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It seems the second student must know about the rules... After all, if the first one were under no obligations to share at all, there would be little reason not to accept the free dollar, handed to the second, just like that... There is a question of the exact rules of the game. One could imagine several possible ones, assuming both participants know all the rules:
  1. Student 1 makes a one off offer to student 2, which student 2 must accept or reject, with the consequences provided by the game
  2. Student 1 makes and offer, and may make additional offers if the first one is rejected, i.e. we have some situation involving negotiation...

Ideas of 'fairness', 'personal gain' and 'utility' should play out under both scenarios, with the second one probably resulting in more deals and a fairer distribution. This is the situation in isolation. If we start considering 'external' factors, outcomes may shift, however. As in, suppose student 2 is really poor, and can normally only afford to eat every other day. In this case, we might expect he would more readily accept even 1 dollar because he would use it to feed himself. Thus, it is a question not just of asymmetric information, the two students may very well both have access to all available data, but also a question of relative (economic) power, and what the two participants know about each other in this area. That's when it starts to get 'interesting'... When the first student know the second cannot afford to reject 1 dollar.

Related to a recent comment on asymmetric power in games:

Jerome a Paris:
A game of bluff and bluster for extravagant reward
Your chances in this world are proportional to the size of your bankroll: the house wins by virtue of being the house.
by someone (s0me1smail(a)gmail(d)com) on Wed Jul 18th, 2007 at 06:07:44 AM EST
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