I come across a question:
Player 1 and Player 2 are to split \$1 in a infinite-horizon game. In each period a fair coin is flipped. If the coin comes up head, then Player 1 proposes a split of the money; if it comes up tails, then Player 2 proposes the split. After a proposal is made, the other player either accepts or rejects the proposal. If the proposal is accepted, then the money is immediately divided as proposed and payoffs are realized. If the proposal is rejected, then in the next period the above procedure is repeated, starting with a fresh flip of the coin. If an aggrement is never reached, none of them receives any money. Player 1's per-period discount factor is $\frac{2}{3}$ and Player 2's discount factor is $\frac{1}{3}$.
When solving the game, I assume that, if a proposal is rejected, Player 1 gets (discounted) present value of money $a$ and Player 2 gets present value of money $b$. Intuitively $a+b\leq1$ because money cannot grow by itself. But is there any way to argue $a+b\not>1$ using a recursive approach or using summation of series?