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When 1 + 1 Doesn’t Equal 2

By Bryan Woolley, JP Wong, and Nick Skiera.

Baseball, like all sports, is exciting because of the concept of variance. No team scores the exact same number of runs every game. That is why the Dodgers (5.82 runs/game) were not 60-0 in 2020. Runs per game strongly correlates with winning percentage for obvious reasons, but a team’s variance (essentially their consistency) plays a crucial role in their ability to win baseball games

Relating to this, we came across an interesting game theory concept. Given certain properties of the run-scoring distributions, the competitor with the lower output can increase their win probability by increasing the variance in their output. Conversely, the competitor with the higher output can increase their win probability by decreasing the variance in their output. Were this to apply to baseball, lower-scoring teams could win more games by becoming more inconsistent. Of course this is all just in theory, so the requirements for it to be relevant in reality to baseball might not be met.

We will examine the importance of variance in baseball both to test the theory and to attempt to uncover interesting trends in the sport. In our analysis we find that variance plays a significant role in a team’s success, suggesting that roster and lineup construction can be optimized by going beyond mean production. So as our title proposes, 1 WAR + 1 WAR and 2 WAR might not always be worth the same amount to a team if they are produced with different consistencies. Read the rest of this entry »

Rearing Back: Pitchers’ Effort in Important Situations

Leading 3-1 and one out away from being a World Series Champion, Los Angeles Dodgers pitcher Julio Urías faces Tampa Bay Rays infielder Willy Adames. The first two pitches of the at-bat, fastballs resulting in a swinging strike and a called strike, clock in at 94.9 mph and 94.1 mph. The last pitch of the at-bat (and subsequently the World Series) comes on the third pitch. Urias fires a third straight four-seam fastball, this time for a called strike three at 96.7 mph. This may not feel particularly fast in a day and age in which some pitchers consistently hit 100 mph, but for Urías, there was a little something extra behind that final pitch. Of the 682 four-seam fastballs that Urías threw in 2020, this pitch was the fastest. While it may have been a coincidence that his hardest-thrown pitch was also in the most important situation, I suspect the significance of the moment was a key factor.

I doubt this claim comes as much of a surprise to anyone. Most people in crucial situations will push a little harder to ensure the outcome is in their favor. To test the theory, I examined pitch velocities from the 2019 regular season. I chose 2019 rather than 2020 to ensure the situations were most similar to a normal year in case any of the irregularities of baseball during COVID influenced the data. In general, it appears that two-strike fastballs are thrown harder than fastballs in other counts. I graphed the respective densities of fastball velocities below. Read the rest of this entry »

Adjusting Batter Performance by the Quality of the Opposing Pitcher

In the 2020 season, American League MVP José Abreu faced 107 different pitchers, including the top four in Cy Young voting point totals (Shane Bieber, Trevor Bauer, Yu Darvish, and Kenta Maeda). Bauer was the only of the four not to allow a home run to Abreu in 2020. In comparison, MVP Runner-up José Ramírez faced 69 of the pitchers that Abreu faced. The third-place DJ LeMahieu faced a completely different set of pitchers, not a single one overlapping with Abreu’s.

While these batters were compared by their offensive production, it appears Abreu faced more challenging pitching. Using FanGraphs’s xFIP- (for which a lower number is better) as a measure of a pitcher’s quality, Abreu was up against a 96.75 xFIP- on average while LeMahieu faced pitchers with at a 105.93 mark. Both LeMahieu’s weighted on-base average (wOBA) of .429 and Abreu’s .411 were exceptional, but is the 18-point difference truly reflective of the difference between the two players’ seasons?


To answer the question, I derived a value with a similar intuition to Baseball Prospectus’s Deserved Run Average (DRA). DRA is a measure that adjusts a pitcher’s performance by the quality of the batters they are facing. This statistic also accounts for numerous context factors to attempt to better isolate the pitcher’s contribution. DRA shows that the quality of the batter can be influential in a pitcher’s performance, so it makes sense that the quality of pitcher is influential in a batter’s performance.

As for the statistic I will be working with, I choose to refer to this as “pitcher-adjusted weighted on-base average,” or pwOBA. The intuition is simple: a batter should get credit for offensive production against challenging pitching. The formula for pwOBA is based on the formula for wOBA. With wOBA, every event has a run value (ex. 1.979 for home runs in 2020) and a batter gets credit for these values accumulated over the course of the season. The sum of these values is then divided by (AB + BB – IBB + SF + HBP). Read the rest of this entry »