This post is concerned with a very interesting problem, called “The Traveler’s Dilemma.” There is a very good article about it, written by it’s creator, Professor Kaushik Basu, in the June, 2007 issue of the Scientific American. The article begins:
“When playing this simple game, people consistently reject the rational choice. In fact, by acting illogically, they end up reaping a larger reward–an outcome that demands a new kind of formal reasoning.”
Please read the article before you read this post.
I want to preface this post by saying that I am fascinated with the Idea of game-theory, and I admire it’s students and professors. I am in no way “officially qualified” to gainsay anything by game-theorists, or by “real” mathematicians or economists.
On the other hand, a little nudge from us plebeians can be good for the aristocracy.
Although the “Traveler’s Dilemma” is an endlessly fascinating subject, I don’t believe the dilemma is the one being presented. Or at least if it is, it is ill-named. Let me explain:
I believe that the problem should be called “The Game Theorist’s Dilemma”, or “The Clever Mathematician’s Dilemma,” or something like that. It certainly doesn’t have all the elements of the traveler’s dilemma. One of the most important elements was left out.
At no point in this article did I read about the actual price of the item. That is something both travelers would know, but is never given as a piece of their dilemma. Therefore the mathematician is playing a significantly different game than the travelers are.
(Note: The assumption that both have paid the same price is derived from the wording of the “Traveler’s Dilemma,” although, as with everything else, I could be wrong about this. I await your comments.)
Rather than gaming the system, their main dilemma is “should I lie or tell the truth?”
If each assumes that the other is basically honest, the game theorist can go home and play with himself. (Absolutely no disrespect intended. Game-theorists are way ahead of me academically – I just liked the sentence.)
If both Pete and Lucy are honest, there is no dilemma – both tell the truth. Game over.
If Pete is honest and Lucy assumes that Pete is not, (or vice-versa) then his or her dilemma is:
“Should I lowball or highball ‘Goody-two-shoes’?”
It seems to me the strategy would be to lowball by one dollar, thereby ending up with one dollar more than the honest price.
If both are dishonest, let the game-theory begin!
One thing is for sure, the first sentence of the last paragraph in the article is most poignant – “If I were to play this game, I would say to myself: “Forget game-theoretic logic.”
Later in the same paragraph, it goes on to state:
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“What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality* attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler’s Dilemma.”
Note: In the sidebar to this article (the Payoff Matrix of the Traveler’s Dilemma link), you can read:
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“This payoff matrix summarized everything game-thorists need to know about Traveler’s Dilemma.”
What a statement! That’s only true if game-theorists are not concerned with anything beyond the narrow scope of staring at their own gamey navels. (Again, please, no disrespect intended.)
The most significant point of all, I think, was the anecdote about the Indian hat-seller. The subsequent discussion about it still, alas, does not bring up the fact that Pete and Lucy knew the price.
I find it odd that game-theorists apparently don’t look at this factor, considering how similar it is to the “Monty Hall Problem,” in which all the difference is made by the fact that Monte knows what is behind the doors.
At the end of the discussion about the Indian hat-seller, Professor Basu says,
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” In my opinion, the common knowledge of rationality assumed by game theorists faces a … demise.”
*I don’t know if I am comfortable with the term “meta-rationality:” for so many reasons. I hope it doesn’t become default. But it would be a fine thing if some study evolved which dealt with these things. Game-theory has proven to be incredibly useful in the real world. Besides its utility, it is a field of endless fascination for your mind. But like the mind, (at least mine) it seems that it can use some maturing. Professor Basu seems very enlightened on this point, and I hope that means that I am on the right track.
So far, I have found nothing that approaches Edux theory. Presenting Edux theory will be the next step after Math Mojo is a completed site. Edux is the brainchild of Dr. Kent Lawson, Professor Emeritus of Physics and Edux at the State University of New York at Oneonta.
You can find more information about Mr. Kaushik at his home page, and at Wikipedia.
You will find more and more references to Edux here and at MathMojo.com. Stay tuned.
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