The king's arena is an amphitheater, an open arena of oval or circular shape surrounded by rising ground with tiers of seats. There is a central area where contests, etc. are held. It has gallleries, which are something like balconies, passageways, and "a mysterious vault," an arched space underground where the tigers are probably kept. This arena is the king's agent of "poetic justice"; that is, it is in the arena where the king's prisoners meet with their punishment. Since he has them choose one of two doors from which a maiden or a tiger can come, the king feels that the prisoner's get what they deserve (poetic justice). The "impartial and incorruptible chance" of the choice by the prisoner is what the king considers justice.
Imagine a large round open arena. Seats are surrounding a large open area. We are told the following about the arena: "the king's arena, a structure which well deserved its name, for, although its form and plan were borrowed from afar ...When all the people had assembled in the galleries, and the king, surrounded by his court, sat high up on his throne of royal state on one side of the arena, a door beneath him.. an amphitheater, on the other side of the enclosed space, were two doors, exactly alike and side by side." Imagine a large arena like the Romans built for their gladiators. Think of a large modern football stadium, only this would have been built of wood and stone.