We already know where the ax is going to fall most harshly in the 18,000 job cuts that Microsoft Corp. (NASDAQ: MSFT) announced Thursday morning. The smartphone business that the Redmond giant acquired from Nokia Corp. (NYSE: NOK) for $7.4 billion will suffer about 12,500 job losses. The company did not say where the rest of the cuts would come from, but there are plenty of suspects to choose from.

On a global basis, as of the end of June 2014 Google Inc. (NASDAQ: GOOG) snags more than two-thirds of the search market. The second largest share goes to China’s Baidu Inc. (NASDAQ: BIDU), with just over 18%. In third place is Yahoo! Inc. (NASDAQ: YHOO) with 6.7%, and Microsoft’s Bing comes in fourth with a 5.6% share, according to research firm Net Marketshare. Note that the combined total for Yahoo and Microsoft still trails Baidu — and Baidu has no penetration at all in the U.S. market.

At the end of March, U.S. users carried out 19.4 billion searches, with Google getting about 13.1 billion, Microsoft nabbing 3.6 billion and Yahoo getting 1.96 billion. Separately, neither Microsoft nor Yahoo has a ghost of a chance to unseat Google’s stranglehold on U.S. search, but combined the two make a reasonable number two with about 5.6 billion monthly searches, or nearly 29% of all U.S. searches.

In 2013 Microsoft’s online services division posted an operating loss of $1.28 billion on revenues of $3.2 billion. That was much lower than the $8.13 billion loss in 2012, which included a $6.2 billion goodwill impairment charge. Part of the improvement was also due to lower guarantee payments to Yahoo as part of the two companies’ search deal.

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Microsoft’s total online advertising revenue in 2013 was $3 billion, up $409 million year-over-year, and the company attributed that to an increase in search advertising revenue. Of the company’s $77.85 billion in 2013 revenues, the online division contributed 3.8%, but the operating loss may be something the company can do without.

It is no secret that Yahoo is not happy with its search alliance with Microsoft, and the coming initial public offering of Alibaba may give Yahoo the ammo to do something about its unhappiness. With a pocketful of cash, Yahoo CEO Marissa Mayer may want to pry Bing out of Microsoft.

The question Microsoft’s Nadella has to answer is does he want to continue losing money on search in order to maintain some kind of foothold there while the company struggles to become a player in mobile devices, or should he strike the best deal he can with Mayer and make search her problem? Our bet is that Bing could go to Yahoo for the right price.

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