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Why did the barbaric king build the amphitheater in "The Lady, or the Tiger?"?

The key is to remember that the king is semi-barbaric. His ways of administering justice are unusual. He needed to build the amphitheater in order to have a place to bring a criminal to choose between the two doors, behind which stood the lady and the tiger. It was a practical reason--he couldn't do this is a small room or out in a forum. He needed a contained area.

An amphitheater also contained seating for the towns people to watch justice being administered. Besides being an entertainment for those not in the arena, the towns folk could witness first hand this unusual for of justice, and it may deter some of them from becoming criminals and suffering a similar fate. Also, punishment is sometimes more cruel and barbaric when an audience is watching.

The king of "exuberant fancy" and "irresistible authority" in "The Lady, or the Tiger?' builds the vast amphitheater, with its encircling galleries and mysterious vault and passages, as "an agent of poetic justice, in which crime is punished, or virtue rewarded...by an impartial and incorruptible chance."

With irony the author describes an autocratic king who satisfies his own fancies, but considers this satisfaction as "impartial and incorruptible" since he leaves the final verdict to Chance.  Of course, Chance offers no real justice; "poetic justice" is the occurrences of what one has hoped would happen. 

This ampitheater affords the cruel king of "irresistible authority" a setting in which he can have his cruelties enacted while calling them "justice," as he gives the accused a chance to select his/her own punishment or reward by pointing to a door.

Like Greek and Roman amphitheaters of the time that were intended for gladiatorial purposes, the amphitheater in this tale is intended for a brutal but absolute administration of justice. The semi-barbaric king enjoyed the spectacle of watching one who is accused deciding their own fate. Behind one gate, a beautiful wife; behind the other, certain death. The outcome of this system is essentially luck of the draw, like flipping a coin.

This historic period saw the rise of many judiciary practices that were, at best, barbaric. The devouring of a human by a tiger is no different. The twist that the author puts into this story lies in the jealous mind of a woman -- did she lead her love to life or death? Intentional ambiguity is used to make the readers decide for themselves.