Please determine whether the soccer referee in this scenario is an
independent contractor, or an employee of one of the parties:
A.  Referee is a member of a local association (provides training,
administration) which appoints a game assignor.  Game assignor works
with soccer league and is paid by soccer league to assign games to
referee. Referee submits availability pre-season and receives schedule
of games. Refusing a game on schedule requires payment of fine to
assignor. Referee is paid by individual teams when he/she arrives at
game. There is no withholding from game fee. Referee sends post-game
report to league following game. Pay rate, fines, and policies and
procedures for game administration are determined by a contract
entered into by the local association and the league.

Is this referee an independent contrator, an employee of the league,
or an employee of the local association?

The ref sounds like an independent contractor on those facts, but the
IRS uses a 20-factor test to make the determination.  See
http://www.toolkit.cch.com/text/P07_1115.asp.  I recommend you run
your facts through the factors and see where you end up.