (5)100.0 (1)50.7 (11)
Baltimore83.7 (6)84.0 (6)69.0 (10)63.2 (2)
St. Louis79.9 (8)82.1 (7)71.2 (9)57.6 (8)
Texas80.9 (7)81.7 (8)63.8 (11)60.0 (7)
Oakland79.0 (9)79.9 (9)81.2 (6)48.8 (13)
Cleveland78.0 (10)78.5 (10)75.0 (8)48.1 (14)
Dodgers71.1 (12)72.3 (12)45.5 (19)62.6 (3)
Cincinnati71.7 (11)73.5 (11)75.4 (7)42.6 (18)
Yankees69.8 (13)70.9 (13)59.0 (12)53.9 (10)
Kansas City69.3 (14)70.4 (14)48.3 (14)60.1 (6)
Toronto61.1 (16)61.8 (16)47.6 (15)48.8 (12)
Arizona61.5 (15)62.2 (15)47.0 (17)47.0 (16)
Washington53.7 (18)53.8 (18)48.5 (13)36.8 (21)
Seattle54.1 (17)53.8 (17)44.6 (20)40.2 (19)
Angels51.1 (19)50.5 (19)43.7 (21)35.3 (22)
Minnesota49.4 (20)50.1 (20)40.2 (25)38.0 (20)
Philadelphia48.3 (21)49.0 (21)43.2 (22)32.7 (23)
San Diego47.8 (22)48.0 (22)47.1 (16)27.3 (27)
Mets47.0 (23)47.7 (23)30.5 (28)46.2 (17)
Colorado46.7 (24)47.1 (24)46.9 (18)27.6 (26)
Cubs46.1 (26)46.6 (26)28.1 (29)47.5 (15)
San Francisco46.6 (25)46.8 (25)41.1 (24)31.3 (24)
Milwaukee43.7 (27)44.0 (27)36.7 (26)31.1 (25)
White Sox39.0 (28)39.7 (28)42.8 (23)21.2 (30)
Miami36.3 (29)36.4 (29)34.6 (27)23.6 (29)
Houston32.6 (30)32.4 (30)22.2 (30)27.0 (28)

When I made this table last week, the Braves were 11th: 1st at home and 18th on the road. They’re now fifth since their road ranking has risen to 11th. That shows what six consecutive road wins will do for you, and also demonstrates some of the fragility of these rankings. Still, the Braves are not ranked as highly as other teams with worse records. That’s the consequence of playing in a crappy division and not beating the crap out of crappy teams, and particularly not beating bad teams on the road. Adjusting for a constant home-field advantage has very little effect, but adjusting as if there were two different teams has very large effects in some cases, particularly the Braves, who still exhibit the biggest quality-adjusted home/road rankings in MLB.

To use the first ranking, the probability of a head-to-head win just follows the first equation: RA/(RA+RB), where RA and RB are the rankings of the two teams. In the second method, the home teams rankings get multiplied by approximately 1.2 before using the equation. In the third method, you use the equation with the ratings for the home team or visiting team, whichever is which.

The interesting thing you can do with this, armed with these values, is to calculate the probability of a team winning the World Series by simulating forward with the probabilities. And I’ll do that as the season winds down with updated values. In the meantime, it’s easy enough to produce the maximum likelihood final record, which currently stands at 101-61.

TeamHomeNumber of GamesProbability Each GameExpected Wins
Milwaukee130.76272.3
St. Louis040.41591.7
Cubs030.64361.9
Cleveland130.67512
Miami040.59442.4
Miami160.80884.9
Mets020.62451.2
Mets130.6842.1
Washington130.7312.2
Washington030.51091.5
San Diego130.78552.4
Philadelphia030.53961.6
Philadelphia170.75345.3
Total31.4

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35 thoughts on “Off-day Math Exercise: Bradley-Terry Baseball Rankings (by JonathanF)”

  1. Note: Alex failed to update the table (but I like him anyway). This is the table as of Sunday, which is why the text doesn’t match.

  2. If tableizer works, this is the correct current table:

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    font-size: 12px;
    }
    .tableizer-table td {
    padding: 4px;
    margin: 3px;
    border: 1px solid #ccc;
    }
    .tableizer-table th {
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    color: #FFF;
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    }

    Team Basic RatingConstant Home FieldSeparate Home/Road Ratings 
    (by average rank)  HomeRoad
    Boston100.0 (1)100.0 (1)86.5 (2)70.6 (1)
    Detroit93.8 (2)94.9 (2)86.3 (3)61.4 (5)
    Tampa Bay90.7 (3)91.4 (3)81.8 (5)61.7 (4)
    Pittsburgh89.3 (4)89.9 (4)83.0 (4)57.2 (9)
    Atlanta84.5 (5)87.0 (5)100.0 (1)50.7 (11)
    Baltimore83.7 (6)84.0 (6)69.0 (10)63.2 (2)
    St. Louis79.9 (8)82.1 (7)71.2 (9)57.6 (8)
    Texas80.9 (7)81.7 (8)63.8 (11)60.0 (7)
    Oakland79.0 (9)79.9 (9)81.2 (6)48.8 (13)
    Cleveland78.0 (10)78.5 (10)75.0 (8)48.1 (14)
    Dodgers71.1 (12)72.3 (12)45.5 (19)62.6 (3)
    Cincinnati71.7 (11)73.5 (11)75.4 (7)42.6 (18)
    Yankees69.8 (13)70.9 (13)59.0 (12)53.9 (10)
    Kansas City69.3 (14)70.4 (14)48.3 (14)60.1 (6)
    Toronto61.1 (16)61.8 (16)47.6 (15)48.8 (12)
    Arizona61.5 (15)62.2 (15)47.0 (17)47.0 (16)
    Washington53.7 (18)53.8 (18)48.5 (13)36.8 (21)
    Seattle54.1 (17)53.8 (17)44.6 (20)40.2 (19)
    Angels51.1 (19)50.5 (19)43.7 (21)35.3 (22)
    Minnesota49.4 (20)50.1 (20)40.2 (25)38.0 (20)
    Philadelphia48.3 (21)49.0 (21)43.2 (22)32.7 (23)
    San Diego47.8 (22)48.0 (22)47.1 (16)27.3 (27)
    Mets47.0 (23)47.7 (23)30.5 (28)46.2 (17)
    Colorado46.7 (24)47.1 (24)46.9 (18)27.6 (26)
    Cubs46.1 (26)46.6 (26)28.1 (29)47.5 (15)
    San Francisco46.6 (25)46.8 (25)41.1 (24)31.3 (24)
    Milwaukee43.7 (27)44.0 (27)36.7 (26)31.1 (25)
    White Sox39.0 (28)39.7 (28)42.8 (23)21.2 (30)
    Miami36.3 (29)36.4 (29)34.6 (27)23.6 (29)
    Houston32.6 (30)32.4 (30)22.2 (30)27.0 (28)

  3. Can’t trust that Remington guy to get anything right.

    In other news, the Nats fans over at BTF are officially discussing off season acquisitions – Jacob Elsbury! Robbie Cano! – so that’s a fun thing.

    Also, in a “feel dirty dirty dirty” sort of way, I’ve officially started rooting for the Mets to take 2nd place in the division.

  4. Wow. That is an impressive piece of work! (And saves me from having to come up with an off-day post idea! :P) Thanks, Alex – I really look forward to your updating this as we move forward.

  5. Really neat post.

    Forgive me if this has been discussed, but Andrelton is closing in on a statistically historic season. The record for DRS (defensive runs saved) is 35 in a season and Andrelton is already at 32 with 47 games left to play. His dWAR is 4.3 which is only 1.1 off the all-time mark. Something fun to watch down the stretch.

  6. @5 – The dead-ball era defensive seasons are a little more impressive on a game by game basis, as the players there had fewer games to rack up dWAR. But Simmons is well on his way to challenging the all time modern record for “best defensive season ever.”

  7. Good point Sam, I was going to mention that.

    Sad thing is I think there’s a good chance he doesn’t even get the Gold Glove as they may give it to Tulo. The best tools as voted on by managers and they picked Tulo and Desmond over Andrelton as best defensive SS and Desmond over Simmons for best arm.

    Obviously, the defensive stats aren’t everything but DRS for 2013: Andrelton 32, Tulo 11, Desmond -2

  8. The Nats entire starting lineup is under contract for next year. Laroche is owed 12M plus a 2M buyout for ’15. The only way to get a 2B upgrade would be at the expense of dropping Rendon out of the lineup or eating a ton of Adam’s contract, Zim to 1st and Rendon to 3rd. They are already looking at an estimated 116M for 2014 – Ellsbury/Cano would be north of another 17M per. They are kinda stuck I think. 2014 Lineup looks ok actually with Rendon, but LaRoche continuing to fade at the plate and Zim physically not being able to play 3rd are very distinct possibilities.

  9. @9, Rendon can play 3rd, so he’s probably the 3B of the future no matter what. One interesting question, in light of Zim’s throwing difficulties, is whether they would ever consider teaching Zim to play second, or if they’ll just make him a 1B after LaRoche leaves.

  10. Agreed, spike. My point is more “the Nats fans are resigned to rosterbation for 2014” more than if their rosterbation fantasies are realistic.

  11. I know his shoulder is shredded, but man, that’s an awfully good glove to waste at first.

    As for the Nats’ free-agency plans, well, one of the only bats that makes sense for them is…Brian McCann.

  12. @16

    Yeah, I think they will have to wait a year to make the move.

    It’s kinda sad. Zimmerman looks like a wounded brid when he throws.

  13. Well it isn’t out of the realm of possibility that Zimmerman’s shoulder improves a bit next season, as he’ll be 16 months removed from surgery. On the other end, it’s not impossible that he’ll undergo extensive surgery, rather than the arthroscopic type he had this past offseason, and that theoretical surgery will answer the question for him (fix him, or relegate him to 1B.)

  14. Jonathan – do you then prefer the wins-based model to those that use runs? Or are you just presenting an alternative? In the past, I’ve tended to think that college football in particular needs more Margin Of Victory models (or models that include MOV), because of the noise in such a small sample, but perhaps baseball, with its (much) longer season, is better off with wins. Not sure, though.

  15. @19

    In a way, it makes a lot more sense to shut down Strasburg this year than it did last year. If they get to early September and still aren’t challenging for anything, shut him down and save some innings on his arm, since it doesn’t matter anymore anyway. That’s way preferable to severely limiting one’s chance to win the World Series in a year where they have the best record in the league. I still can’t believe how stupid that decision was, and how many people lapped it up as if it was some kind of forward-thinking brilliance. If fate has any sense of irony at all, the Nationals won’t have another legitimate chance to go to the World Series for the next 20 years. I wonder how brilliant Rizzo will think he is then.

  16. Adam M: I haven’t really done a comparison in baseball. I have done a comparison in college hockey that compares a Poisson-based goal model with a Bradley-Terry wins model and find the two yield pretty similar results. (That’s in the context of a 30 or so game season.)

    I think the sample size in college football is such a big problem, as is the extremely limited number of interregional games that you probably don’t want to throw away the information that scores give you if the NCAA doesn’t make you. Even there, though, you probably want to truncate runaway scores, because 56-0 and 42-0 probably aren’t telling you very different things.

    Pythagorean rankings in baseball tell you that run differences and win differences are giving you pretty much the same information, though. But a lot of that comes from the fact that baseball plays so many more games and the team mixing is so thorough.

    The real question is whether wins predict wins better than net runs predict wins. Barring the effect of blowouts, it seems clear to me that net run differences ought to work better than wins, since wins are just a game-by game collapsed information set of runs, so there’s almost certainly something to be derived from the extra information you’re throwing away. BUt that involves modeling the extra information, raising hte possibility of the introduction of modeling error in runs-based schemes which is absent from in wins-based schemes. The blowout game problem is just one species of that. If one team is 15-5 in one run games, while another team 5-15, that’s only a ten run net difference, but it’s a heck of a lot of games… lucky or good? The wins-based method says good. The runs-based method says lucky.

  17. Playin’ at the Clermont Lounge tonight if any of you guys want to make constructive use of the off day….

  18. 8 — I think this year is the first where they were going to add some measure of statistics to the Gold Glove selections? If so it’s almost impossible to see anyone else winning it at short.

  19. Reynolds has something like a .180/.200/.220 line for about his last 300 ABs. Something is really broken there.

  20. @27 So now there are three things in life that are certain: death, taxes, and the Atlanta Braves winning the NL East in 2013? I can live with that.

    If we go on a 20 game losing streak now and things get interesting, I am totally blaming BP!

  21. Got my post season tickets invoice today from the Braves. I guess it’s official: the Braves must be going to the post season.

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