## The Pittsburgh Pirates have hit the second fewest home runs in the MLB this season. Many have cited this as a major factor for the club’s anemic offense this season.

The point certainly seems valid. The Pittsburgh Pirates register just one player in the top 50 for home runs with Andrew McCutchen sitting at 48th, and just three in the top 150 (Josh Bell 67th, Josh Harrison 141st). It doesn’t seem likely that a team that has so little power in their lineup would be capable of winning a lot of games.

All else being equal, if the Pirates just hit more home runs they would obviously score more home runs than if they didn’t.

That being said, that’s not practical.

For the Pirates to have hit more home runs this season, they would have had to do so at the expense of other kinds of hits. In other words, for the Pirates to hit more homers, they would have had to have not hit as many singles, doubles, or triples in order to achieve that goal of more home runs. Additionally, since hitting home runs would mean fewer, say, singles, there would be fewer batters on base, so each home run would score fewer runs.

This is because there are a finite number of plate appearances that teams have to work with in order to get these hits. So for a team to hit more HRs, they have to give up plate appearances where they might have hit a single or double, in order to hit extra home runs.

This is true for every kind of hit; a team that hits more singles as the expense of doubles, triples, and homers would score fewer runs because of the lack of power. The same goes for doubles and triples as well and indeed, even walks.

### The tradeoff

A simplified way of making this point would be to say that there is some tradeoff of the number of each kind of hit because it means not having the other kinds of hits.

Thus we have this idea that there is some optimal number of singles, doubles, triples, and home runs that a team needs in order to score the most runs, given their fixed number of plate appearances. This optimal amount is where the costs of not hitting some other kind of hit are equal to the benefits of the additional runs scored from that particular kind of hit.

That being said, we have no idea what the optimal quantity of each kind of hit are or how close to the optimal level teams are.

we can find this out using something known as a Quadratic Regression.

As a quick refresher on the math, a quadratic equation is something like Y=X2 where we have a variable “X” raised to some power, in this case 2. The shape of a quadratic equation is a U shape. The curve either bends upward if the sign on the X2 is positive or downward if the sign of the X2 is negative. Here’s a quick visual example of both where the blue curve is a positive X2 and the red curve is a negative X2;

**The point that we will be interested in** is either the top on the red curve, or the very bottom of the blue curve. If, as an example, Runs is our “Y” and Home Runs is our “X” and we have the equation Runs= -Home Runs^{2}, then we would want to know the “maximum point” at the top of the red curve, where a team is getting the maximum number of runs from the long ball, while sacrificing the fewest runs from trying too hard to hit them.

Using data from each team between 2015-2017 we can get this quadratic regression;

Giving us this equation;

**Runs= -1402.804 +2.326*1B -.0010*1B ^{2 }+2.307*2B -.0020*2B^{2 }+2.438*3B -.0290*3B^{2+1.958*HR -.0008*HR2 }**

If you notice, the sign on all the squared terms is negative, meaning that the curves all look like the red one from above. This is really important for two reasons. For one, this confirms that there is some optimal number of each kind of hit, so teams shouldn’t seek to only get certain kinds of hit, like just home runs or just singles, but rather should be able to hit all kinds. Secondly, it means we can find those maximum points, giving us an optimal number of each of these hits.

To find these maximizing points, we have to take the partial derivative of this equation with respect to each of the types of hits. I’ll spare you the calculus and just give you the numbers.

### Optimal numbers

The optimal number of singles is 1127, doubles 587, triples 42, and home runs 1220. Clearly these are purely theoretical numbers. This team would average more than seven home runs per game and over 18 hits per game. These are video-game-like numbers and not really useful to what we’re talking about

If we bound each type of hit by the highest number of singles, doubles, etc. that any team hit over the past 3 years we get slightly more realistic numbers, though still pretty extreme. The optimal number of singles is 1031, doubles 343, triples 42, and home runs 253; all but triples are at the upper bound of what has been achieved in recent years.

What this tells us is that, since most teams are not at the theoretical optimal number of these hits, getting more of them is always better than having fewer of them until you approach that theoretical number. A way of thinking of this graphically is that every team is on the left side of the red curve, so each additional hit moves them rightward on the curve, which also pushes them upward along the curve, meaning more runs.

The only exception to this is triples, which do have a feasibly achievable inflection point at 42. What that means is that teams that try to hit more than 42 triples end up losing the benefits of additional runs scored and actually negatively impact runs scored. Since they are giving up the offense of the other types of hits, their attempts to hit more triples cost their teams runs.

One other point to note is that as teams approach this point of 42 with their “upwards” movement slows down along the curve, or their additional runs scored become fewer and fewer. This means that teams that are near this maximum point would actually see more benefit from instead trying for more singles, doubles and home runs instead. Economists refer to this phenomenon as “diminishing marginal returns”.

### No offense at all

What this also tells us is that the Pittsburgh Pirates’ problem this year isn’t that they haven’t hit enough home runs; their problem this year was that they didn’t have enough offense of any kind.

The home run was not the downfall of the offense.

The Pirates were below the league average in singles, doubles, home runs and walks. The only hitting outcome that the Pirates are above league average in this year is triples; coincidentally, that is effectively the only hitting outcome with diminishing benefits to offense, meaning that they would have been better off trading those triples for more singles doubles or homers.

The Pittsburgh Pirates needed more Home Runs this year, but they also needed more singles and doubles and walks. The home run was not the downfall of the offense.

As an example, the team with the 4^{th} fewest home runs in the league is the AL East leading Boston Red Sox. The team is in the top 10 in singles doubles and walks, as well as total runs scored, meaning that a team can have a high performing offense without overwhelming power.

For those of you who are not satisfied by that answer, the Pirates do seem to have a budding homerun hitter in their midst. Josh Bell’s 24 home runs puts him at 2^{nd} among NL rookies and 4^{th} overall. As Bell continues to improve and mature as a major leaguer, expect that number to rise accordingly.

It’s tempting to believe that more of the instant offense that the home run provides would have solved the Pirates offensive issues this year. For as much as the Pirates miss the power that someone like Kang would have provided them; they have equally missed all the singles and doubles that someone like Starling Marte has not produced even while playing this year.

While the Pittsburgh Pirates offense would certainly benefit from the addition of a power bat this coming off-season, they would really benefit from the addition of any productive bats. The looming possibility of losing the team’s most productive hitter in McCutchen either this year or next, means that the Pirates need the production from players like Gregory Polanco and Starling Marte to increase over their dismal campaigns this year as well as finding production from external additions through free agency and trades. The offensive issues of this team run deeper than just a lack of home runs and must be addressed this offseason if the Pirates hope to compete in 2018.