Now is why the July trade market went bananas

Admit it, you spit your coffee onto your computer screen when you first saw the details of the trades for the likes of Andrew Miller, Aroldis Chapman, Will Smith, and if we expand to the last calendar year, Ken Giles and Craig Kimbrel.

Just like for every other position, we have a standard perception of how much relievers are valued in the trade market based on past history, and this past year's deals took that perception and shattered it into pieces smaller than Big Papi's MVP hopes.

See, if you get lucky, this reliever for which you just gave up a king's ransom will pitch you 30-40 innings from August 1 through the end of the postseason. Small sample size means high variance, and there's no guarantee that these will even be quality innings. And even if they were, in many cases, it didn't matter in the standings one bit. Look at the three best relievers that were moved at this deadline -- Aroldis Chapman, Andrew Miller, and Mark Melancon. Their new teams clinched the division by so much that it's almost mathematically impossible for their new reliever acquisition to have made a difference. In addition, Chapman and Melancon are going to be free agents at year's end. Will Smith's Giants barely made the playoffs; Giles' Astros didn't. So between the seemingly excessive acquisition cost and the limited impact of many of these relievers, it would seem intuitive that these deals weren't worth it. However, the real value of the trades still hasn't come to fruition.

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Anyone who follows baseball somewhat closely is going to tell you that the point of most of these trades was "for the playoffs", and I agree. But what may not have been in consideration is that a trade "for the playoffs" means something different depending on the position of the acquired player.

First, consider an everyday position player, such as Jonathan Lucroy or Eduardo Nunez. They play roughly every day in the regular season and in the playoffs, so the value of a position player in the playoffs is roughly the extrapolated value a team gets from them in the regular season, ignoring the fact that playoff games "count" for more than regular season games.

Next, think about a starting pitcher. He will pitch about once every five games in the regular season and roughly once every four in the playoffs. That would be a 40-start regular season extrapolation instead of about 32 starts, so using simple math, he provides value to his team about 25% more often in the playoffs.

Now consider an elite, high-leverage relief pitcher such as Andrew Miller. Because of the more frequent rest days and the importance of the playoff games, managers will use their best relievers more often and their worse relievers less often in the postseason, especially in high-leverage, late-game situations. For reference, let's look at the best reliever from each of the four Championship Series teams last season. Roberto Osuna pitched in seven of the Blue Jays' 11 postseason games last season, which would be a 162-game pace of about 103 appearances. He had an outing of two innings and another outing of five outs recorded, so on a per-inning pace, that'd extrapolate to about 123 innings. Hector Rondon's 2015 postseason would equal 90 appearances and 90 innings over 162, while Wade Davis' would be 81 appearances and 108 IP. Are you ready to hear Jeurys Familia's usage last postseason? Familia appeared in 12 of the Mets' 14 games, throwing more than one inning in five of those appearances. That extrapolates to a full season pace of 139 games and 170 innings.

Relief innings such as these are generally more valuable than starting pitchers' innings because the manager can choose to deploy them at any time in the game, ideally in the highest-leverage situation, not to mention the fact that these innings pitched by elite relievers are usually of higher quality than the starting pitcher's. Now, if we assume that the average relief ace pitches about 60 innings in 60 appearances during the regular season, we are getting at least 1.5x their relative value in the playoffs. In Familia's case, that multiplier becomes over 200%.

At this point, one way or another, you should already be convinced that an elite reliever brings an insane amount of value to a team's playoff chances. However, there's one more kicker (especially in regards to the Cubs and the Nationals). The inclusion of additional elite relievers to a team that already has some is exponentially inverse to the value they provide. For example, Wade Davis appeared in only half of the Royals' postseason games because Kansas City had plenty of other awesome options such as Kelvin Herrera and Luke Hochevar. The Mets, on the other hand, had no reliever that was even somewhat comparable to Jeurys Familia, which helps explain his ridiculous workload.

These elite relievers are clear upgrades over the alternatives. The drop-off between them and the next-best reliever (Hector Rondon for the Cubs and Shawn Kelley for the Nationals) is fairly sizable, and the difference between Rondon/Kelley and the third-best reliever is even larger. Other than a ROOGY or LOOGY appearance, rarely will a team need more than two high-leverage firemen in a given playoff game, so the marginal upgrade that these relievers represent is massive both in terms of quantity and quality. In fact, underusage of a team's shiny new relief weapon would be the only reason that they would "lose" their respective trade.

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Austin Yamada is a contributing writer for Beyond the Box Score.