Kenneth G. Wilson (1923). The Columbia Guide to Standard American English. 1993.
arbitrate, adjudicate, mediate (vv.)
To arbitrate is to have a third party, chosen by the disputants, decide a dispute; to mediate is to serve as mediator, a third party who will attempt to conciliate the disputants, to find some compromise agreement, and to adjudicate is to serve as a judge, to hear and decide a case. Adjudicate suggests that neither conciliation nor compromise is a goal.