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An Arbiter in Sehto is a human being chosen as an interface to the human world by the Legendaries.
An Arbiter fields two primary purposes:
For one, he is designated as the one person allowed to pass final judgement on his fellow human beings when they're at the mercy of the legendary Council. The Sehto legendaries discuss decisions largely amongst themselves, but they will look to the Arbiter for the final call. Since this privilege is granted by the legendaries, it can, of course, equally be revoked at any time, effectively nilling its power in theory, though in practise the legendaries are quite good at taking advice - from their current Arbiter, Jagdish Tsukinaka, at least.
For two, rather unlike what the term might imply, the Arbiter is responsible for the execution of the will of the legendaries once a consensus between him and the legendary Council is attained.
While a strong supporter of principle of an Arbiter, Jagdish Tsukinaka only reluctantly fills the role himself, because the second chore is not entirely to his liking - while he fully agrees the victims of the Council deserve their (arbitrated) fates, emotionally, he would rather delegate the task of its execution to someone else. Rationally, he deems himself the best candidate, as it is precisely his lack of sadism that makes his actions so meticulously and precisely applied.