Stockton leaves us hanging, ending on the following question:

And so I leave it with all of you: Which came out of the opened door,—the lady, or the tiger?

We know the lady is much like her father, which is to say "semi-barbaric," with his practices strongly favoring his barbaric side. It is, therefore, highly likely that she would do what was in her own interest, rather than sacrifice her needs for another's well-being. We learn that

had it not been for the moiety of barbarism in her nature it is probable that lady would not have been there, but her intense and fervid soul would not allow her to be absent on an occasion in which she was so terribly interested.

In other words, she is completely capable of watching her lover being torn to shreds. At this point, it becomes a mind game. If her lover picks the door she indicates, we might assume a tiger will come leaping out, as she would rather see him die than see him married to her rival. However, the lover would likely know the princess's nature and therefore not pick the door she indicated. In this case, he would choose the door from which would come the lady.

However, what if she anticipated that he would pick the opposite door? Would she indicate the door with the lady, knowing he wouldn't pick it and therefore have the enjoyment of watching him be devoured as a result of distrusting her?

Weighing the odds, the princess's barbarism points in the direction of her not overthinking the situation. Like her father, she probably reasons simplistically, so we could assume she would point her lover to the door with the tiger, and he would be fully aware of this.

Stockton, the author of the work, deliberately did not give us the "right" answer to this question.  Given that, we can guess and surmise and speculate; however, the answer is still, ultimately, a matter of opinion.

Think about this people. The story explains about her barbaric nature like her father’s. We are also told that her lover “knows” her true nature. If these things are true, the princess would have directed him to the door with the tiger, but her lover would have known that. Consequently, he would have chosen the door she did not indicate, and he and the beautiful girl would live happily ever after.

With all due respect, I think #3 needs to look at the quoted section of text more closely. " How in her grievous reveries had she gnashed her teeth, and torn her hair, when she saw his start of rapturous delight as he opened the door of the lady!" The princess did not want him to find happiness with anyone else. She had a barbaric nature, and she knew that the beautiful young woman had cast glances upon her lover, and she had perceived those glances to be returned. This would not been taken lightly, and it is safe to say it would probably not be forgiven. In addition, let us not forget the "savage blood" that coursed through her and the "barbaric" ancestry she came from, and the fact that she hated the woman behind the door. It would seem that due to her barbaric nature she would have not wanted her lover to find happiness with another woman, and so we could assume she guided him to the door with the tiger.

Now with all that being said, we are told also that her lover ". . . understood her nature. . .", so if that is true then perhaps he would know she might not want any other woman to have him, and he would choose the door opposite the one she directed him to. So if the lady came out, perhaps it was because he outwitted his barabic lover.

After reading this story, it is my opinion, based on the text quote below that the young man chose the door with the Lady behind it. Here is why.  The Princess can't bear to see the young man die at her instruction.

"How her soul had burned in agony when she had seen him rush to meet that woman, with her flushing cheek and sparkling eye of triumph; when she had seen him lead her forth, his whole frame kindled with the joy of recovered life; when she had heard the glad shouts from the multitude, and the wild ringing of the happy bells; when she had seen the priest, with his joyous followers, advance to the couple, and make them man and wife before her very eyes; and when she had seen them walk away together upon their path of flowers, followed by the tremendous shouts of the hilarious multitude, in which her one despairing shriek was lost and drowned!" (Stockton)

The Princess has gone through a great deal to discover the identity of the two doors.  She loves the young man, but she knows that she can never marry him.  The author gives us much more detail about the Princess's reaction to the marriage of the young man and the lady. 

Even though she questions her decision, I think that she signaled him with the door where the lady stood,  because she loved him, and could not bear to lead him to his death.  She would be far more haunted by this idea, knowing that she was responsible for him being torn apart by the tiger.

Ah, that's the question now, isn't it?  It makes the reader wonder, which was the intent of the author.  By not telling you the ending, he let you create your own (which is why this story is so often used in schools, to generate discussion and even as the source of a writing assignment to provide an ending).  All authors have ways of drawing the reader into the story and making him a co-creator in the story-telling process.  Some are subtle, some (like this) are really "out there."

Sometimes the author will give clues, allowing the reader to draw a reasonable conclusion.  Yet in "The Lady or the Tiger?" no such clues are given, which was deliberate on the part of the author.  Either one could have come out.  It is up to the reader to decide, .  Or else not decide, and wonder about for a long, long time....