Judge, adjudge, adjudicate, arbitrate mean to decide something in dispute or controversy upon its merits and upon evidence.
All these words imply the existence of a competent legal tribunal or of its equivalent. Judge implies mainly the investigation of evidence on both sides, a comparison of the merits of each case, and a decision as to where the truth lies.
Adjudge stresses decision by a court either at the end of a trial or during a legal process.
Adjudicate, on the other hand, stresses formal deliberate determination of an issue by or as if by a court and often the pronouncing of a judgment, sentence, or decree.
Arbitrate implies deliberate determination of a matter in dispute by one or more persons who constitute a tribunal that may or may not be legally recognized and who are usually acceptable to both sides to the controversy.