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Weak Strategyproofness in Randomized Social Choice

Felix Brandt, Patrick Lederer

TL;DR

This paper revisits strategyproofness in randomized social choice, introducing weak strategyproofness where manipulation must improve a voter's expected utility for all utilities consistent with their ordinal preferences. It develops a broad class of score-based SDSs that are weakly strategyproof on strict preferences and can simultaneously achieve ex post efficiency or Condorcet-consistency, highlighting a richer design space than under strong strategyproofness. It also provides a tops-only characterization and analyzes even-chance SDSs, showing tight possibilities and limitations. For weak preferences, the paper proves two sweeping impossibility results: no attractive anonymous, neutral SDS can be ex ante efficient, and any weakly strategyproof, ex post efficient, even-chance SDS must be dictatorial or bidictatorial, underscoring a stark contrast between strict and weak preference domains with significant practical implications for designing fair randomized voting rules.

Abstract

An important -- but very demanding -- property in collective decision-making is strategyproofness, which requires that voters cannot benefit from submitting insincere preferences. Gibbard (1977) has shown that only rather unattractive rules are strategyproof, even when allowing for randomization. However, Gibbard's theorem is based on a rather strong interpretation of strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for at least one utility function consistent with his ordinal preferences. In this paper, we study weak strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for all utility functions consistent with his ordinal preferences. We show how to systematically design attractive, weakly strategyproof social decision schemes (SDSs) and explore their limitations for both strict and weak preferences. In particular, for strict preferences, we show that there are weakly strategyproof SDSs that are either ex post efficient or Condorcet-consistent, while neither even-chance SDSs nor pairwise SDSs satisfy both properties and weak strategyproofness at the same time. By contrast, for the case of weak preferences, we discuss two sweeping impossibility results that preclude the existence of appealing weakly strategyproof SDSs.

Weak Strategyproofness in Randomized Social Choice

TL;DR

This paper revisits strategyproofness in randomized social choice, introducing weak strategyproofness where manipulation must improve a voter's expected utility for all utilities consistent with their ordinal preferences. It develops a broad class of score-based SDSs that are weakly strategyproof on strict preferences and can simultaneously achieve ex post efficiency or Condorcet-consistency, highlighting a richer design space than under strong strategyproofness. It also provides a tops-only characterization and analyzes even-chance SDSs, showing tight possibilities and limitations. For weak preferences, the paper proves two sweeping impossibility results: no attractive anonymous, neutral SDS can be ex ante efficient, and any weakly strategyproof, ex post efficient, even-chance SDS must be dictatorial or bidictatorial, underscoring a stark contrast between strict and weak preference domains with significant practical implications for designing fair randomized voting rules.

Abstract

An important -- but very demanding -- property in collective decision-making is strategyproofness, which requires that voters cannot benefit from submitting insincere preferences. Gibbard (1977) has shown that only rather unattractive rules are strategyproof, even when allowing for randomization. However, Gibbard's theorem is based on a rather strong interpretation of strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for at least one utility function consistent with his ordinal preferences. In this paper, we study weak strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for all utility functions consistent with his ordinal preferences. We show how to systematically design attractive, weakly strategyproof social decision schemes (SDSs) and explore their limitations for both strict and weak preferences. In particular, for strict preferences, we show that there are weakly strategyproof SDSs that are either ex post efficient or Condorcet-consistent, while neither even-chance SDSs nor pairwise SDSs satisfy both properties and weak strategyproofness at the same time. By contrast, for the case of weak preferences, we discuss two sweeping impossibility results that preclude the existence of appealing weakly strategyproof SDSs.

Paper Structure

This paper contains 29 sections, 19 theorems, 1 equation, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Related work.
  4. Preliminaries
  5. Strategyproofness
  6. Further Axioms
  7. Anonymity.
  8. Neutrality.
  9. Ex post efficiency.
  10. Condorcet-consistency.
  11. Results
  12. Possibility Theorems for Strict Preferences
  13. Remark 1.
  14. Remark 2.
  15. Impossibility Theorems for Strict Preferences
  16. ...and 14 more sections

Key Result

Theorem 1

Every score-based SDS on $\mathcal{L}^N$ satisfies weak strategyproofness.

Figures (2)

  • Figure 1: Weighted majority relations for the profiles $R^2$, $\hat{R}^2$, $R^3$, and $R^4$ in the proof of \ref{['thm:impSDS']}. An arrow from $x$ to $y$ with weight $w$ means that $w$ more voters prefer $x$ to $y$ than $y$ to $x$. Pareto-dominated alternatives and their corresponding edges are colored in gray to improve readability.
  • Figure 2: Profiles used in the proof of \ref{['thm:SDimpossibity']}. Preference relations highlighted in red indicate manipulations and horizontal lines indicate the three steps of the proof.

Theorems & Definitions (36)

  • Theorem 1
  • proof : Proof sketch
  • Theorem 2
  • proof
  • Theorem 3
  • proof : Proof Sketch
  • Theorem 4
  • proof : Proof Sketch
  • Theorem 5: BBEG16a
  • Theorem 6
  • ...and 26 more