"The Sox should bid on (player X) to raise the price for the Yankees to get him."

It's not best for the Red Sox to raise the price on a free agent whom the Yankees are going to get to make the Yankees pay a higher price for the player. Doing this would hurt the Red Sox because agents of future players will use these players as comps. For example, it's not a coincidence that the Yankees' first offer to Sabathia was a little higher than what Santana got with the Mets. We don't want that to happen with future players the Red Sox want when the Yankees are either out of the picture or not interested. The main, if not the only, example of when the Yankees are out of the picture for a potentially expensive free agent is when the Red Sox are going for one who plays a position the Yankees already have filled, like when the Red Sox signed J.D. Drew while the Yankees already had Cabrera and Damon for CF and Abreu for RF.

12/09 edit: When the Red Sox signed John Lackey during the 2009-10 offseason, his agent used A.J. Burnett's contract as a comp. Had the Red Sox driven up the price of Burnett during the 2008-09 offseason, it likely would have cost the Red Sox more to sign Lackey.

Before you decide the Sox should sign free agent X...

Before you decide that the Sox need to sign any free agent, one of the many issues you have to remember is the Yankees. If the Yankees have a need, an interest and a spot available for a free agent to get a lot of playing time, chances are that free agent won't be signing with the Red Sox. Steinbrenner simply will not let the Red Sox outbid the Yankees for a free agent both teams want.

"But what if the Sox offer $X million for him?", you might ask. The answer is that Steinbrenner will offer more. Whether it's a starting pitcher whom any team would want if money weren't an issue, or whether it's a position player of a position both the Yankees and Red Sox need to fill (examples: Damon for CF in the '05-'06 offseason, A-Rod for 3B in the '07-'08 offseason), the Red Sox won't offer more than the Yankees if the Yankees want that player.

Some potential examples for the 2008-9 offseason:
Sabathia
Teixeira (if you're under the impression that Lowell won't be healthy enough for the Sox in '09 and you want Youkilis to play 3B)

I'm not saying these 2 players will definitely be Yankees, as opposed to Vladimir Guerrero or Carlos Beltran when they signed longterm contracts with other teams. I'm saying they won't sign with the Red Sox if the Yankees want either or even both of these players. Steinbrenner has already made it clear that he's more willing to let a valuable free agent go to another team than he is for that free agent go to Boston.

"But the Sox outbid the Yankees for Dice-K", you might say. This isn't a valid argument because that was a secret bid. Unlike free agency, the Yankees didn't have the opportunity to be told, "The Red Sox bid $51,111,111.11. Will you bid more?".

If you don't believe this post, feel free to try to think of the last time a free agent who fit the criteria mentioned above signed with the Red Sox. At best you might be able to come up with one who re-signed with the Red Sox.