Closer

I’m not much of a fan of the saying “the exception that proves the rule,” but here is an exception that proves the rule: During the 2013 Major League Baseball All Star Game, something very unusual happened. The manager of the American League Jim Leyland decided to pitch legendary closer Mariano Rivera in the eighth inning, rather than the conventional ninth. Leyland explained that since Rivera, who is retiring at the end of the year, was appearing as part of his unofficial farewell tour, it would be too risky to wait until the bottom of the ninth inning, which may have been omitted should the home National League take the lead. Usually a manager will only bring in the closer when leading in the ninth, rarely when tied, and almost never when trailing. What is the rational for this accepted strategy? Even if closer (who usually throw hard and don’t get as much rest as starting pitchers) are limited to one inning per game, shouldn’t you want your best pitcher in the game as much as possible, regardless of the score? That is, if your team is behind by one run, a comeback is easier if you don’t make the deficit bigger.

I think part of the solution is the way runs are scored in baseball… in bunches. Since runs in an inning are not independent, the chance of scoring two runs in an inning is more likely than the chance of scoring one squared. A simple example is a batter who comes up with a runner on first. If he hits a double, the runner scores and there is now  a runner in scoring position, making it likely that he will record a run as well. Hits both drive in existing runners as well as creating new ones. Add to this the possibility of non-solo homers, and you can see the correlation between having a big inning given at least one run is scored.  According to the Baseball Prospectus, the probability of a team of scoring a certain number of runs during it’s half-inning is strongly non-linear. In fact, it is close to being exponential:

0 – 73.0%

1 – 14.8%

2 – 6.8%

3 – 3.1%

4 – 1.4%

5 – 0.6%

6 – 0.2%

7 – 0.1%

Imagine a team playing at home (that is, batting last) with an unhittable closer, in the sense that he is guaranteed to pitch one perfect inning, but then his arm gets tired and he has to hit the showers. If his team is leading going into the top of the ninth, their victory is assured. The value of having this super-closer available is increase in the probability of ultimately winning the game. In the case of a one-run lead, the change is from about 85% to 100%. (That is, the home team can win by not conceding a run in the top of the ninth, which for a mortal pitcher would occur 73% of the time, or in the bottom of the ninth or extra innings if the visitors do tie the game, about 0.148*(0.148+0.73*0.5) = 0.076). So the benefit in win-percentage is about 0.15. Contrast this with a game in which the home team is trailing by one run going into the ninth. Preventing the deficit from getting bigger is not as valuable, since the home team still needs to score in the bottom of the inning no matter what.

P{Home team wins after trailing by one going into the bottom of the ninth} =

P{Home Team Scores Two or more in the Ninth} + P{Home Team Scores One in the Ninth then wins in extra innings} =

0.116 + 0.73*0.5 = 0.19

VS:

P{Home team wins after trailing by two going into the bottom of the ninth}

P{Home Team Scores Three or more in the Ninth} + P{Home Team Scores Two in the Ninth then wins in extra innings}

0.048 + (0.068*0.5) = 0.08

Sometimes managers get flak for strategies that do not maximize the chance of winning, but it appears that this aspect of bringing in closer in basically sound. If a closer’s innings need to be conserved over the course of a season, it makes more sense to use them to cement a win when ahead, rather than make a comeback slightly more likely.