I’ve been busy at FEW the past few days, but thanks to everyone who has responded to my previous post. Anyway, in the airport on the way back from Pittsburgh, I saw that the current issue of Scientific American has several philosophically interesting articles, including ones about the origin of life (did it start with a single replicating molecule, or a process involving several simple ones?) and anesthesia (apparently, the operational definition of general anesthesia isn’t quite what you’d expect, focusing on memory blockage more than we might have expected). (It looks like you’ll have to pay to get either of those.)
But I want to discuss an interesting article by economist Kaushik Basu on the Traveler’s Dilemma (available free). This game is a generalization of the Prisoner’s Dilemma, but with some more philosophically interesting structure to it. Each player names an integer from 2 to n. If they both name the same number, then that is their payoff. If they name different numbers, then they both receive the smaller amount, with the person who named the smaller number getting an additional 2 as a bonus, and the one with the larger number getting 2 less as a penalty. If n=3, then this is the standard Prisoner’s Dilemma, where naming 2 is the dominant strategy. But if n≥4, then there is no dominant strategy. However, every standard equilibrium concept still points to 2 as the “rational” choice. We can generalize this game further by letting the plays range from k to n, with k also being the bonus or penalty for naming different numbers.
Unsurprisingly, in actual play, people tend not to actually name k. Interestingly, this is even the case when economics students play, and even when game theorists at an economics conference played! Among untrained players, most play n, which interestingly enough is the only strategy that is dominated by another (namely, by n-1). Among the trained players, most named numbers between n-k and n-1.
In the article, this game was used to suggest that a better concept of rationality is needed than Nash equilibrium play, or any of the alternatives that have been proposed by economists. I think this is fairly clear. The author also uses this game to suggest that the assumption of common knowledge of rationality does a lot of the work in pushing us towards the choice of k.
I think the proper account of this game may bear some relation to Tim Williamson’s treatment of the Surprise Exam Paradox in Knowledge and its Limits. If we don’t assume common knowledge of rationality, but just some sort of bounded iteration of the knowledge operator, then the backwards induction is limited.
Say that an agent is rational0 only if she will not choose an act that is dominated, based on what she knows about the game and her opponent’s options. Say that an agent is rationali+1 iff she is rationali and knows that her opponent is rationali. (Basically, being rationali means that there are i iterations of the knowledge operator available to her.) I will also assume that players are reflective enough that there is common knowledge of all theorems, even if not of rationality.
Now I claim that for ii, then when she plays the Traveler’s Dilemma, she will pick a number less than n-i.
Proof: By induction on i. For i=0, we know that the agent will not choose any dominated strategy. However, the strategy of picking n is dominated by n-1, so she will not pick n=n-i, as claimed. Now, assume that it is a theorem that if an agent is rationali, then when she plays the Traveler’s Dilemma, she will pick a number less than n-i. Then the agent knows this theorem. In addition, if an agent is rationali+1, then she knows her opponent is rationali, and by knowing this theorem, she knows that her opponent will pick a number less than n-i. Since she is also rationali, she will pick a number less than n-i. But given these two facts, picking n-(i+2) dominates picking n-(i+1), so by rationality0, she will not pick n-(i+1) either, proving the theorem, so the induction step goes through, QED.
Thus, if an agent picks a number n-i, then she must be at most rationali-1. But based on what Williamson says, iterations of the knowledge operator are generally hard to come by, so it should not be a surprise that even game theorists playing with common knowledge that they are game theorists will not have very high iterations of rationality. I wonder if it might be possible to use the Traveler’s Dilemma to estimate the number of iterations of knowledge that do obtain in these cases.