A harmonic take on Immaculate Grid

Immaculate Grid is a game for baseball trivia nerds. Put out by Baseball Reference, it asks you to name 9 different players in a grid, each having two characteristics, such as “Played for the Rockies” and “Hit over 300 career HR”, which you would promptly miss because noone has any idea what the hell is going on with the Rockies. One of my solved grids is below. The percentages at the upper right of each players image are the fraction of people that used that player. The sum of the percentages is your score, the lower the better. So there’s an incentive to try to think of strange players that meet each category, kind of a reverse Keynesian beauty pageant, if you will.

Most puzzles will have at least one row and one column which are individual teams, which means that you need to find a player who was (as in the above example) on both the Guardians and the Yankees. If you miss one, or you get a high score on a player, you can click through and Baseball Reference will show you everyone who met the criterion, with a list of the Top 5 players ranked by career WAR, or Wins Above Replacement. WAR tries to put a number to how useful a player was relative to a player that basically any team could get at any time, a “replacement-level player”.¹ Notionally, the Top 5 ranking is by total career WAR, but Knucksie doesn’t show up on the Guardians/Yankees list in spite of having more WAR than everyone on it.²

What this is lacking is a ranking of which players might be the most readily associated with both teams. Let’s stick with Guardians/Yankees for now. C.C. Sabathia (who I used above for a different combination, Brewers/Yankees) is fairly well-associated with both teams, coming up with Cleveland and winning his Cy Young there, then playing the second half of his career in the Bronx, and winning the 2009 Series in pinstripes. By contrast, Gaylord Perry was a great pitcher overall who also won a Cy Young in Cleveland³, but only played a forgettable half-season in New York. Both are ranked in the Top 5 for the Guardians-Yankees franchise pair, but Sabathia is much more associated with both teams.

There is a great way to assess this, using a type of average called the harmonic mean. Baseball already uses the harmonic mean for a different statistic called the Power-Speed Number (PSN). To compute it,

where HR is the number of home runs hit by the player, and SB is the number of stolen bases. The PSN can be computed over a career ⁴ or over a season⁵. Why it’s called the harmonic mean is not germane to the present discussion, but what is interesting about it is the way it biases toward the lower number without completely ignoring the higher one.

You can probably see where this is going. It is possible to compute a similar number for every combination of two teams. The Guardians and the Yankees yields the Guardians-Yankees number. The Tigers and the Reds have the Tigers-Reds number, etc. There are 435 of these numbers: one for each combination of 30 teams, not counting the trivial combinations where the team is paired with itself. Since we know the characters already, we start with the Guardians-Yankees number:

where is the WAR a player totaled for the Guardians, and is the same for the Yankees. The top two players are C.C. Sabathia with a CLE/NYY of 28.4, and Graig Nettles with 25.5. I would call the general name of this stat the “harmonic”. As in, the highest harmonic for the Guardians and the Yankees is C.C. Sabathia.

Both of the players in the above example are in the Top 5 by total career WAR among players that spent time with both franchises, but this is not always the case. Consider the Tigers/Red Sox number, DET/BOS. The Top 5 listed for this franchise pair are Dizzy Trout, Bobby Veach, Fred Lynn, Johnny Damon, and Dick McAuliffe. The highest DET/BOS scores are Howard Ehmke (15.8) and Dutch Leonard (14.2), two pitchers that ended their careers before WWII.⁶ Number 3 for this franchise pair is still active: J.D. Martinez scores a 13.5 on the back of an exceptionally balanced 13.2/13.7 WAR for Detroit and Boston, respectively. He’s no longer active with either team, but an additional 1.7 WAR for the Tigers would push him over Leonard.

What does this average mean? I think it’s a way of measuring the following: if you paid some attention to both teams during a player’s career, who would jump out at you the most? It obviously can’t account for recency, or extremely short notable stints with a team, like Doyle Alexander on the Tigers in 1987, or Sabathia’s preposterous half-season on the Brewers in 2008. Not all of the combinations are interesting, in the sense of showing you players who aren’t famous.

If you have a metric that never matches up with the eye test, it’s probably wrong. And if it never surprises you, it’s probably useless. But if four out of five times it tells you what you know, and one out of five it surprises you, you might have something. — Bill James

I hesitate to call the harmonic “useful”. But the point of Immaculate Grid is to Remember Some Guys, but I think it’s nice to have a number that occasionally highlights someone that did some good work for both franchises, but was never going to get his jersey retired in either place. With some abuse of terminology, you could say he resonated with both franchises.

(I haven’t computed every combination yet, so I’m planning to plug away at these and do little writeups as I get them done. See below!)

(Featured image original from Cooperstown Expert, modified by Aren, questions best directed to Bluesky)

• Bluesky

Completed to date

Guardians-Yankees: C.C. Sabathia (28.4) and Graig Nettles (25.5). Both are in the Top 5 for the franchise pair. At time of writing, Sabathia is not yet in the Hall of Fame, but has been inducted in his first year of eligibility. Nettles probably should be in the Hall, but a combination of a long career, a low batting average and a lot of his value being added defensively led to him getting bounced from the ballot after his 4th year.

Tigers-Red Sox: Howard Ehmke (15.8) and Dutch Leonard (14.2). J.D. Martinez (13.5) is still active, and passes Dutch with another 1.7 WAR for Detroit.

Tigers-Reds: Sam Crawford (11.6) and Harry Heilmann (8.3). Both are in the Top 5 for the franchise pair, and both are in the Hall of Fame. Each had more than 60 WAR for the Tigers. Nick Castellanos (4.64) is still active at time of writing, and would need 6.1 more wins for the Reds to tie Heilmann.

Diamondbacks-Rangers: Doug Davis (5.7) and Rick Helling (3.5). Both are in the Top 5 for the franchise pair. Max Scherzer is still active at time of writing, and would pass Helling with another 4.0 WAR for Texas. He played for Texas in 2024, but is a free agent at time of writing.

Braves-Giants: Darrell Evans (22.1) and Felipe Alou (19.3). Evans is in the Top 5 for the franchise pair. Neither is in the Hall of Fame, but Evans probably should have gotten more votes than the 1.7% he received in 1995. No active player is particularly close.

Reds-Yankees: Paul O’Neill (16.8) and Carl Mays (11.8). Both are in the Top 5 for the franchise pair. Neither is in the Hall of Fame. O’Neill won a World Series with each team–1990 with the Reds and 1996, 1998-2000 with the Yankees. Mays won four himself, picking up three with the Red Sox in 1915-6 and 1918, then the 1923 title with the Yankees. He is perhaps best known for other work. Aroldis Chapman (8.65) is the nearest active player.

Angels-Cubs: John Lackey (6.7) and Jose Cardenal (6.2). Lackey is in the Top 5 for the franchise pair. Neither is in the Hall of Fame. Lackey won a World Series with both teams: 2002 with the Angels as a rookie, and 2016 with the Cubs in his penultimate season. No active player is particularly close.

Twins-Orioles: Goose Goslin (20.6) and Sam West (16.2). Goslin is in the Top 5 for the franchise pair, and is in the Hall of Fame. Neither one ever played in Minnesota or Baltimore. West’s WAR was exceptionally balanced between the teams, with 16.7 for the Senators (Twins) and 15.7 for the Browns (Orioles), and only 0.3 career WAR outside the two franchises. No current player is particularly close.

Braves-Tigers: Darrell Evans (17.8) and Bill Bruton (11.4). Evans is in the Top 5 for the franchise pair, and is the only player so far to have the highest harmonic for two different franchise pairs. Neither is in the Hall of Fame. No active player is particularly close. Late 80’s Tigers fans’ bête noire / hero Doyle Alexander (5.3) is the highest-scoring pitcher.

1. More details for the interested.
2. Ah, the knuckleball.
3. Ah, the spitball.
4. Highest career: Barry Bonds, 613.9
5. Shohei Ohtani, 56.4 in 2024, also the highest season mark of all time.
6. Full disclosure: Ehmke looks to have slightly more career WAR than Dick McAuliffe and should probably be in the Top 5 despite not being listed. I sympathize, especially if these Top 5 lists are done manually. I’ve yet to automate things at time of writing, and I could miss someone.