I was recently watching Yale's open course on game theory. One game presented involved the following: the game starts with 1 dollar "on the table." In each round, a player offers the other player a certain amount of the dollar, which the other player can either accept or reject. If the game continues another round, the amount on the table decreases by 10% (due to the time preference of the players).
Each player must offer the other player some of the money in each round. For simplicity, offers are only in whole pennies (no fraction of penny offers permitted). Obviously, they must be at least 1 and at most the amount of money that's currently on the table - so, for example, in the first round Player 1 could offer Player 2 anywhere from 0.01 to the entire dollar. In the second round, there's only 0.90 on the table, so Player 2 could offer Player 1 anywhere from 0.01 to 0.90.
In a one-round game, player 1 should clearly offer player 2 0.01 and keep 0.99 for themselves.
In a two-round game, I'm a little more confused about the exact amount to offer. Clearly if Player 1 makes Player 2 a bad offer, the game will go to round 2, in which case player 2 will offer player 1 0.01 and keep 0.89 for themselves. So, player 1 should offer player 2 just enough for them to accept the offer and keep the rest for themselves.
This may be a nit-picky point, but should Player 1 offer 0.89 or 0.90? Would player 2 accept the 0.89 because 0.89 now is better than 0.89 later, or would offering 0.89 just make player 2 indifferent because the discounting already accounts for time preference? In the first case, clearly Player 1 should offer 0.89; in the second case, they should offer 0.90 to guarantee that Player 2 will accept the offer.
Can someone set me straight in this area?