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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.

Let $P_k$ be the initial proposer's share, measured as a fraction of the pie currently available, with $k$ rounds remaining. The last-round argument in part (e) gives $P_1=1$. Rejecting an offer leaves the responder as next proposer with a pie smaller by factor $\alpha$, so its continuation value is $\alpha P_{k-1}$. The current proposer offers exactly that amount, which is accepted by the continuous-offer bargaining acceptance lemma. HenceThe formula follows by summing a finite geometric series, or by induction in the recurrence. Agreement is immediate in every subgame, so this share is player 1's actual initial payoff.

For odd $k$, $P_k=(1+{\alpha }^k)/(1+\alpha )$ lies above $1/(1+\alpha )$ and decreases with $k$. For even $k$, $P_k=(1-{\alpha }^k)/(1+\alpha )$ lies below that value and increases with $k$. Thus prefer odd to even, fewer rounds among odd choices, and more rounds among even choices. For the given $k_1<k_2$, this simplifies to choose $k_1$ if $k_1$ is odd, and choose $k_2$ if $k_1$ is even. All comparisons are strict because $0<\alpha <1$. This is the finite-horizon bargaining parity effect.