3 reasons I can think of for going the x86 route
1. The developer tools for x86 are phenomenal... I mean, it is THE platform to target. There is (almost) no library or language that doesn't support x86. ARM is up there, Power is less, and everything else is a crapshoot.
2. Swappable components. They are going with amd today, probably because AMD offers a good price/performance for them, they have decent GPU performance and good enough CPU performance. However, if one day intel offers a dirt cheap x86 processor with great power savings and a super fast GPU, well, the cost of switching over wouldn't be all that huge.
3. Future backwards compatibility. Like it or hate it, x86 will probably be here in 10 years. That means that the PS5 or the XBox tew will have a pretty easy time (well, not terrible at least) being able to support PS4 and XBox one games.
Microsoft gets the added bonus that the original XBox was x86 based, which means they MAY be able to support the old XBox games.
Developers get the benefit that their game should work pretty seamlessly against either the xbox or the PS4... This is an interesting thing... We may even see the birth of an XBox/PS4 compatibility library, since the two platforms are almost identical such a thing make be possible for the first time ever.
I never would have predicted that the two console giants would go for nearly the exact same hardware... Insane. AMD must have had a pretty good sales team working for them.