Arbitrator who will hear Grievance in bounty case gets fired by the MLB

This is the scenario I've read horror stories about with respect to forced binding arbitration clauses when purchasing a new car or house. The arbiter is hired from a pool designated by the company. Arbiters who routinely rule against the company simply don't get more work...

Which is why any arbitration clause should have a mutually agreed arbiter. That solves the problem. Sure the losing side in any arbitration would be incentivized to not rehire someone who ruled against them but moving forward both sides of an arbitration have the opportunity to select an unbiased arbiter.