= Finite-horizon alternating-offer bargaining
Two players alternate proposals for dividing a pie, with a fixed final round. Agreement ends the <game>; rejection shrinks the next available pie by a factor $0<\alpha<1$. Final rejection gives both players zero. Offers range continuously over the entire available pie. <Backward induction in games> determines immediate agreement and the proposer's share through the <continuous-offer bargaining acceptance lemma> and the <finite-horizon bargaining parity effect>. Discrete offer grids can change both the boundary behavior and the exact shares.
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