Who finds the accused guilty, the king or the accused?In one part of the story it says "There was no escape from the king's arena," and in another...

Chance is the judge of the accused in "The Lady or the Tiger?"  The king employs the public arena where he feeds his "exuberant and barbaric fancy."  In this amphitheater

was an agent of poetic justice, in which crime was punished, or virtue rewarded, by the decrees of an impartial and incorruptible chance.

When a subject is accused of a crime of "sufficient importance to interest the king," this person is forced into the arena where he has the choice of his fate.  That is, the person points to one of two identical doors.  The prisoner must choose; thus, it is pure chance which one holds the fate of this prisoner.  If a hungry tiger emerges from the door, it tears the person to pieces as "punishment for his guilt."  However, if a fair lady emerges, he is immediately married to her "as a reward of his innocence." 

Thus, Chance is the "poetic justice" of the king.  The prisoner has "the whole matter in his own hands" in the sense only that he is the one to choose the doors. This statement of course, is ironic since the choice is so limited:  Death or marriage to someone he does not know.