The Lady, or the Tiger? Game Theory

Frank Stockton’s short story The Lady or the Tiger apparently is a classic for high school freshmen. Or so my mom told me. It came up because my little sister just read it for freshman literature. I had never heard of it. (see also Wikipedia link)

It’s short. Go ahead and read it. You can come back here in five minutes. Don’t be surprised if you have to look up a few annoying words, but once you get past that, it’s really simple to understand the puzzle he lays out.

Stockton’s semi-barbaric king uses a court of justice where a 50/50 chance decides the guilt of the accused. The criminal has a choice between two doors, one containing a tiger who will kill him (thus, guilty) and one containing a lady who will marry him (innocent).

Stockton says the semi-barbaric king thinks this game is a model of true justice. He says, “Its perfect fairness is obvious.”

As the story goes, the king’s beloved daughter takes a lover far below her station. He is discovered and sent to await the king’s justice. The princess finds out which fate lies behind each door and has a chance to send her lover a signal. She tells him to choose the door on the right, which he does. Stockton then asks the reader which fate she sent him to.

What does a game theory description of this puzzle tell us?

The commoner lover chooses from one door or the other:

At first, you would probably be tempted to describe it like this:

You might think this especially if you’ve taken seriously the notice that the princess suspects her lover of flirting with this particular beautiful lady behind one of the doors.

Stockton asks which fate the princess chose for her lover, when either choice represents a loss for herself. With her suspicions about the lady’s flirting, she would view the outcomes like this:

In the case that he likes the other lady, there is some positive utility in his ending up with her, assuming that the two princess’s outcomes are equally undesirable to her. Is it a question of her nature, then? Is she an altruist (or utilitarian), wishing a positive future on her lover while she stews in jealousy? Or a murderer, sending him to his death to prevent him from loving another, and as Stockton offers, “Would it not be better for him to die at once, and go to wait for her in the blessed regions of semi-barbaric futurity?” (a case that imagines a negative utility of his leading a long life with the other lady.

I’d like to jump in at this point to say that my little sister’s assignment here was to write a sequel to the story. It seemed that most people in her class would try to “beat” the game, maybe to figure out some kind of solution to the puzzle that would bring about a happy ending.

I don’t think such a sequel would match the original, though. The story presents a conundrum, a game under rules that make it perfectly unwinnable. Wouldn’t a true follow-up present a similar puzzle to the reader rather than breaking the king’s rules. After all, Stockton says of the king, “He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts.” He fancied there being no way to defeat his perfect justice, and so I think you have to assume he gets his way here.

But in all this, does Stockton give no credit to the common lover, who has had just as long to think about the princess’s decision as she has? Does he not consider her options? What does he suspect she signaled?

But then, consider, would he think abandoning his (semi-barbaric) princess lover and marrying this other beautiful lady would be preferable to death?

Now, that question is what I think a sequel should be about. But there are even other factors. Since the princess’s signal and the lover’s decision to open one door or the other armed with the information, what message would he be sending with each choice? (And how would his choice signal his interpretation of her motive?) Stockton says he moves without hesitation to the door she signals, so we can assume he decided in advance to follow her advice.

This cuts down the probabilities in the large table to two. Either he would be happy with the other lady and also thinks the princess would want him to be with her, or he wouldn’t want to live without the princess yet also suspects she would send him to his death. He entrusts his lover to choose the right door for him in either case. Just as her decision makes her a murderer or perhaps some kind of altruist, because she loses in every possibility, we can imagine trying to choose the tiger (suspecting it is in the door opposite her signal) to be an example of “true love” yet risks being wrong about her signal and them both looking like asses, as he ends up with the other woman while she had sent him to die.

And really, if you go back and think about how she might also attempt to second-guess his interpretation of her signal, does it even still matter what she signals or what door he chooses? The king’s game seems more like his ideal perfect random justice after all.

…

What is truly annoying to me though is that half of the Google hits for “the lady or the tiger game theory” end up being a stupid Monty Hall math question, adding a third door like in the game show Let’s Make a Deal. That is completely unrelated and invalidates the entire conundrum in the story.

This all said, Stockton did write a follow-up: Discourager of Hesitancy. I haven’t read that yet. We’ll see if he continues as I think would be more fitting than ye olde happy ending of yore.

P.S. Hooray for public domain content.