Well, Google and Yahoo go under the same owner, so theirs no point of them having a competition.
On the other hand theirs Google and Bing, Microsoft and Facebook have sided with Bing, their sorta like enemies with Google, when I mean sided with bing I mean that when you try a search on Microsoft products, eg: msn, it redirects to bing, same thing with facebook, if you enter text on the search box, results from bing would show up.
Now google wins all because it has the most traffic ever I saw in a blog by Mashable..
~Mashable~ Trusted content
Anyways, just a note..
Google < Most Traffic
Microsoft < 2nd most
Facebook 3rd most