Quote Originally Posted by NeonGod
It can be rationalised that the upkeep cost ties into the effort of maintaining a full-strength force, which would include small reinforcements due to desertion.
It's not how the ancient (and even later) armies operated: no small reinforcements (few man), only the major ones - remember it requires lot of advanced management to do so: see some of reinforcements Alexander received etc. (exception - army in home, like the Roman one). But the troops were most likely dispersed among more experienced troops anyway.

And deserters were hunted for, but for army new large body of troops/mercenaries were enlisted (so no notification like "you are in the army now"). Same for later ages (great examples during 30 years war, where attrition was very high - desertion, diseases, underpayment, long sieges etc.)