I think it's very solvable: Given N numbered judges, you have both sides, perhaps plus an allegedly neutral third party, submit a number from 1 to N in a sealed envelope. Then, in full view of the public, someone opens the envelopes, and you add up the numbers and take the result mod N.
If even one of the parties submits a fairly chosen random number, and manages to conceal it from the others until they've submitted their numbers, then the result will be a fairly chosen random number. Then no one involved will have standing to complain that the result wasn't chosen fairly enough.
If even one of the parties submits a fairly chosen random number, and manages to conceal it from the others until they've submitted their numbers, then the result will be a fairly chosen random number. Then no one involved will have standing to complain that the result wasn't chosen fairly enough.