## Traveler's Dilemma and a new kind of formal reasoning in game theory

What is rational and what is irrational?

That question lay at the core of a mountain of letters we received about our June article "The Traveler's Dilemma" by Kaushik Basu. We ran a small selection of the letters here in the blog along with responses from Basu. A letter from Adam Brandenburger of New York University appears in the October print edition of SciAm. Here is a somewhat longer version of his letter:

In "The Traveler's Dilemma" (June 2007), Kaushik Basu describes an intriguing game he introduced into the game theory literature some years ago. (The Traveler's Dilemma bears some similarity to the famous Prisoner's Dilemma.) In the game, two players must each choose a number between 2 and 100. As Basu explains, the game is constructed so that there is a unique Nash equilibrium—at which each player chooses the number 2 (and each then receives \$2). Yet, when the game is actually played, much higher choices are often seen. Often, both players choose numbers close to 100, in which case they both receive much higher "payoffs" than in the Nash equilibrium.

Basu is right that experience with games such as the Traveler's Dilemma poses a serious challenge to the use of the Nash-equilibrium concept in game theory. However, game theory is not synonymous with Nash equilibrium. There are now theorems in formal game theory giving conditions under which Nash equilibrium emerges (see [1]). The conditions are very stringent: In particular, the assumption that the players in a game are rational is far from sufficient to yield Nash equilibrium. This is good news. There is no conflict between game theory and what we observe in games like the Traveler's Dilemma—only between Nash equilibrium and what we observe.

Basu asks that a "new kind of formal reasoning" be developed to deliver more satisfactory analyses of many games. In fact, over the past two decades, a subfield of game theory—called interactive epistemology—has emerged on precisely this topic. It is now possible to analyze mathematically what it means for the players in a game to be rational or irrational, to think that other players are rational or irrational, and the like. (See [2] for a recent survey.) This is different from the classical Nash-equilibrium analysis of games, and often yields the more intuitive answers Basu wants.

J.P. Valles Professor
New York University

[1] "Epistemic Conditions for Nash Equilibrium," by Robert Aumann and Adam Brandenburger, Econometrica, Vol. 63, pages 1161-1180 (1995). [Also available: Unpublished 1991 version (pdf).]

[2] "The Power of Paradox: Some Recent Developments in Interactive Epistemology," (pdf) by Adam Brandenburger, International Journal of Game Theory, Vol. 35, pages 465-492 (2007).

View
1. 1. NiveusCella 05:13 PM 1/5/10

Well, no matter what the other person picks (besides \$2 or \$100 in which case only one of the following results is possible) you will always get either \$1 more than what they say or \$2 left. For example, If they pick \$44 the most you can get is \$45 from picking \$43. The least you can get (if you pick higher than them) would be \$42. Therefore, the best choice would be to pick high, right? If you pick lower than them, you are bound to receive from \$4 (a fixed amount) to \$1 more than what they chose. If you pick higher than them, you are bound to get two dollars less than what they have picked. You have a much better chance of getting a reasonable amount if you pick higher than them. To clarify on my previous statement about having a better chance of getting a reasonable amount; if they chose \$44 and you chose something beneath them, you could possibly get \$1 more than that (\$43), \$0 more (\$42), or anywhere from \$1 less to \$40 less (\$41-\$2). If you pick higher you will only get \$2 less at most. Because you do not know what they are going to pick, the best choice would be \$100 (possibly \$99). I think that is fair reasoning for one hoping for the highest outcome.

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