What does he mean by poetic justice?

In "The Lady, or the Tiger?," author Frank Stockton defines poetic justice as follows:

This vast amphitheater, with its encircling galleries, its mysterious vaults, and its unseen passages, was an agent of poetic justice, in which crime was punished, or virtue rewarded, by the decrees of an impartial and incorruptible chance.

This is the classic definition of poetic justice: virtue is rewarded and crime is punished.
In the story, the amphitheater is understood as the vehicle of poetic justice. An accused criminal is put before two doors and asked to choose one. That door opens. The accused is faced with either a ravenous tiger, which eats him in front of a vast audience, or a beautiful maiden, to whom he is married with great fanfare. There is no trial, and no evidence is brought forth. In this society, every one relies on the power of poetic justice: the strong conviction that whatever is, is right. In this story, the two doors are considered a more reliable guide to justice than empirical evidence is. They are not simply considered the toss of a die but a way of ensuring that a person's guilt or innocence will expose him.