The mathematics of baseball are linear.
Perhaps this is a no brainer of a statement to you or perhaps, and maybe more likely, you haven’t spent a great deal of time thinking about this. But the linearity of baseball’s math is a critical underpinning to thinking about successful team construction.
I have two primary theses related to that opening statement. First, the valuation of baseball players is a linear exercise. By this I mean that you can calculate a player’s approximate worth on the diamond
by a series of first order equations that, in the grand scheme of things, aren’t terribly complicated. And second, the valuation of a baseball team is also a linear equation. Any one player can be expected to produce and/or prevent a certain number of runs. The sum of all runs produced and prevented (or allowed) will lead you to a pretty good understanding of how many wins a team will have at the end of the year. And generally speaking, how you accomplish the stacking of wins is a matter of flexibility.
Let’s assume you are a President of Baseball Operations and want to build a 90 win team. You might start by looking at history to see how this is done and you would find in your research that there have been 90 win teams who weren’t very good at hitting or not very good at pitching or not very good at defense. A team’s deficiencies can be counterbalanced elsewhere. Similarly, you would find successful teams in the past whose best player was a corner outfielder and others who were built around a shortstop, and others still with a lockdown ace as their best player. It’s a puzzle game with very few parameters.
As is often the case, popular narratives can distract us from thinking clearly about the real issue. “Pitching and defense wins championships” is a maxim that is technically true but not exclusively true. In the dynastic run of the New York Yankees that began in the mid-1990s, an opportunistic sportswriter noticed that the mainstays of that team happened to play up-the-middle defensive positions (i.e., Jorge Posada, Derek Jeter, and Bernie Williams) and posited that the way to build a lasting winner was securing talent at those positions. Doing so isn’t a bad idea, but it’s a concept that potentially distracts from the central goal: collect wins.
So, back to the baseball executive with a dream of a 90 win ballclub. Since Wins Above Replacement (WAR) is the metric du jour and since the replacement level in WAR has been calibrated to produce a theoretical 48 win team if all players were exactly replacement level, the question of “How do I build a 90 win team?” can be restated as “How can I aggregate a roster that will produce 42 WAR?”.
Side note: I will be relying heavily on WAR in this article. Unless otherwise noted, all WAR figures will be from Baseball Reference. Also, the various issues and objections to the calculation of WAR are generally known and understood by the author. Rather than relitigate those items, I’ll note that the sum of a team’s WAR correlates closely to a team’s win total and that’s going to be good enough for me for the purposes of this article.
The genesis of this article is from a thought that is not original and has been expressed many times in various corners of the internet. It’s something along these lines: The Reds have two bona fide stars and both of them are being paid a relative pittance…how is the team not any better? I’m talking about Elly De La Cruz and Chase Burns, of course; some versions of the above lament include Sal Stewart. Either way, it’s a completely understandable frustration and a question worth pondering if you care about the success of the Reds.
Rather than specifically trying to answer that common question, I focused my mind on a slightly different query and it led me to a truly surprising discovery.
A long, long time ago, I was getting my bachelor’s degree in mathematics and in one particular statistics course, I encountered a concept that fascinated me: conditional probability. Standard probability gives us the odds of an outcome if we are coming in blind (what is the likelihood that I will have three-of-a-kind on this next poker hand?), but conditional probability asks the question with some information already baked in (I drew a pair of Kings pre-flop…what is the likelihood that I’ll see a third King on the flop?). I decided to apply this thinking to the 2026 Reds.
The methodology: I pulled WAR data for the top two position players and top two pitchers from each franchise for every season since 1996 (excluding the pandemic-shortened 2020). Why 1996? I saw that season as the demarcation point for ever-expanding team payroll disparities from the wealthiest to the poorest. The economic landscape of 2026 is considerably different than 1996, of course, but broadly speaking it’s useful to think of 1996-present as a modern baseball era. That gave me nearly 900 data points to ask questions of. Let’s dive in.
Question #1: How good would we expect a team to be if all we knew was how good their best player was?
There will be several charts in this article. For each of them, the number of team wins will always be plotted on the vertical axis and the metric that I’m testing against will be on the horizontal axis.
What do we make of the above chart? For one thing, there’s clearly a positive relationship between having a really good player and having a good team, but the relationship is not ironclad. I count 26 instances of a team having a losing record and also having a player with at least an 8 WAR season. That’s maddening!
The chart gives us a visual that makes intuitive sense. The majority of outliers from the big cluster in the middle of the chart are below the trend line. In other words, it’s easier to be a bad team with a great player than it is to be a very good team without a great player. The most extreme negative outliers is a murderer’s row lineup of excellence shining through a sea of putrescence: 2004 Diamondbacks (51 wins, Randy Johnson with 8.4 WAR), 2004 Mariners (63 wins, Ichiro Suzuki with 9.2 WAR), 1996 Giants (68 wins, Barry Bonds with 9.7 WAR), 2009 Royals (65 wins, Zack Greinke with 10.4 WAR), 2016 Angels (74 wins, Mike Trout with 10.4 WAR), and the 1997 Blue Jays (76 wins, Roger Clemens with 11.9 WAR).
Maybe the best example of the opposite circumstance was with the 2017 Dodgers, a team that won 104 games without having a single player reach 6 WAR.
Back to the chart. The correlation coefficient (denoted as “R2” in the chart) tells us how much of the variance in the dependent variable (in this case, number of team wins) can be explained by the variance in the independent variable (in this case, WAR from a team’s best player). For this particular comparison, we’re told that 23% of a team’s win total can be explained by how good their best player was. I want to keep part of the focus on the 2026 Reds throughout this article, so let’s apply the slope equation to the Reds and Elly De La Cruz. If all you knew of the Reds was that their best player was worth 5.6 WAR in 2026, the above model would suggest an expected team win total of 80. A nearly average team!
And based on the empirical data, we could call back to our original question. Of all the teams in our data set whose best player had between 5-6 WAR, 24% of them won 90 or more games.
Question #2: How good would we expect a team to be if all we knew was how good their best two players were?
We’re making the model just a bit more complex. There are two predictable changes to the chart: the R-squared figure is higher but there’s a diminishing return to the added information.
In other words: whereas with one single player we can explain 23% of a team’s win total, with two players we can explain 33% of the win variance.
I guess it’s probably an OK time to interject that while wins are the obviously the goal of a team, there will always be noise behind the math that approximates wins. This is true at an aggregate level (witness the year to year variances between a team’s win total and their expected win total based solely on runs scored and runs allowed), and at the individual level (WAR is best described of an approximation of a player’s value rather than a precise calculation). The 1997 Blue Jays won just 76 games despite Roger Clemens and Pat Hentgen contributing nearly 18 WAR. That should probably be an impossibility. The fact that it’s merely an outlier should tell us that WAR will never be a perfect descriptor of wins but it’s a pretty good approximation.
Applying this to the Reds and our favorite duo of EDLC and Chase Burns, their combined 10.3 WAR would give an expected team win total of 80. Of all teams that had their top two players with a sum of 9.5-11.5 WAR, 25% won at least 90 games.
Question #3: How good would we expect a team to be if all we knew was how good their best position player and best pitcher were?
This is a slight wrinkle on the previous question, and curiously enough there is some minor incremental improvement in the model when we know the best bat and the best arm on the team, rather than just the two most valuable players irrespective of position.
For the 2026 Reds, of course, the two players in question don’t change at all but the expected win value goes up to 82 wins based on the combined 10.3 WAR of Elly and Chase. From my perspective, the slight changes in the shape of this graph aren’t quite as interesting as the next query.
Question #4: When thinking about a team’s best hitter and best pitcher, does balance matter?
OK, things get a bit more math heavy on this one. To build this view, I multiplied the best hitter’s WAR by the best pitcher’s WAR to come up with a comparison of product of those two figures against team wins. Why do this? Let’s use an example to explain.
Picture a team whose best two players are perfectly balanced. A hitter worth 6 WAR and a pitcher worth 6 WAR. Then assume a second team who has a stratospheric star in left field worth 10 WAR, then maybe some additional good-to-great hitters, but their best pitcher is worth a middling 2 WAR. The top hitter + pitcher are equal to 12 in both instances, but which team would you expect to be better, based just on the balance between top hitter value and top pitcher value? By multiplying the two figures together, team 1 would have a score of 36, whereas team 2 would have a score of 20. That creates differentiation from which to look at win expectations.
However, because we’re now comparing a linear number (i.e., team wins) against a non-linear number, we don’t use a linear trend line anymore but a logarithmic one.
And, long story short, we can see that 36% of a team’s win total can be explained if you just know how good (and balanced) their top position player and top pitcher are. And having Elly De La Cruz at 5.6 WAR and Chase Burns at 4.7 WAR, we can say that based on solely those two data points, the Reds would have an expected win total of 84.
Of course, not all star players are created equally. Or paid equally for that matter. The era since 1996 might accurately be labeled as the “market size era” of baseball, one in which the gap between haves and have-nots is ever growing and a team’s access to local wealth is a reasonable (but not perfect) predictor of success.
The holy grail for smaller market teams is dependable star talent that is paid way below market value.
Consider, again, the 2026 Reds.
In Elly De La Cruz, the Reds had a star shortstop who aggregated 5.6 Wins Above Replacement, a level that signals that he is showing Hall of Fame potential at age 24. He was paid a salary of $800,000, an amount that equates to roughly 0.5% of the Reds total payroll this year.
Chase Burns was a top 10 pitcher in the National League, accumulating 4.7 Wins Above Replacement. However you define the nebulous term “staff ace”, Burns qualifies. He is 23 years old and was paid $785,000.
Combined, the two stars contributed 10.3 of the necessary 42 WAR to reach 90 wins. They accounted for 1.1% of the Reds payroll. This would, on paper, seem to be a rather academic exercise to add another 32 wins over 24 roster spots (and more than $140M with which to do it).
But let’s frame the question in terms of conditional probability.
We’ve already established that you can explain a good deal of team quality based solely on the quality of their top position player and top pitcher. What if we added another variable, namely what proportion of a team’s payroll is made up by those two players?
The hypothesis, I suppose, would be that percentage of payroll made up by the top two players will be inversely proportional to the number of wins that team ultimately finishes with. In other words, if you find yourself in a situation where your top two players are paid relative peanuts, you should expect even more wins than a team whose two best players are among the highest paid on the team.
It makes intuitive sense. It’s also not supported by historical data.
In part 2 of this study, I’ll look at how team payroll data integrates with the above questions and how that might impact how we think about the 2026 Reds.
