Something unarbitrary is deliberate or logical — it's not random. If a kindergarten teacher keeps a list of whose turn it is be the line leader, updating it daily, that's an unarbitrary method of giving everyone a turn.
The adjective unarbitrary is formed by adding the prefix un- ("not") to arbitrary, which means "done at random." So when you do something in a methodical way, on on purpose, it's unarbitrary. Instead of letting friends arbitrarily take a swing at the piñata, letting them fight over control over the bat, find an unarbitrary way to do it. Perhaps you'll decide to have them line up in alphabetical order.