In 2062, the Cape Fear Swamp Foxes' then-rookie reliever, Kalaila Cobar, won the JL Jason Egan Reliever of the Year Award. This was an unexpected development, even by the Swamp Foxes' GM, who didn't think he was even in the running.
Cobar's achievement was brushed off by many GMs as "a kid having some fluke year," "Atlantic favoritism," or "Where even is American Samoa?" But I'm here to provide a deep dive into his accomplishment, to see if we can understand more about its significance, and how it came to be.
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The Historical
Cobar never held the "closer" role for Cape Fear. He filled in for closers David Wade or Jian-Cheng Shi in a few isolated save situations, racking up a total of five saves on the season. As progressive as the league has become, its GMs can still be very traditional in some ways, rarely considering a non-closer for the league's top reliever award.
The last pitcher to win one of the league's Egan awards with 10 or fewer saves was Long Beach's Scott Everard, who had 8 saves in 2056, and 10 saves in 2057, winning the award both years.
Another notable comparison may be Lorenzo de' Medici, who won the award with as few as three saves in his 2050 campaign. For de' Medici, it took an ERA in the '1's, with over 200 strikeouts, more than the total of most starting pitchers. But like Cobar, de' Medici was a lefty super-reliever whose team reserved him for later-inning "stopper" situations, but not to finish games.
The main difference, though: all the players mentioned were huge "innings" guys, pitching 100+ per year. And Cobar wasn't, with 80.1 innings pitched in 2062, he barely scratched the top twenty in terms of total workload. So how did he win the Egan?
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The Voting
Cobar won the award with 119 points received in voting, and 8 first-place votes, more than any other candidate.
I think the key to understanding how the voting shook out is to classify the candidates under certain "camps", thereby understanding the motives of the voters who voted for them. From that, we'll see how Cobar rose to the top.
So let's review all the other candidates who received multiple first-place votes, and see how they stacked up.

Dougherty led the league in IRS% (12.5%), and was second in saves (44), only one behind leader Sergio Tormes. He also had the benefit of playing for the JL pennant-winning team, a result which may have factored into the voting. Quite fairly, it's hard to imagine Charlotte having the success they did last year without Dougherty at the back of the bullpen. I would classify his proponents as being in the "team success plus irreplacability" camp.

Stone led JL relievers in ERA (1.27), batting average against (.156), WHIP (0.69), and ERA+ (332). His performance was so dominant in the situations in which he pitched, that there would've been no question as to him winning the award, but for his lack of traditional stats, with only 1 save, and 0.6 WAR on the season. His first-place votes (which included the one from Cobar's own GM), were clearly from those in the "dominance regardless of game situation" camp.

He led the JL in saves with 45, so his first-place votes were likely from the "saves über alles" advocates. His 4.54 ERA shows that he had more than his fair share of struggles, and in fact was a slightly below league average pitcher, according to his 98 ERA+. The voters in this camp likely sorted by saves and voted down the list.

Bristcoe was the ultimate setup man for Montreal, notching 37 holds, which is one of the top-ten single season marks of all time. He was 2nd in the league in innings pitched for relievers (132.1), which led to him also being 2nd in WAR (2.3). He should've easily finished top three in voting, but was left off several ballots, presumably due to his good-but-not-great ERA (3.26), and lack of saves (2). I doubt that any of his voters were aware of his holds achievement, but I know that BBA voters often check for super-relievers on the ballot, sorting by innings. I'd guess that Bristcoe's two first-place votes came from voters who value innings above all else, and who consider a 130-inning reliever worth double that of a 65-inning reliever. And they're not wrong in that regard.

A great dark horse candidate if I've ever seen one, McFarland converted 26 saves in 26 chances, for a perfect (1.000) save percentage. He also led the league in WPA (3.3), and finished 4th in ERA (2.47) and ERA+ (178). His chances were likely hurt due to his relatively low innings count (58.1), and therefore his lower WAR (0.8). You could argue that McFarland was the most dominant JL reliever who actually was his team's closer, though astute voters should have noticed his 50% IRS% and noted that he wasn't actually as dominant as his ERA would seem to indicate. But still, that perfect save record is sexy. In fact, let's just call this camp "saves are sexy".

Her two first-place votes were the only votes she received, and we are obliged to mention her here. She finished 15th among JL relievers in ERA+ (140), and 16th in ERA (3.15). I would define her voters as "Sean Marko".
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Conclusion
Cobar won the Egan because he appeased both the sabermetric side, finishing 3rd in WPA (2.7), 2nd in ERA+ (207), and fifth in FIP (3.13) — and the traditionalists, finished T-3rd in WAR (2.0), 2nd in wins (12), and 2nd in ERA (2.13). He likely lost some votes from the pro-saves camps, but won over some voters with his low ERA and respectable innings total. And Cape Fear making the playoffs probably gained him some consideration from the "team success" camp, too.
So how can you, reader at home, have your very own Egan-winning pitcher? Well, from what I can deduce about the voters, most of them are sorting by WAR, ERA, or saves, and using innings as a sort of qualifier. So you want your candidate to rank high in as many of those categories as possible, and be above some arbitrary innings threshold of 60-70 or so. Making the playoffs doesn't seem to hurt either. So I suppose the answer is that you want a little bit of everything.
I also want this to be a community forum discussion, so if you want to share your voting agenda or which "camp" you subscribe to, please do so. It's always enlightening to hear about how GMs value different stats.