MINNEAPOLIS — In the end, the Courtship of Roger Clemens (Part III) came down to want versus need.

The Red Sox wanted to add Clemens to their starting rotation; the Yankees, by contrast, needed to add him.

That sense of desperation, as reflected in the Yanks’ $28-million, pro-rated offer, won the day. The Red Sox, according to a source with knowledge of the negotiations, offered a prorated $18-million deal.

The $10-million discrepancy was too great for Clemens to pass up.

“Can you blame him? said one Red Sox player yesterday.

Outbid and outfoxed by the Red Sox on Daisuke Matsuzaka, the Yankees were determined not to be outspent again. In the first five weeks of the season, the Yankees have already used an astounding 10 different starting pitchers.

Carl Pavano and Jeff Karstens are likely lost for the season. Phillip Hughes will be sidelined until mid-June at the earliest. Mike Mussina and Andy Pettitte are neither as durable nor as effective as 2003, when they were last teammates.

That predicament left the Yankees with Chien-Mong Wang as their only proven, successful starter.

“Having Clemens is a plus,” said Sox slugger David Ortiz. “But I think they needed him more.”

That’s why the Yankees agreed to reward Clemens with the biggest single-season contract in the history of the game, a contract that will actually cost the Yanks a stunning $40 million (again, pro-rated) when the luxury-tax ramifications are factored in.

The Red Sox’ inflated payroll — approximately $135 million — left them with less flexibility than a year ago, when under similar circumstances they offered Clemens a pro-rated $21-million salary.

But after adding J.D. Drew, Julio Lugo and Daisuke Matsuzaka, the Sox self-imposed budget constraints had them proposing a somewhat more modest figure this time.

The Red Sox entered yesterday among the top-three teams in the league in virtually every pitching category —staff ERA, starters ERA, bullpen ERA, shutouts, saves, on-base percentage allowed — but their downsized offer was hardly the mark of a cocky team.

Five weeks do not a season make, and the Sox are under no delusions about their pitching invincibility. Sooner or later, injuries will strike their staff and the team will need to rely on its organizational depth. It’s quite likely that some night this summer, on a night when the Yankees are pitching a seven-time Cy Young Award winner, the Red Sox will be sending out Kason Gabbard or Devern Hansack or Kyle Snyder as a fill-in for a fallen starter.

But given the economic stakes, the Red Sox are willing to take that gamble.

There are risks involved for the Yankees, too. For the last three seasons, Clemens has pitched in the weaker league and, last year anyway, arguably the weakest division in baseball.

Returning to the American League and the A.L. East in particular will represent a significant upgrade in competition for Clemens, who will turn 45 in August. Clemens maintained yesterday at Yankee Stadium that he can still do things at 45 that he could at 25, but that remains to be seen.

Back in the A.L., Clemens won’t have the luxury of facing pitchers every three innings, and he’ll find the emphasis on offense a challenge he may not relish. Clemens hopes to cap his career with a third championship, but in so doing, he puts his legacy on the line. It’s doubtful that he’ll finish this season with a sub-2.00 ERA, as he did just two seasons ago in Houston.

The Red Sox reluctance to go all-out for Clemens may haunt them in October, should they face the Yanks in the ALCS for the third time in the last five seasons. But if the loss of Clemens to their archrivals was demoralizing, the Red Sox were keeping their hurt well-hidden yesterday.

“We’re doing OK right now, don’t you think?” said Josh Beckett

“It’s a good choice,” concluded Ortiz. “But we’ll be fine.”

smcadam@projo.com