The big three cell phone carriers have a paradoxical relationship with the iPhone: each carrier needs the device to survive, but the more iPhones they sell, the more money they seem to lose. This morning Sprint released its fourth quarter results; Verizon and AT&T quarterly announcements came out last month. The reports all tell a similar story: Each company sold a lot of iPhones; each company lost money.

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Here's how the numbers break down, Sprint, Verizon and AT&T each signed up 1.8 million, 4.2 million and 7.6 million iPhone users during the holiday season, respectively. And for the quarter, the companies each lost $1.3 billion, $2.02 billion and $6.68 billion, respectively. And a big part of these losses come from iPhone sales. In order to offer the phones at reasonable and competitive prices, these companies subsidize the price of phone, losing money on each sale. The idea is that they'll make up offering the devices at a lost by locking in subscribers into their services. But the front-end costs are enormous. Verizon, for example, loses an estimated $3 to $5 billion on this buyer-support system.

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