Billy Hamilton has played in 537 games for the Cincinnati Reds over the past five years. In that time, Hamilton has accumulated a career on base percentage of .298, putting him at #256 on the list of 289 qualified hitters in the league during that time. His career slugging percentage is .334, putting him at #280 out of 289 hitters in the league. He has been, by almost every metric, one of the worst hitters in all of baseball in the last 5 years. He has also started 364 games as the leadoff hitter for the Reds in that same time.

I would say that opinions are mixed on the issue of Billy as a leadoff hitter, but that would probably be a bit generous. As the years have worn on and Billy has continued to hit like Billy, even his most ardent supporters have conceded that his place in the lineup is likely not at the top. And with the arrival of Jesse Winker, the best on-base machine in the Reds system whose name is not Joseph, not only is there a clear need to remove Hamilton from the leadoff spot, but a clear replacement for him there as well. This appears obvious to most everyone in the Reds community, except perhaps, the guy crafting the lineups every day.

Ã¢â‚¬Å“I’m still looking to see the evolution of Billy as a leadoff man, I’d love him to hold down that spot, if we’re a better team with someone else hitting up a little higher, that’s the way we’ll go.Ã¢â‚¬Â Ã¢â‚¬â€œ Bryan Price

IÃ¢â‚¬â„¢ll ignore the first part of that sentence, where he believes that somehow, after four full seasons of non-progress and a 2017 campaign almost on the dot with his career averages, Price is still looking for the evolution of Billy Hamilton. IÃ¢â‚¬â„¢ve spent enough time waiting for that evolution, and I along with many others have come to accept that Billy is what he is. ItÃ¢â‚¬â„¢s the second part of that quote that IÃ¢â‚¬â„¢m going to dissect: Ã¢â‚¬Å“if weÃ¢â‚¬â„¢re a better team with someone else hitting up a little higher, thatÃ¢â‚¬â„¢s the way weÃ¢â‚¬â„¢ll go.Ã¢â‚¬Â Well, if you donÃ¢â‚¬â„¢t mind, IÃ¢â‚¬â„¢d like to demonstrate to you that the Reds are a better team with someone else hitting leadoff.

The first thing weÃ¢â‚¬â„¢re going to look at is Run-Expectancy charts. If youÃ¢â‚¬â„¢re unfamiliar with the concept, fangraphs has a great primer on them here. The basic idea is that, in any of the 24 base-out situations (first and third with no outs, second and third with two outs, etc), there is an average number of runs the offensive team would expect to score in that inning, given the situation. For example, from the chart in the article, with a runner on second and one out, the team on offense can expect to score 0.644 runs in that inning, on average. You can measure a playerÃ¢â‚¬â„¢s contribution to the situation by subtracting the run-expectancy before they hit from the run-expectancy after they hit, and adding any runs that scored.

This is easy to do for a leadoff hitter, since the run-expectancy before they hit is always the same: 0.461, with nobody on, nobody out. LetÃ¢â‚¬â„¢s use Billy Hamilton and Jesse WinkerÃ¢â‚¬â„¢s 2017 seasons as our numbers in this hypothetical. For Winker, weÃ¢â‚¬â„¢ll extrapolate to 633 plate appearances, the same as Billy had last year.

These both seem like reasonable stat lines to expect out of a full season from each player. Now, letÃ¢â‚¬â„¢s scale them to 140 plate appearances, one for each time they would lead off a game in our hypothetical season:

Again, pretty reasonable numbers. Now, for every time they lead off, it is going to end in one of five ways: either they get out, end up on first, end up on second, end up on third, or hit a home run. The change in run expectancy for each of these situations is -0.218, 0.370, 0.607, 0.965, and 1.000, respectively. To find each playerÃ¢â‚¬â„¢s total contribution to run expectancy in the leadoff position over a whole season, we simply multiply the change in run expectancy for each outcome by the number of times that outcome occurs. For example, Billy will end up on first 35 times (25 by a single, 10 by walking), so we multiply 0.370 by 35 to get 12.95 expected runs added. For the sake of ease, weÃ¢â‚¬â„¢ll assume all stolen bases result in the player on second, and all caught stealings end with the player out.

When you total up all the contributions over a whole season, the numbers come out like this: Billy Hamilton will add 5.5 expected runs over 140 leadoff appearances, while Jesse Winker will add 6.4 expected runs.

Now, that is a relatively small margin, but at the end of the day, baseball is a game of inches, and any expected advantage, no matter how small, has to be taken. No matter how you shake it, Jesse Winker will provide more value to the team in the leadoff position than Billy Hamilton would, according to run-expectancy.

But wait, thereÃ¢â‚¬â„¢s more! Hitting leadoff isnÃ¢â‚¬â„¢t *just* about who gets to hit first. If that were all it were, people would not be as upset by Billy leading off as they are. No, the biggest component of being a leadoff hitter is that you get the most plate appearances of anyone on the team. Again, itÃ¢â‚¬â„¢s an easy and intuitive concept Ã¢â‚¬â€œ youÃ¢â‚¬â„¢ll go around the order four and some change times in an average game, and as a leadoff hitter, the order will come back around to you a fifth time more often than it will any other player.

So why on earth is the worst hitter on the team Ã¢â‚¬â€œ one of the worst hitters in all of baseball Ã¢â‚¬â€œ getting the most plate appearances? I ran a few more numbers, this time using this chart of expected plate appearances per game as my guide. In 140 games started, from the leadoff position, Billy can expect to get about 651 plate appearances in a season. Now, letÃ¢â‚¬â„¢s compare that to if he were hitting, say, ninth. ThatÃ¢â‚¬â„¢s not a bad place for him to hit Ã¢â‚¬â€œ not only does it give the worst hitter the fewest plate appearances, but Billy still functions as effectively a leadoff hitter every time through the order but the first, coming after the pitcher and before the thumpers. From the ninth spot, Billy would get about 528 plate appearances in a 140 game season. Given his .299 on base percentage last season, Billy would make an out 456 times in the leadoff position versus 370 times in the ninth spot.

Now letÃ¢â‚¬â„¢s say we replace him with Jesse Winker. On days that Billy was in the lineup, Winker would bafflingly hit 7^{th} last year, so weÃ¢â‚¬â„¢ll use that as our baseline. From the 7^{th} spot, over 140 games, Winker would get 561 plate appearances, compared to 651 from leadoff. With WinkerÃ¢â‚¬â„¢s .375 OBP last year, he would make 351 outs batting 7^{th} and 407 outs batting leadoff.

Now letÃ¢â‚¬â„¢s add the two scenarios. With Billy hitting 1^{st} and Winker hitting 7^{th}, the two will combine for 807 outs made on the season. Compare that to Winker hitting 1^{st} and Billy hitting 9^{th}, where the two would combine for 777 outs on the season. ThatÃ¢â‚¬â„¢s 30 fewer outs made by the two. Over a 600 plate appearance season, 30 fewer outs is a boost of 50 points to your OBP. For a team that totaled 6,213 plate appearances in total last year, it would raise the team OBP 5 points, from .329 to .334, bumping us from 11^{th} to 6^{th} in all of baseball.

Imagine that! Imagine if you could arbitrarily add 50 points to the OBP of anyone in your lineup, simply by changing the order in which you bat people! This is the magic of mathematics. In a game where the most important thing is not making an out, Bryan Price could make a dramatic improvement to his team by doing nothing other than changing where people hit in the lineup.

LetÃ¢â‚¬â„¢s circle back to the beginning here. Bryan Price said:

Ã¢â‚¬Å“if weÃ¢â‚¬â„¢re a better team with someone else hitting up a little higher, thatÃ¢â‚¬â„¢s the way weÃ¢â‚¬â„¢ll go.Ã¢â‚¬Â

Jesse Winker would add more expected runs in the leadoff position than Billy Hamilton would over a full season. Jesse Winker hitting leadoff and Billy Hamilton hitting 9^{th} would take away 30 outs made over the course of the season. From every angle you cut it, the Reds are a better team with Jesse Winker hitting leadoff, not Billy Hamilton.