We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ('disagree strongly'), 2 ('disagree'), $\ldots,$ 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour's opinion, if different. For opinions drawn from a set $\{1,2,\ldots,k\}$, the opinion of the vertex would change by $+1$ if the opinion of the neighbour is larger, or by $-1$, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as clique graphs $K_n$ and random graphs $ G_{n,p}$ for suitably large $p$. If the average of the original opinions satisfies $i \le c \le i+1$ for some integer $i$, then the asymptotic probability that opinion $i$ wins is $i+1-c$, and the probability that opinion $i+1$ wins is $c-i$. With high probability, the winning opinion cannot be other than $i$ or $i+1$. To contrast this, we show that for a path and opinions $0,1,2$ arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions $0$ and $2$ is initially supported by a constant fraction of vertices.

翻译：我们考虑一种适用于可在线性尺度上比较的离散数值意见（例如，1（“强烈反对”）、2（“反对”）、$\ldots$、5（“强烈同意”））的拉取式投票。在观察到随机邻居的意见后，如果不同，顶点会逐步将其意见朝邻居意见的值改变。对于从集合$\{1,2,\ldots,k\}$中抽取的意见，如果邻居的意见更大，则顶点的意见会改变$+1$，如果更小，则改变$-1$。目前尚不清楚如何预测此过程的结果，但我们观察到系统的总权重（即所有顶点个体意见之和）是一个鞅。这使我们能够分析该过程在某些稠密扩展图类（例如，团图$K_n$和随机图$G_{n,p}$，其中$p$适当大）上的结果。如果原始意见的平均值满足$i \le c \le i+1$（对于某个整数$i$），则意见$i$获胜的渐近概率为$i+1-c$，意见$i+1$获胜的概率为$c-i$。以高概率，获胜意见只能是$i$或$i+1$。与此对比，我们证明对于一条路径，以及初始沿路径按非递减顺序排列的意见$0,1,2$，结果则截然不同。只要两个极端意见$0$和$2$最初均由恒定比例的顶点支持，则任何意见都可能以恒定概率获胜。