The Benefits of Being Beta...

The Alpha will always fall, the Beta will always be there...

Being the alpha male is stressful...and stress is not always good, and genetic studies show the most successful breeding males are actually those who are nice to the ladies.

There are two options for an alpha male who is deposed:  A quick trip to the omega position of the troop and likely exile (or possibly an early death) when every other male inflicts payback for all the crap that was dumped on them when the deposed alpha was in charge.

But if a male baboon maintains friendships with females on his rise to the top, they will generally hang out with him when they are not in peak estrus when he loses his position.  Because of friendships formed he is not dropped to omega status and will generally be left alone by the other males.  And guess what:  when the new alpha is fighting off the challengers, those females will often sneak off with the friendly beta male.  

Baboon troops also have culture, which can change over time.  At least one troop of baboons seems to to give thanks.

The lesson of all of this: