They may, but wisely. You are Bryce or Manny. Do you want money, chance to win, or both? The White Sox right now offer money and a possibility in a year or three to be that team year in and year out. What does a team like the Yankees offer? Money, win now, win future. Very attractive. What would it take for the White Sox to make it better for the player? CASH. And it's already way too much if the rumors of 30+ per season are true for either of them. By not being smart the White Sox run the risk of losing a decade in progress.
Tanaka would have cost $200-225 for the White Sox to land him. For a guy that has never throw 200 innings and has been a 1-2 WAR player the last two season, yes, we did great by losing out.