I think if a pro sports team could figure out how to make it so the opposing team couldn't tell which one of their players is which they would choose to. Good luck running a shift in the MLB if you don't know if the batter is swinging lefty or righty.
Also I think pro sports teams is probably one of the best examples that public information doesn't make perfect decisions. Anybody that seriously follows a team can tell you about numerous bonehead (perhaps bonehead in hindsight) trades that their team made. Or players that were offered a deal well beyond what they were worth.
Athletes leave teams all the time over disagreements over compensation. At least in the NHL it's not uncommon for a player to take the team to arbitration over their salary [1].