In the Traveler's Dilemma (TD), I pick 100. I hope the other person is meta-rational and at least picks 99.
I can't point to exactly why the game theory is wrong here, but I can try to show why this is very different from the Prisoner's Dilemma (PD) (which is what TD becomes if the choices are only "2" or "3").
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In PD, if I choose "2", I get either "4" or "2". If I choose "3", I get either "3" or "0". Clearly '"4" or "2"' is a better option for me than '"3" or "0"'. This is a true dilemma, because no matter what probability distribution I use for the other person's choice, my choosing '2' still maximizes my EV. Of course, the other person realizes the same thing, so he also picks "2", resulting in both of us getting 1 less than we could have.
The TD is very different. If I assign even a modest probability to the chance that the other person picks a high (90+) number, then I can maximize my EV by also picking a high number. When I consider the fact that there is not a lot of relative difference between "101" and "97" (our results if one of us picks "100" and the other "99"), I might as well pick "100" and hope the other person isn't extremely greedy (to get 1% more by having me get 3% less).
The author himself says he would pick a high number: "probably 95". Why 95? That's irrational. I will pick 100, and you will get 97 instead of the 100 or 101 you could get by picking 100 or 99 respectively.