Arbitration was used on Ventax II. The first example of its use with an alien claimant was a dispute between a Klingon craftsman and the owner of a home over services rendered during construction. The two parties selected an arbitrator who then resolved the matter. They felt that this would be better than using the courts due to the involvement of an alien.
That example was used as a precedent by Captain Jean-Luc Picard in 2367 during a dispute with a person claiming to be Ardra, a figure from Ventaxian mythology. This impostor claimed that she had arrived on the planet to enslave the population per the terms of the Contract of Ardra. Picard, suspecting that the woman was a con artist, challenged the validity of the contract and requested an arbitration. He, along with the impostor, agreed to have Data serve as an arbitrator, as he would not be affected by emotion or bias. The proceedings did expose the impostor as a con artist, and resulted in the dissolution of the contract. (TNG: "Devil's Due")