Over the past decade, a tremendous amount of progress has been made in quantifying if and when managers should do everything from sacrifice bunt to issue an intentional walk, but one area that has been (at least to my knowledge) overlooked is when to pinch run. One of the main reasons pinch running hasn’t been investigated to the same degree as these other strategies is that there are a lot of moving parts to look at- how much more valuable is the pinch runner than the previous runner? Will removing the starter hurt the team’s defense? What’s the opportunity cost of losing an available bench player? While the answers to each of these questions are needed for a complete analysis of when to pinch run, there is one question we can answer relatively simply- how often can we expect the pinch runner’s spot in the order to come up again later in the game?

The more often a pinch runner can be expected to have to come to the plate, the less appealing pinch running is likely to be, as the players who are most often removed for pinch runners are generally very strong hitters (or catchers), and pinch runners are generally light hitters.

Take for example the White Sox-Twins game on April 9th of last year. Tied in the bottom of the 8th inning, the White Sox elected to have Mark Teahen pinch run for Paul Konerko following Konerko’s 1-out single. The next two White Sox hitters went down in order, and two innings later Teahen stepped to the plate in the bottom of the 10th with men on first and second and one out. Teahen grounded into a 6-4-3 double play and the White Sox went on to lose in 11 innings.

To look at how often a pinch runner’s spot in the order can be expected to come up, I used retrosheet data and looked at all American League games over the past two seasons, tracking all the times a pinch runner was used in close games (what I defined as a game within three runs and in the seventh inning or later) and then how often that spot in the order came up again. I limited the sample to only American league games so as not to have to deal with the complexity of tracking double switches, so this analysis may not be applicable to National League games.

Here is the average number of times the pinch runner’s spot in the order came up after pinch running:

(Click for clear view)

I filled the squared where the sample sizes were extremely small, but as you can tell from some of the round numbers, sample sizes in the 7th inning were also pretty small. There are some other anomalies in the chart. With more data I expect we would find that pinch runners used by the home team when down by 1 would be more likely to hit later in the game than those used when the home team was down 2 and 3 runs. But for the purposes of this post, the above numbers should suffice. If there is interest, I can add years to the data to come up with more reliable numbers.

So are pinch runners being used effectively?

For a pinch runner to be used effectively, the value of the pinch runner on the base-paths + the difference in the defense between the pinch runner (or whoever enters on defense) and the previous player – the opportunity cost of using bench players must be equal or greater to the run difference of the two players at the plate multiplied by the number of times that spot in the order is expected to come up again.

For now, let’s ignore defense and the opportunity cost of losing an available bench player and just focus on how much is gained on the base-paths and lost at the plate by looking at an example Angels fans became very familiar with last season- having Reggie Willits run for Hideki Matsui.

To estimate the relative run values of having each player on base, I took each player’s BSR ratings for the the year in which the pinch running occurred (2010 in this case) and the two years prior, and divided the baserunning total by a best estimate of times on base (hits + walks + intentional walks + HBP). I used a standard 3/4/5 weighting scale to give more weight to more recent performance (using this method will likely overstate how valuable having a player who is often called upon to pinch run, as they will be on base more than the total given by hits and walks, but this method will also likely understate how bad players who are often get pinch ran for are, so the bias should be minimal).

Using this method, I calculated Willits as being .027 runs above average per time on base and Matsui as being .008 runs below average. From these values, the Angels expected to gain .035 runs by bringing Willits off the bench to run.

How much do they stand to lose at the plate?

Using the same weighting procedure, Willits’ wOBA comes to .265 and Matsui’s is .361, giving us a difference of .096. Converting this difference to runs gives us a run difference of .083 runs per PA.

To find the point at which pinch running and not pinch running yield the same expected value is given by .035 = .083*(expected number of future PA). Solving the equation, when a pinch runner can be expected to bat .42 more times a manager should be indifferent between using Willits as a pinch runner or leaving Matsui on the bases.

Going back to our chart, the change should be made for all situations listed in the 9th, extra innings, and for home teams in the 8th inning.

Of course, the expected number of times the pinch runner’s spot in the order will come up is based on situations in which a pinch runner was actually used. By not using a pinch runner, teams that trail in the game may decrease the chances they score a run in that inning, thereby decreasing the chances the spot in the order comes up again. Even taking this into account, I think this analysis is worth while, as gives us a rough guide as to when pinch running may be a good idea, and when it is in the team’s interest to let the slower runner stay on base.

From this analysis, it appears that teams are already pinch-running fairly optimally, as very few pinch runners are used in the 7th inning, with an increasing number as the game goes on.