To see why 2 is the logical choice, consider a plausible line of thought that Lucy might pursue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy. (If the antique actually cost her much less than $100, she would now be happily thinking about the foolishness of the airline manager's scheme.)
Soon, however, it strikes her that if she wrote 99 instead, she would make a little more money, because in that case she would get $101. But surely this insight will also occur to Pete, and if both wrote 99, Lucy would get $99. If Pete wrote 99, then she could do better by writing 98, in which case she would get $100. Yet the same logic would lead Pete to choose 98 as well. In that case, she could deviate to 97 and earn $99. And so on. Continuing with this line of reasoning would take the travelers spiraling down to the smallest permissible number, namely, 2. It may seem highly implausible that Lucy would really go all the way down to 2 in this fashion. That does not matter (and is, in fact, the whole point)--this is where the logic leads us.
Jason Kuznicki has an interesting response:
The smallest figure only becomes a reasonable strategy when neither Pete nor Lucy have any guidance whatsoever about how the other traveler might respond. In the world of actual prices, this never happens. In other words, the $2 solution is only plausible when it is entirely divorced from economics, and when neither player has any cues at all for giving an answer.
I don't think market cues are relevant here. There's a perfectly salient "default option" even in the abstract case, namely, the maximum value of $100. Suppose the market price is $50. The recursive "race-to-the-bottom" reasoning will apply just as disastrously from 50-49 as it originally did from 100-99. Changing our starting point doesn't really change anything. It's the reasoning that's the problem.
The real solution, then, is to affirm norms of global rationality: look at the big picture, and reason according to a decision-procedure that will predictably yield better results. That means ditching the economist's "local rationality" of backwards induction and its race to the $2 bottom. The SciAm writer has it right:
If I were to play this game, I would say to myself: "Forget game-theoretic logic. I will play a large number (perhaps 95), and I know my opponent will play something similar and both of us will ignore the rational argument that the next smaller number would be better than whatever number we choose." 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.