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Manny Machado and paying for the bell curve

Paying for a player isn’t just paying for expected value, but the entire distribution of performance

San Diego Padres v Chicago Cubs Photo by David Banks/Getty Images

Among baseball fans’ favorite pastimes, very few rival snickering at recent free agents. Whether it was after Albert Pujols signed his deal and quickly declined, or watching Ryan Howard fall flat on his face immediately after signing his contract, there is often a wide chorus of “see-I-told-you-so” finger-pointing that occurs the season after, or maybe two seasons after, a massive contract is inked.

This year there aren’t many contracts to focus on, and very few of the above-age-30 variety, so we’re stuck discussing Manny Machado and Bryce Harper ad infinitum. Harper has had his ups-and-downs but is hitting at a 119 wRC+ clip and is projected to finish the season with 3.9 fWAR; Machado is two points higher in wRC+ and is projected to finish at 4.2 fWAR.

At first glance, that seems a hair low for a collective $660 million in total salary. There are currently 25 players by ZiPS projected for more value than Machado, and 36 for Harper. Players like Ketel Marte, DJ LeMahieu, and Marcus Semien could all be more valuable players in 2019, and they were paid a collective $22.9 million in average annual value.

Yet we full well know which player we’d prefer. I talked briefly about this on Twitter, and sabermetrician Sky Kalkman posed a good question: “Why is a narrower value probability curve more valuable?”

While the mathematics will have to escape me because full performance distribution curves aren’t available (BP’s percentiles are currently down), suffice it to say there are a couple of reasons a team would choose a closer guarantee of good performance than an outside chance of great and bad performance, even with a similar expected value.

The first reason is that some teams are incredibly risk averse. Consider you, ceteris paribus, have a team projected to win 85 games and are looking to add another player. Do you add player A, who has an expected value of three wins but can equally likely produce zero or six, or do you add player B, who has an expected value of three wins but could equally likely produce two or five?

This is a false choice in that all players’ distributions are different, but in this thought experiment, I would personally choose the latter, for a couple of reasons. The first is a higher guarantee of certainty; your range of possibilities is now 87-90 wins instead of 85-91, which gives you a better chance of being competitive for longer in the season, despite the fact higher-end projections give you a better chance of a division or playoff spot if you hit the lottery. There’s a better chance you find yourself in an OK spot at the trade deadline, for example, and that could buy you time to make another improvement

The other reasons aside from risk aversion is that you’re not just buying a single year with a Machado or Harper, right? You’re really buying a Markov chain of projections, where the n+1 year depends upon the outcome of the prior. Each n+1 year projection depends on what value you receive, not what you might receive. Sure, you could get the higher number and the long-term projections scale upward, but you are also more likely to get a lesser-than-ideal number than the Machado/Harper, meaning there is no guarantee you can rely on those players in year n+2 and beyond.

Consider the Yankees, for example. They eschewed Machado and instead opted for signings like LeMahieu, Gio Urshela, and Luke Voit. Those cost them essentially nothing, but we know for a fact that even despite LeMahieu’s stellar performance, these collective players have wider distribution curves than a perennial All Star like Machado. I would not bet good money that Urshela is hitting at a .300 clip as a regular starter in 2021, while Machado I can reasonably assume he will be an excellent player then.

And when you’re a team that makes a ton of revenue and makes future business decisions based on the certainty you’ll be good, having players essentially baked into good performance allows you to take risks elsewhere, and to map out your expenses in a more predictable way. If those guys get hurt and it forces them to miss a postseason spot or make a rash decision in 2020 or 2021, it could have more irreparable harm to their revenue and outlook than if they just paid the big bucks from day one.

This is all subjective, of course, and it’s also team-dependent. When you’re a smaller market team, it makes more sense to get players with more upside and possibly more risk; if fans are expecting you to possibly perform poorly, like the Rays, you can try to find players who have the possibility of a star season but could also bust. A lot of this depends on what your revenue-based win curve looks like, in a word. Yet we’re talking about Machado here, not waiver wire selections.

When you buy an expensive, but young, free agent, you are getting a much more narrow curve than most players, and you’re getting a much better guarantee for good performance than almost any investment in the sport. There are arguments here and there why you shouldn’t, but with the money flowing into the sport at the rate is, and the number of teams that could miss the postseason by just the amount Machado would provide, you could say they would have loved that high-probability outcome.