= Finite-horizon bargaining parity effect
{title2=$P_k=\frac{1-(-\alpha)^k}{1+\alpha}$}
In <finite-horizon alternating-offer bargaining>, the proposer with $k$ rounds left retains fraction $P_k$ of the currently available pie. Terminal acceptance gives $P_1=1$. After rejection, the responder becomes proposer and can obtain fraction $\alpha P_{k-1}$ of the current pie. The <continuous-offer bargaining acceptance lemma> therefore gives $P_k=1-\alpha P_{k-1}$. Summing the resulting finite <geometric series> proves the displayed formula.
Odd horizons give shares above $1/(1+\alpha)$, decreasing toward it; even horizons give shares below it, increasing toward it. Between two available horizons $k_1<k_2$, choose $k_1$ if $k_1$ is odd and $k_2$ if $k_1$ is even. This concerns the initial proposer and immediate agreement, rather than waiting until the selected final round to obtain its share.
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