We study the voting problem with two alternatives where voters' preferences depend on a not-directly-observable state variable. While equilibria in the one-round voting mechanisms lead to a good decision, they are usually hard to compute and follow. We consider the two-round voting mechanism where the first round serves as a polling stage and the winning alternative only depends on the outcome of the second round. We show that the two-round voting mechanism is a powerful tool for making collective decisions. Firstly, every (approximated) equilibrium in the two-round voting mechanisms (asymptotically) leads to the decision preferred by the majority as if the state of the world were revealed to the voters. Moreover, there exist natural equilibria in the two-round game following intuitive behaviors such as informative voting, sincere voting [Austen-Smith and Banks, 1996], and the surprisingly popular strategy [Prelec et al., 2017]. This sharply contrasts with the one-round voting mechanisms in the previous literature, where no simple equilibrium is known. Finally, we show that every equilibrium in the standard one-round majority vote mechanism gives an equilibrium in the two-round mechanisms that is not more complicated than the one-round equilibrium. Therefore, the two-round voting mechanism provides a natural equilibrium in every instance, including those where one-round voting fails to have a natural solution, and it can reach an informed majority decision whenever one-round voting can. Our experiments on generative AI voters also imply that two-round voting leads to the correct outcome more often than one-round voting under some circumstances.

翻译：我们研究具有两个备选方案的投票问题，其中投票者的偏好取决于一个不可直接观测的状态变量。尽管单轮投票机制中的均衡能够产生良好决策，但这些均衡通常难以计算和遵循。我们考虑一种两轮投票机制，其中第一轮作为民意调查阶段，而获胜方案仅取决于第二轮的结果。我们证明，两轮投票机制是进行集体决策的有力工具。首先，两轮投票机制中的每个（近似）均衡都会（渐近地）产生多数人偏好的决策，仿佛世界的真实状态已向投票者揭示。此外，两轮博弈中存在遵循直觉行为的自然均衡，例如信息性投票、真诚投票[Austen-Smith and Banks, 1996]以及令人惊讶的流行策略[Prelec et al., 2017]。这与先前文献中的单轮投票机制形成鲜明对比，后者尚未发现任何简单均衡。最后，我们证明标准单轮多数投票机制中的每个均衡都会在两轮机制中产生一个均衡，且其复杂度不高于单轮均衡。因此，两轮投票机制为每个实例提供了自然均衡，包括那些单轮投票无法获得自然解决方案的情形，并且只要单轮投票能够达成知情多数决策，两轮机制同样可以实现。我们在生成式AI投票者上的实验也表明，在某些情况下，两轮投票比单轮投票更频繁地产生正确结果。