The Bullpen Tax On A Team Total Under

Every team total under is sold to you as a bet on a starting pitcher. Yoshinobu Yamamoto has held opponents to three runs or fewer in 20 of his 23 starts this season, so the Atlanta under 3.5 tonight looks like an 87 percent proposition against a price that needs 59.2. That is the pitch, and it is wrong by nine percentage points before anyone throws a ball.

The correct statistic is not what the starter allowed. It is what the opposing team finished with. Those are different numbers, and the gap between them has a name worth using: the bullpen tax.

Take the six pitchers starting tonight's three most heavily bet games. For each one, count how often he personally allowed three runs or fewer, then count how often the team he faced finished the whole game with three or fewer. Same games, same sample, one number counts his innings and the other counts all nine.

Nobody in this group escapes the tax. The smallest is nine points and the largest is twenty. If you have been pricing team total unders off a starter's own run distribution, you have been running ten to twenty points hot on every one of them.

Chris Sale has allowed three runs or fewer in 22 of 23 starts. Teams facing him have finished with three or fewer 19 times. Photo: Jsayre64, Wikimedia Commons, CC BY-SA 4.0

The obvious explanation is that some clubs have better relievers. It is a reasonable guess and the data does not support it as the main driver. Cole pitches in front of a staff carrying a 3.23 ERA, the best of any club in these three games, and he has the second largest tax on the board.

Sort the same six by innings per start instead and the picture snaps into focus.

That is the mechanism. A team total under does not survive on how good the starter is. It survives on how few outs are left over for somebody else, because those leftover outs are where a 1 to 0 game becomes a 4 to 0 game and a ticket that was winning in the seventh is a loser in the eighth.

Yamamoto sits at the friendly end of that curve for a boring reason. He goes 6.51 innings, which leaves seven and a half outs. Cole averages 5.67, which leaves ten. Three extra outs of exposure is worth roughly nine points of team total probability in this sample, and that is close to the entire difference between the two of them.

Atlanta under 3.5 is priced around -145, which needs 59.2 percent. Two different ways of reading Yamamoto give two different answers, and only one of them is the right question.

Those last two rows are the actual argument and they point opposite directions, which is what makes the ticket interesting rather than obvious. The pitcher based read says the under clears the price by nineteen points. The opponent based read says Atlanta at home has been held under this number about once every four games. One of those samples describes who is pitching and the other describes who is hitting, and a real position has to weigh both instead of quoting whichever one agrees with it.

Three rules fall out of the table, and none of them need a model.

One. Never price a team total under off a starter's own run distribution. Use the opponent's full game finals in his starts. It is the same query with one different column and it is the only version that matches what the ticket actually grades on.

Two. When the full game version is not available, discount the starter number by ten to twenty points and lean toward the larger discount for anybody under six innings a start.

Three. Prefer innings per start over ERA when choosing which unders to shop. A 2.61 ERA that goes six and a half is worth more to this bet than a 3.11 that goes five and two thirds, and the market prices the ERA far more aggressively than it prices the workload.