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 input for the next round. Existing theory by Fain et al.~\cite{fain2017sequential} treats the current outcome solely as the disagreement alternative in bargaining. Yet the existing outcome might also carry social influence and anchor participants' positions toward the status quo. In this paper, we introduce \emph{anchored sequential deliberation}. At each round, two participants with bliss points $U$ and $V$ shift their positions toward the previous outcome $O_{t-1}$ with anchoring strength $0\leq λ<1$. They then bargain using $O_{t-1}$ as the disagreement alternative. For the sake of analysis, we assume that the decision space is one-dimensional, the anchoring effect is linear, and participants use Nash bargaining. The update simplifies to $O_t=(1-λ)Med\{U,V,O_{t-1}\}+λO_{t-1}$. Our analysis reveals a trade-off. Through a coupling of two outcomes, we find that the process contracts in $1$-Wasserstein distance with a factor of at most $\frac{1+λ}{2}$, which implies that stronger anchoring slows mixing. On the other hand, stationary distortion weakly decreases with $λ$, although the worst-case distortion remains $\frac{1+\sqrt{2}}{2}$ for every feasible $λ$. We also identify a unique \emph{deliberative fixed point}, at which the expected movement is zero, and prove that the stationary distribution concentrates around it as $λ$ increases. For symmetric populations, we further provide a tighter bound on the stationary variance around this fixed point. For the uniform distribution, we establish upper and lower bounds on stationary distortion, both of which approach $1$ as $λ$ increases.

翻译：暂无翻译