The king in this story is described as "semi-barbaric," meaning that he is, in his essence, barbaric by nature, but has borrowed some more civilized ideas from some of the gentler nations which surround his kingdom. The public arena which he builds is actually, in its origin, one of these "borrowed notions" intended to reduce the barbarism of the nation by offering a space to display exhibitions of valor, which would supposedly refine and culture his subjects.
However, being at heart a barbaric king, the king warped this idea in the arena which he built. He doesn't have any interest in showcasing gladiators or religious people fighting with tigers in the hopes of improving their minds this way, as the Romans did. Instead, he prefers to use his vast amphitheater as an "agent of poetic justice," used for punishing crimes and rewarding virtues based on chance alone. When an interesting enough crime was committed, the fate of the supposed criminal would be decided in the king's arena.