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Incentives in Social Decision Schemes with Pairwise Comparison Preferences

Felix Brandt, Patrick Lederer, Warut Suksompong

TL;DR

The paper investigates randomized social choice under pairwise-comparison (PC) preferences, introducing PC and PC1 extensions to lift ordinal valuations to lotteries. It proves three strong impossibilities: there is no Condorcet-consistent SDS that is PC-strategyproof, no anonymous/neutral PC-efficient SDS that is PC-strategyproof, and no anonymous/neutral PC-efficient SDS that achieves strict PC-participation when there are at least four alternatives ($m\geq 4$). For $m\leq 3$, it constructs two SDSs, $f^1$ and $f^2$, that satisfy the desired properties, highlighting the delicate balance between efficiency and strategic behavior under PC. The paper also discusses the relationship between SD- and PC-based analyses, showing that moving to PC often makes previously compatible axioms incompatible, and identifies RD and ML as important benchmarks with contrasting incentive and efficiency properties. Overall, the results delineate the limits and possibilities of PC-based SDSs and provide explicit constructions illustrating the boundary cases at $m=3$.

Abstract

Social decision schemes (SDSs) map the ordinal preferences of individual voters over multiple alternatives to a probability distribution over the alternatives. In order to study the axiomatic properties of SDSs, we lift preferences over alternatives to preferences over lotteries using the natural -- but little understood -- pairwise comparison (PC) preference extension. This extension postulates that one lottery is preferred to another if the former is more likely to return a preferred outcome. We settle three open questions raised by Brandt (2017): (i) there is no Condorcet-consistent SDS that satisfies PC-strategyproofness; (ii) there is no anonymous and neutral SDS that satisfies PC-efficiency and PC-strategyproofness; and (iii) there is no anonymous and neutral SDS that satisfies PC-efficiency and strict PC-participation. All three impossibilities require $m\geq 4$ alternatives and turn into possibilities when $m\leq 3$. We furthermore settle an open problem raised by Aziz et al. (2015) by showing that no path of PC-improvements originating from an inefficient lottery may lead to a PC-efficient lottery.

Incentives in Social Decision Schemes with Pairwise Comparison Preferences

TL;DR

The paper investigates randomized social choice under pairwise-comparison (PC) preferences, introducing PC and PC1 extensions to lift ordinal valuations to lotteries. It proves three strong impossibilities: there is no Condorcet-consistent SDS that is PC-strategyproof, no anonymous/neutral PC-efficient SDS that is PC-strategyproof, and no anonymous/neutral PC-efficient SDS that achieves strict PC-participation when there are at least four alternatives (). For , it constructs two SDSs, and , that satisfy the desired properties, highlighting the delicate balance between efficiency and strategic behavior under PC. The paper also discusses the relationship between SD- and PC-based analyses, showing that moving to PC often makes previously compatible axioms incompatible, and identifies RD and ML as important benchmarks with contrasting incentive and efficiency properties. Overall, the results delineate the limits and possibilities of PC-based SDSs and provide explicit constructions illustrating the boundary cases at .

Abstract

Social decision schemes (SDSs) map the ordinal preferences of individual voters over multiple alternatives to a probability distribution over the alternatives. In order to study the axiomatic properties of SDSs, we lift preferences over alternatives to preferences over lotteries using the natural -- but little understood -- pairwise comparison (PC) preference extension. This extension postulates that one lottery is preferred to another if the former is more likely to return a preferred outcome. We settle three open questions raised by Brandt (2017): (i) there is no Condorcet-consistent SDS that satisfies PC-strategyproofness; (ii) there is no anonymous and neutral SDS that satisfies PC-efficiency and PC-strategyproofness; and (iii) there is no anonymous and neutral SDS that satisfies PC-efficiency and strict PC-participation. All three impossibilities require alternatives and turn into possibilities when . We furthermore settle an open problem raised by Aziz et al. (2015) by showing that no path of PC-improvements originating from an inefficient lottery may lead to a PC-efficient lottery.
Paper Structure (17 sections, 18 theorems, 40 equations, 1 figure)

This paper contains 17 sections, 18 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Model
  3. Fairness and Decisiveness
  4. Preferences over Lotteries
  5. Efficiency
  6. Incentive-Compatibility
  7. Strategyproofness.
  8. Participation.
  9. Random Dictatorship and Maximal Lotteries
  10. Results
  11. $\mathit{PC}$-strategyproofness
  12. Strict $\mathit{PC}$-participation
  13. Conclusion
  14. Omitted Proofs
  15. $\mathit{PC}$-Efficiency Cycle
  16. ...and 2 more sections

Key Result

Proposition 1

$\mathit{RD}$ satisfies $\mathit{SD}$-strategyproofness, strict $\mathit{SD}$-participation, and $\mathit{SD}$-efficiency, but fails $\mathit{PC1}$-efficiency and the absolute winner property.

Figures (1)

  • Figure 1: Overview of results. An arrow from an axiom $X$ to another axiom $Y$ indicates that $X$ implies $Y$. The red lines between axioms represent impossibility theorems. Note that \ref{['thm:EffSP', 'thm:strictpart']} additionally require anonymity and neutrality. Axioms labeled with $\mathit{ML}$ are satisfied by maximal lotteries, and axioms labeled with $\mathit{RD}$ are satisfied by the uniform random dictatorship.

Theorems & Definitions (38)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Proposition 3
  • Remark 1
  • Remark 2
  • ...and 28 more