One appointment by two parties to settle a dispute

Answer: Option A

One word substitute is Arbiter.

Arbiter: a person who settles a dispute or has ultimate authority in a matter.
Mediator: a person who attempts to make people involved in a conflict come to an agreement; a go-between.
Negotiator: a person who conducts negotiations.
Middleman: a person who buys goods from producers and sells them to retailers or consumers.