The Art of Two-Round Voting

2025-05-15

AI Summary

This paper studies binary collective decision-making under unobserved states, where voter preferences depend on latent state variables, rendering single-round voting incapable of achieving equilibrium or optimal outcomes. To address this, we propose a two-round voting mechanism: the first round serves as information aggregation (akin to an opinion poll), and the second round enables decisions based on updated posterior beliefs. We prove that this mechanism admits a natural Bayesian Nash equilibrium for any parameter configuration and asymptotically converges to the majority-correct decision achievable under full information. Notably, it is the first framework to unify informative voting, sincere voting, and “surprisingly popular” behavior within a provably equilibrium-consistent setting. Moreover, we show that all equilibria of any single-round mechanism can be succinctly embedded into our two-round design. Generative AI–based simulations demonstrate substantial improvements in correct decision rates—particularly pronounced under strong information asymmetry.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Humans and AI: Voting

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Studies two-round voting with unobservable state variables
• Compares equilibria in one-round vs two-round voting mechanisms
• Demonstrates two-round voting achieves informed majority decisions

Innovation

• Two-round voting mechanism for majority decisions
• Natural equilibria with intuitive voting behaviors
• Outperforms one-round voting in correct outcomes