By Rob Cox
The author is a Reuters Breakingviews columnist. The opinions expressed are his own.

Now that Yahoo has fired its chief executive, anything could happen to the rudderless Internet hodgepodge. Private equity firms, one of Yahoo’s founders and even AOL are said to be mulling bids. But consider a more radical option: a takeover by the rival most responsible for Yahoo’s fall from grace — Facebook.

It’s of course easy to marshal arguments why Facebook’s creator, Mark Zuckerberg, should avoid staining his company Yahoo purple. The social network is already growing rapidly. Revenue doubled in the first half to $1.6 billion with profit of nearly half a billion.

Moreover, Facebook is a private company without the $20 billion or so of cash needed to buy Yahoo. Since Facebook is just starting to profitably harvest its audience of 750 million users, the firm should stick to its knitting, or so the argument goes.

But Facebook has the ingredients to make Yahoo succeed, starting with a clear mission. Yahoo has struggled to articulate a vision beyond being the first page people see when they open a browser. Beyond that, nothing binds Yahoo’s pieces — news, photo albums, stock quotes, email, job listings and entertainment — together. They look like orphaned applications for a social network.

What unifies Yahoo’s bits and bobs is a relatively robust display advertising platform. In an overall crummy second quarter, display revenue increased 5 percent to $467 million. Facebook is still building out its capacity to sell such ads. A combination would make a compelling pitch to advertisers.

In search, both have a common nemesis in Google. They also have a shared partner in Microsoft, which owns a piece of Facebook and whose Bing search engine collaborates with Yahoo.

A deal would nevertheless be complex. A reverse merger, where Yahoo shareholders would be issued Facebook stock, isn’t usually done on such a scale and would almost have to be agreed by Yahoo’s board. Facebook is valued at around $80 billion. It also has no experience of M&A generally.

But it does have a management team led by Chief Operating Officer Sheryl Sandberg, whom Yahoo shareholders would be delighted to put in charge. That alone should be sufficient incentive to at least study such a seemingly outlandish idea.