Anchored Sequential Deliberation

Sequential deliberation is a mechanism for collective decision making: at each round, a uniformly randomly selected pair is asked to revise a collective outcome, which then becomes the reference point for the next round. Existing theory by Fain et al.~\cite{fain2017sequential} treats the current outcome solely as the disagreement alternative in bargaining. Yet an existing draft, policy, or proposal might carry social influence and anchor participants' expressed positions toward the status quo. We introduce anchored sequential deliberation on a one-dimensional decision space. In each round, two participants with bliss points $U$ and $V$ shift their positions toward the previous outcome $O_{t-1}$ with anchoring strength $λ$, then Nash-bargain using $O_{t-1}$ as the disagreement alternative. The update simplifies to $O_t=(1-λ)\mathsf{Median}\{U,V,O_{t-1}\}+λO_{t-1}$. We establish a convergence--stability trade-off. For every population distribution and $λ<1$, the process has a unique stationary distribution. A monotone coupling yields a $1$-Wasserstein contraction factor of at most $\frac{1+λ}{2}$ and at least $λ$; thus, stronger anchoring slows mixing. On the other hand, stationary social cost weakly decreases with $λ$, although the worst-case distortion remains $\frac{1+\sqrt{2}}{2}$. We also identify a unique \emph{deliberative fixed point}, where the expected unanchored movement is zero, and prove that the stationary distribution concentrates around it as $λ\to 1$. For the uniform population, stationary distortion lies between $1+\frac{1-λ}{9+7λ}$ and $1+\frac{1-λ}{6(1+λ)}$, with both bounds approaching $1$ as $λ\to1$. Simulations for uniform and Beta populations show that stronger anchoring slows mixing, concentrates the stationary distribution, and lowers stationary distortion in these instances.