In the classical Approximate Majority problem with two opinions there are agents with Opinion 1 and with Opinion 2. The goal is to reach consensus and to agree on the majority opinion if the bias is sufficiently large. It is well known that the problem can be solved efficiently using the Undecided State Dynamics (USD) where an agent interacting with an agent of the opposite opinion becomes undecided. In this paper, we consider a variant of the USD with a preferred Opinion 1. That is, agents with Opinion 1 behave stubbornly -- they preserve their opinion with probability $p$ whenever they interact with an agent having Opinion 2. Our main result shows a phase transition around the stubbornness parameter $p \approx 1-x_1/x_2$. If $x_1 = \Theta(n)$ and $p \geq 1-x_1/x_2 + o(1)$, then all agents agree on Opinion 1 after $O(n\cdot \log n)$ interactions. On the other hand, for $p \leq 1-x_1/x_2 - o(1)$, all agents agree on Opinion 2, again after $O(n\cdot \log n)$ interactions. Finally, if $p \approx 1-x_1/x_2$, then all agents do agree on one opinion after $O(n\cdot \log^2 n)$ interactions, but either of the two opinions can survive. All our results hold with high probability.

翻译：在经典的具有两种意见的近似多数问题中，存在持有意见1和意见2的代理。目标是达成共识，并在偏差足够大时同意多数意见。众所周知，该问题可以通过未定状态动力学（USD）高效解决，其中与相反意见代理交互的代理会变为未定状态。本文考虑一种具有偏好意见1的USD变体。即，持有意见1的代理表现得固执——每当与持有意见2的代理交互时，他们以概率$p$保持自己的意见。我们的主要结果展示了围绕固执参数$p \approx 1-x_1/x_2$的相变。如果$x_1 = \Theta(n)$且$p \geq 1-x_1/x_2 + o(1)$，则所有代理在$O(n\cdot \log n)$次交互后同意意见1。另一方面，对于$p \leq 1-x_1/x_2 - o(1)$，所有代理在$O(n\cdot \log n)$次交互后同意意见2。最后，如果$p \approx 1-x_1/x_2$，则所有代理确实在$O(n\cdot \log^2 n)$次交互后同意一种意见，但两种意见中的任意一种都可能存留。我们所有的结果均以高概率成立。