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The Art of Two-Round Voting

Qishen Han, Grant Schoenebeck, Biaoshuai Tao, Lirong Xia

TL;DR

This paper addresses collective decision-making when voter preferences depend on an unobservable world state by proposing a two-round voting mechanism that first polls voters and then decides the winner in a second round. The authors show that, under δ-approximate strong Bayes Nash equilibria, the mechanism yields the informed majority decision with fidelity approaching 1 as ε→0, and they identify natural equilibria such as informative+sincere and informative+threshold. They prove general results linking two-round equilibria to one-round equilibria, establish the mechanism’s greater expressive power, and demonstrate its robustness under general valuation utilities. Theoretical results are complemented by experiments using generative AI voters, which indicate that two-round voting often outperforms one-round voting in achieving the correct outcome under biased signals, suggesting practical relevance for AI-augmented or AI-proxy voting systems.

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.

The Art of Two-Round Voting

TL;DR

This paper addresses collective decision-making when voter preferences depend on an unobservable world state by proposing a two-round voting mechanism that first polls voters and then decides the winner in a second round. The authors show that, under δ-approximate strong Bayes Nash equilibria, the mechanism yields the informed majority decision with fidelity approaching 1 as ε→0, and they identify natural equilibria such as informative+sincere and informative+threshold. They prove general results linking two-round equilibria to one-round equilibria, establish the mechanism’s greater expressive power, and demonstrate its robustness under general valuation utilities. Theoretical results are complemented by experiments using generative AI voters, which indicate that two-round voting often outperforms one-round voting in achieving the correct outcome under biased signals, suggesting practical relevance for AI-augmented or AI-proxy voting systems.

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.
Paper Structure (64 sections, 20 theorems, 16 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 64 sections, 20 theorems, 16 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work and Discussion
  4. Preliminaries
  5. Alternatives and World States.
  6. Private Signals.
  7. Utility and Types of Agents.
  8. Informed Majority Decision.
  9. Sequence of Environments.
  10. One round voting
  11. One-round voting.
  12. Strategy in the one-round voting game.
  13. Expected utility
  14. Sequence of Strategy Profiles.
  15. $\varepsilon$-strong Bayes Nash Equilibrium (BNE)
  16. ...and 49 more sections

Key Result

Proposition 1

Suppose $\alpha_{C} = 1$. Then for any sequence of environments $\{\mathcal{I}_n\}$ such that $P_{hL} > 0.5$ or $P_{lH} > 0.5$, no sequence pure symmetric strategy profiles $\{\hat{\Sigma}_n\}$ of the one-round voting game satisfies that $\hat{\Sigma}_{n}$ is a $\varepsilon$-strong BNE for every $n$

Figures (2)

  • Figure 1: The information structure of the world state and private signals.
  • Figure 2: Illustration of the range of the second round threshold for an "informative + threshold" equilibrium.

Theorems & Definitions (29)

  • Example 1: Failure of informative voting being a Nash equilibrium Austen96:Information
  • Example 2: Information Structure
  • definition 1: Informative Voting
  • Proposition 1
  • Theorem 1: All $\varepsilon$-BNE are good
  • Theorem 2: "Informative + sincere" equilibrium
  • Example 3
  • Example 4
  • Definition 1
  • Theorem 3: "Informative + threshold" equilibrium
  • ...and 19 more