There is no way in hell that Microsoft would sell the Xbox division. If anything, they would shut down the REST of the hardware and devices division and only keep Xbox.
Off the top of my head, here are five big reasons:
1. It's a proprietary box that sits in front of your tv. Microsoft are currently salivating over Apple's "app store" and iTunes profits. The Xbox, sitting right there in front of your tv, is their strongest and best hope for getting people to buy into the Microsoft proprietary hardware/services pipeline. Windows 8 phone, Windows Surface, Windows 8 desktops, etc...on those devices the app store and subscription services have not been nearly as successful as the company would hope (outside of Office 365), but people actually buy, use, and buy content on their Xboxes. Apple has came to huge profitability through selling content on proprietary devices. Microsoft wants to be like Apple, and the Xbox is their best foothold in the consumer market right now. Getting rid of their most successful hardware product and brand, which helps them build their proprietary store and services profits across a variety of devices, would be suicide. It goes against the long term vision of the company.
2. Hell, they've integrated the Xbox brand into Windows 8 itself at this point. It is THE Microsoft brand for entertainment content. Xbox Music and Xbox Music are built into Windows 8 as the successor to "Zune Music" or whatever the name was before. They just recently integrated the brand into their flagship product (Windows 8) because it is successful. Getting rid of it now, when their strategy is obviously to capitalize on the popularity of the brand, would be stupid.
3. Xbox Live subscriptions. A future where income increasingly comes from software and services subscriptions (Azure server farm rentals, Office 365, etc) is a stated goal of the company. They've already got the "Azure" server platform and server farms built. They've got to keep them running regardless for other business purposes, and Xbox Live (and subs) usage helps them maximize the value of this large investment.
4. Who would buy the Xbox brand? And, who has the money and infrastructure to take over something as big as Xbox Live and manage it properly? Few companies have server farms that big, so they would have to lease them and likely pay for support services as well. Running Xbox Live isn't as simple as paying Amazon to set up a few cloud servers. What about the proprietary technology in Xbox One? e.g. Kinect, or even the server features. I would imagine that some of the server features, like cloud support with Azure, are set up to run on specialized proprietary server setups... they could likely run on other servers as well, but the whole thing sounds like a nightmare. Also, what about content licenses? What about developer support for game studios? ...All that could be worked out, but it sounds like a real headache for any buyer. A really expensive headache.
5. Looking back at point #2, what would Microsoft replace the brand with? They've spent years building up the Xbox brand's success. They've integrated it into Windows 8. For Microsoft, establishing popular brands, much less popular proprietary devices, seems difficult. I doubt they would unload the only real success they have.
