标题： 有限时间妥协过程中的共识 显示英文标题

标题： Finite-time consensus in a compromise process

摘要： 一个妥协过程描述了通过二元交互演化的意见。 意见是实数，在每一步中，两个随机选择的代理通过获得他们交互前意见的平均值来达成妥协。 我们证明，如果代理的数量$N$是2的幂，则在有限数量的妥协事件后以概率1达成共识；否则，如果初始意见处于一般位置，则无法在有限步骤内达成共识。 此外，达到共识所需的步骤数对于$N=2^k$是随机的，其中$k\geq 2$。 我们证明，通常情况下，达到共识的最小步骤数是$k\cdot 2^{k-1}$。 对于$N=4$，我们确定了步骤数的分布：我们证明它具有纯指数尾部并计算了所有累积量。 显示英文摘要

摘要： A compromise process describes the evolution of opinions through binary interactions. Opinions are real numbers, and in each step, two randomly selected agents reach a compromise by acquiring the average of their pre-interaction opinions. We prove that if the number $N$ of agents is a power of two, then consensus emerges after a finite number of compromise events with probability one; otherwise, consensus cannot be reached in a finite number of steps, provided the initial opinions are in a general position. Furthermore, the number of steps required to reach consensus is random for $N=2^k$ with $k\geq 2$. We prove that typically, the smallest number of steps to reach consensus is $k\cdot 2^{k-1}$. For $N=4$, we determine the distribution of the number of steps: we show that it has a purely exponential tail and compute all cumulants.