But even he has a problem with the statistic that has so helped his bottom line.

“I’m thankful for the saves stat the previous few years,” Reed said. But even so, he added, “the save stat is overhyped.”

It indeed may be, but the way bullpens are structured nowadays, the closer and the save receive the most attention. All-Star rosters are stuffed with closers, not middle relievers or setup men. And which factoid can fans recall more: that Jeurys Familia had a 2.55 earned run average in 2016, or that he had a major-league-leading 51 saves for the Mets?

“A save — it is an arbitrary number,” Los Angeles Angels Manager Mike Scioscia said over the winter . “But at least it’s a guideline that gives you a little base line of a player’s performance.”

The save became a statistic in 1959 thanks to the baseball writer Jerome Holtzman. He once said that the term “save” had already been in use by the early part of that decade and that he had only developed the first formula for it.

In that era of baseball, the complete-game starters were beginning to fade, and Holtzman wanted to recognize the growing number of relievers who logged important innings in games. The team’s best reliever was usually called the stopper or fireman and often enough asked to work multiple innings. With the save, there was a way to quantify those who finished victories.

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Goose Gossage made the Hall of Fame on the strength of his feats as a closer.

Credit
Ron Frehm/Associated Press

By 1969, the save was officially recognized by Major League Baseball and later refined to its current requirements.

Rollie Fingers, Goose Gossage, Bruce Sutter, Lee Smith and Dan Quisenberry brought notice to the statistic in the 1970 and ‘80s, often recording multi-inning saves. As relievers grew more prominent, 40 saves in a season became routine in the 1990s and 2000s because of closers such as Dennis Eckersley, Trevor Hoffman and Mariano Rivera, the career saves leader with 652. Francisco Rodriguez, a former Mets closer, set the single-season saves record with 62 with the Angels in 2008. As those number rose, so too did their financial value to the players who piled them up.

Now, modern bullpens are built around the one-inning closer, normally the team’s best and highest-paid reliever. Those bullpens are also built around the save, for teams try to get their closer as many save chances as possible — as opposed to assigning them what might be the most difficult situation of the game, regardless of the inning in which it comes along.

“If we played in a nonemotional, noncompensated, neutral environment, I think we would flush that saves stat down the toilet,” Houston Astros Manager A. J. Hinch said during the off-season.“But we don’t.”

“There is always going to be a constant pull and tug between players, compensation, manager and analytics about what’s the smart thing to do,” he continued. “But by far the smartest thing to do is win the game. That makes people like me happy.”

So is it time to rethink the save, or simply get rid of it, to let managers approach the game feeling less restricted in how they use their bullpens?

“Doing away with it is not the right thing,” said Drew Storen, a former Washington Nationals closer and setup man and currently a reliever for the Cincinnati Reds, a team that has shown a willingness this season to dispatch with traditional bullpen roles. ’“Guys are rewarded because of the save.’’

One issue with saves is that they are arbitrary. Why does a save have to be a three-run lead or less? Why not rely on advanced statistics that attempt to quantify the difficulty of a situation a reliever may find himself in? Why does pitching the final three innings of a game, regardless of the score, also count as a save?

Gossage, a Hall of Fame closer, said in February that there was nothing particularly impressive about holding a three-run lead for a single inning. Any reliever who “can’t save a three-run lead for one inning, they shouldn’t even be in the big leagues,” Gossage told N.J. Advance Media.

Those who disagree with his notion argue that there is an emotional component of save situations that cannon be discounted.

“It’s the ninth inning,” said Sandy Alderson, the Mets’ general manager. “You’re three outs away. I know how I react emotionally in the seventh, eighth, versus the ninth inning.”

Scioscia called closers a “special breed.”

“The last three outs, although logic tells you it’s just another three outs, my experience tells me it’s not,” he said. “And some guys have trouble pitching in tight games.”

But the biggest moment of a game comes not always at its end. In fact, the average game last season featured 11.3 base runners, meaning that the leadoff hitter in the eighth inning would be the opponent’s No. 3 or 4 hitter. The ninth would more likely start with the bottom of the order. So shouldn’t the setup men assigned to the eighth inning be better rewarded than they are? Shouldn’t they be treated almost as closers even if they are not piling up saves?

“To me, I don’t know why the save is such a big stat,” Reed said. “It’s such a weird topic.”

Saves are important, Reed said, but the heart of the order coming to bat in the middle of the game may be the more crucial moment.

“I honestly think that if you have a one-run lead in the fifth and you get out of it, that should be the save,” said. “And if you come in in the eighth with a three-run lead and nobody on, I think what happened in the fifth inning was more of a save than in the eighth or ninth.”

Then there is the even less-heralded work of a reliever who comes in without the benefit of a lead. Preventing your team from falling further behind in a close game could be considered a vital statistic. Right now, there is no stat that recognizes such an accomplishment, but Storen said there should be, maybe something called a “deficit hold.’’

“A lot of times, the biggest innings I pitched were keeping the score 6-5 and we’re losing,” added Miami Marlins closer A. J. Ramos, a former setup man. “I come in and keep the score close, and the next inning we take the lead or tie.”

To quantify the work of nonclosers, another statistic — holds — was developed in the 1980s for setup relievers who enter in a save situation and maintain the lead while recording at least one out. But even that does not necessarily satisfy Reed.

“I think the stats you should look at are the times you got out of a jam and your E.R.A., ” Reed said. “If you have a low E.R.A., you’re doing your job. When I had 32 saves, my E.R.A. was in the high fours.”

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Mariano Rivera of the Yankees after setting the major league career saves record in 2011. Many say a team’s closer can often be best used earlier in games.

Credit
Jason Szenes for The New York Times

Saves occupy such a rarefied place in baseball that they affect how the game is played. In the ninth inning of tie games on the road, or in extra innings, many managers decline to use their closer, preferring to send out other relievers and wait until the closer can potentially enter the bottom of the inning with a save opportunity.

The Cleveland Indians demonstrated notable flexibility with relief roles after acquiring the star reliever Andrew Miller from the Yankees midway through last season and turning him into a multipurpose fireman. It also helped that Miller was durable and had already received his big payday — a four-year $36 million contract signed before the 2015 season.

Miller had been handsomely rewarded by the Yankees for his previous bullpen work, even though he had very few saves before 2015. But when one of Miller’s former bullpen mates in the Bronx, Dellin Betances, tried to follow Miller’s lead and get $5 million in salary arbitration last winter, he was sharply criticized by the Yankees’ president, Randy Levine.

Betances is very good, but he has only been a fill-in closer for the Yankees. That was enough for Levine to argue that Betances did not deserve such a big payday.

“It’s like me saying, ‘I’m not the president of the Yankees; I’m an astronaut,’” Levine said at the time. “No, I’m not an astronaut, and Dellin Betances is not a closer.”

In the end, the arbitrator agreed with the Yankees and Betances ended up with $3 million for 2017 despite his three All-Star appearances from 2014 to 2016, and a 1.93 earned run average in that span — the fifth best in baseball.

So it’s no wonder that everyone wants to be a closer, with all those opportunities to pile up saves.

And as long as baseball’s compensation system continues to lavishly reward saves, Hinch said, managers will care about them. If it’s important to players, it should be important to the manager, too.

If the save ever goes out of fashion, Alderson said the financial dynamic surrounding it would change. For now, though, the smart money is on being a closer and coming into the ninth with as many three-run leads as possible.

“As long as the game is making those choices and those designations, so is an arbiter,” he said.

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