Just a quick history lesson to alleviate some misinformation going around: The Rare buyout was the product of many different factors, and their sale and subsequent downward spiral cannot be blamed on any one party.
Firstly, Rare was sold because their contract to work exclusively with Nintendo expired. When that happened, Rare gave Nintendo two options: renew this contract and continue their partnership as a second party developer, or Nintendo could buy them out and make them a first party. Nintendo chose to do neither, effectively allowing Rare to go up for sale.
Why did Nintendo let that happen? Well, it coincided with a lot of change in leadership at both the Japanese and NoA branches of Nintendo. Without some of the higher ups that had previously been a huge part of the relationship with Rare, their relationship became more difficult to work with. The relationship slowly deteriorated, and by the time their contracts had come to an end, it was pretty much mutual that the two companies were ready to part ways. The new leadership (led by Mr. Iwata) didn't think they needed Rare anymore.
Nintendo didn't get along well with Rare's development philosophy (they didn't follow a "gameplay comes first" philosophy, instead have a more loose and artistic approach to creating concepts and seeing where they went), and Rare felt creatively constrained under Nintendo's guiding hand. Unfortunately, neither party realized how well their relationship actually worked until it was over.
Once Nintendo allowed Rare to be sold, Microsoft obviously won the bidding war. At first, both Microsoft and Rare were happy and excited to begin their relationship. Microsoft gave Rare the creative freedom they didn't get under Nintendo, which was what Rare wanted. Unfortunately, as is often the case with former-Nintendo partners, new found freedom turns out to be a bad thing. Throughout their years at Microsoft, Rare struggled to get their many random ideas to come together into realized game concepts. They also struggled to adapt to the new hardware, and to adapt to modern game development. Finally, they struggled to find an audience on Microsoft's adult-oriented gaming consoles.
After awhile, Microsoft realized that leaving Rare to their own devices wasn't working... without a corporate guiding hand, they worked hopelessly on prototype after prototype and never arrived at profitable and marketable end-products. They were hopelessly inefficient and without any strong direction, so Microsoft put their boot down and made them do something profitable- Kinect Sports games. Only now, after many years and a change in leadership under Phil Spencer, are we starting to see the classic Rare come out of their shells and become something the Microsoft company are proud to advertise.
Also, for anyone confused on the distribution of Rare and Nintendo IPs after the buyout, here's how it went:
-Most IPs that were created originally by Rare were owned 100% by Rare, despite Nintendo publishing them. The notable exception was, interestingly, Banjo-Kazooie, an IP that Rare created wholly FOR Nintendo and was out and out OWNED by Nintendo while Rare was with them.
-The Donkey Kong IP was always owned by Nintendo, except for, for some reason, the original Kongs created for Donkey Kong 64.
-That's right, as part of the distribution of IP ownership, NINTENDO owned Banjo, and Rare owned Tiny, Chunk, and Lanky Kong. When they split up, they made a trade, in which Nintendo allowed Rare to keep Banjo and related properties, while Nintendo owned everything Donkey Kong related. Nintendo didn't want their Donkey Kong IP to be divided, and they had no use for Banjo at the time because Iwata was shortsighted.
-Diddy Kong Racing was mostly made up of Rare made characters, but it did include some Donkey Kong elements. Nintendo retained ownership of Diddy Kong and the Kremling species, but everything else in the game, from unique characters to the environments and location names, stayed owned by Rare.
-Goldeneye 007 was complexly divided in rights between Rare, the developer, Nintendo, the publisher, and Activision, who ended up buying the James Bond video game rights.
Hope this cleared up some confusion for people.