None of these three ever were campaigning for the honor. In fact, they would rather answer 10 questions about the White Sox surprise success than one about their bounce-back performances.
"It's something that certainly I don't believe any of us are worried about, I know that," Peavy told MLB.com during an interview prior to the season's conclusion. "But it's a good problem to have."
"You know what, I don't think about awards or anything like that," Rios said. "Awards are just like for bragging rights. So, I'm here to play the game right and help the team win games."
That winning focus was there for Rios, Peavy and Dunn in 2011. The results were not.
Dunn turned in a thoroughly forgettable .159 average, with a career-low 11 homers and 42 RBIs. Rios' struggles were almost as pronounced through a .227 average, 13 homers and 44 RBIs.
As for Peavy, his 2011 season came to a close on Sept. 6 and left him with a 7-7 record and 4.92 ERA over 19 appearances of which 18 were starts. Those numbers came in his first full season following major surgery in July 2010 to reattach his right lat muscle to his posterior shoulder.
Three integral players dropping off so severely caused plenty of doubts around the White Sox entering the 2012 season. Until these veterans returned to their previous level of career-long excellence, those doubts weren't going away. Those same doubts didn't exist within the White Sox clubhouse and that belief soon played out on the field.
Peavy finished at 11-12, but could have easily won another four or five games with a little offensive support. He proved infinitely durable by making 32 starts, throwing 219 innings, striking out a team-best 194 and holding the opposition to a .234 average.
Doctors told the 31-year-old he would be as good as he's going to get 18 months after surgery. Peavy's 2012 campaign proved that assessment true, and set him up strong for free agency during this present offseason, assuming the White Sox don't pick up his $22 million contractual option and use their $4 million buyout instead.
Rios stood up a little more in his plate approach, making his hands go through the zone a little quicker. He also focused less on mechanics and more on simply making solid contact, leading to individual career-highs with a .304 average, 25 homers and 91 RBIs, while also playing in 157 games during a more comfortable move from center field back to right field.
"Basically, I'm trying to have a good approach at the plate and see good pitches and be a little more, if I can say selective, yeah, being a little more selective on my pitches that I'm going to swing at," Rios said. "Besides that, I'm just playing the game the way I used to play it."
With 41 homers and 96 RBIs, Dunn had the most pronounced change among this group. He still hit .204 and fell one strikeout short of tying the Major League single-season record with 222, but Dunn's job is to hit homers, drive in runs and get on base. He topped all Major League hitters with 105 walks and provided strong clubhouse leadership.
Whereas Dunn was striking out in tight game situations last August or September, he was delivering a game-winning, three-run homer off of Indians reliever Vinnie Pestano to keep the White Sox in the American League Central lead on Sept. 24.
"Adam Dunn, the person, is one of those off-the-chart teammates," White Sox general manager Ken Williams said.
"From Day 1 [of Spring Training], I said my goal was to have fun win, lose or draw. I didn't care if I hit 10 homers or 110," Dunn said. "You play so many games, there's enough anxiety. You don't need to add more. I wasn't going to go through that again. I had a great time but the goal was to make it to the playoffs and win a ring, and we didn't do that."
A playoff chance existed for the White Sox, who spent 117 days in first place, thanks in large part to Dunn, Rios and Peavy. Only one of them will be in line for AL Comeback Player, although the award legitimately could be split three ways.
Of course, the Comeback honor is another way to still talk about 2011's struggles, talk that should change for the better once the 2013 season begins.
"Really there's no way I could give it to one guy outright, but I don't think either one cared," Peavy said of the White Sox team-first approach. "I shouldn't say didn't care to win that award, because any time you are talked about in the light that you are talking about those guys in, it's an honor."
"I was focusing on this year," Rios said. "We've had a pretty good run. But last year is in the past and I've been focused on some other things that helped me have this good of a season. I took that year and erased it from my head and I focused on new things."