BTW, the "right" answer according to the game theorist is "2". I think most folks here went the direction I expected, and frankly that most folks do (higher numbers).

The logic for choosing 2 is explained here:

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.

Game theorists commonly use this style of analysis, called backward induction. Backward induction predicts that each player will write 2 and that they will end up getting $2 each (a result that might explain why the airline manager has done so well in his corporate career). Virtually all models used by game theorists predict this outcome for TD--the two players earn $98 less than they would if they each naively chose 100 without thinking through the advantages of picking a smaller number.

Whole article on this is here:

http://www.sciam.com/article.cfm?chanID=sa006&articleID=7750A576-E7F2-99DF-3824E0B1C2540D47&ref=rss