Generally 37 year old players are less concerned about their careers, and more about stability and their family. Looking at it realistically, there are currently 8 major league caliber relievers who will be fighting for 7 spots.
Bailey, Lackey and Buchholz will probably get hurt or become ineffective early on in the season, and be replaced by Morales, Aceves, and the pitcher who was moved to the minors. That leaves two open spots for relievers.
Athicson would have to compete with De la Rosa, Webster, Bard, Wilson, and Mortensen for those spots, but not all of them will be relievers, and more players could get injured