Not sure the specifics of the Amazon rumor, but as was pointed out Amazon has dual sourced for their prime business for some time now. With the additional DC's that they are adding across the country, they have transitioned more and more of their "Prime" business from Air to Ground which you could assume would result in lower revenues across the board from both FedEx and UPS.
Keep in mind that FedEx Home Delivery delivers on Saturday instead of Monday which would enable shippers that dual source like Amazon to ship with one carrier for Thursday/Friday shipments that would be delivered on Saturday when Time-in-Transit is important to their customers.
As for time-in-transit, there will always be areas where one carrier is superior over another. There are companies that are well aware of which carrier has better Time-in-Transit to certain lanes and if they are dual sourcing and they were paying the same price, it would be foolish for them not to use the carrier with the lower transit time.
Price has been mentioned quite a bit in this thread. If you purchase an item on Amazon Prime you are paying an annual fee for shipping ($79 last I checked), but nothing for shipping and no minimums. This means you could purchase a $5 item no shipping. Do this enough and that $79 which also pays for other things on Amazon in theory (Think Kindle Library, Instant Video, and Cloud Services) no longer covers your shipping costs. If you read some of the recent postings about their stock, they are hemorrhaging profits in their transportation costs. This is the main reason for opening the DC's which forced them in many states to collect sales tax, something they tried a long time to avoid. Investor news would also point out a sizable bond offering to raise money for, opening yet more DC's.
So if you were Amazon, you would move to the low-cost provider no matter what the reason. UPS delivered more additional packages last Thursday from a normal day that FedEx does on a normal day. FedEx has been opening new Ground facilities across the country and has to fill them. Despite their news releases to the contrary, it's obvious they are not concerned about margins when they cannot fill capacity.
The question is, does UPS really want this business? If we are losing money, definitely not. We are not the Post Office and do not lose $25M daily. Do we need to do a better job in the next contract controlling costs? I would concur with previous comments in this thread that stronger 9.5 language that results in the addition of more routes could take care of that, but at what cost to those employees?
A $28/hour employee who receives 15 hours of OT for 45 weeks (Vacation, Holidays, Option Days, etc) is making an extra $28K a year. What happens when we add more routes, providing better service to our customers and eliminate that OT effectively cutting annual salary by a third? Sure more time with the family is worth it in the long run, but I fear too many people have come to depend on this additional money.