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Unravelling Expressive Delegations: Complexity and Normative Analysis

Giannis Tyrovolas, Andrei Constantinescu, Edith Elkind

TL;DR

The paper analyzes binary decisions under an expressive liquid democracy where agents can submit monotone Boolean delegation functions. It establishes complete complexity dichotomies for the MinSum (minimizing total certificate rank) and MinMax (minimizing the maximum rank) unravellings, showing NP-hardness when nontrivial functions are allowed and tractability only under restricted classes, with nuanced results for the classic model of delegation to individuals. It provides near-optimal algorithms for the classic model, efficient methods to realize unravellings that maximize the chance of a given alternative winning, and tie-breaking rules that can bias outcomes toward a status quo when desirable. A new cast-monotonicity axiom is introduced, showing MinSum and a lexicographicMinMax refinement satisfy it, while MinMax itself does not, offering a normative lens on these rules. Overall, the work blends algorithmic, complexity-theoretic, and normative analyses to advance understanding of expressive delegation in liquid democracy and to inform practical tie-breaking and policy-design choices.

Abstract

We consider binary group decision-making under a rich model of liquid democracy recently proposed by Colley, Grandi, and Novaro (2022): agents submit ranked delegation options, where each option may be a function of multiple agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such ballots are unravelled into a profile of direct votes by selecting one entry from each ballot so as not to introduce cyclic dependencies. We study delegation via monotonic Boolean functions, and two unravelling procedures: MinSum, which minimises the sum of the ranks of the chosen entries, and its egalitarian counterpart, MinMax. We provide complete computational dichotomies: MinSum is hard to compute (and approximate) as soon as any non-trivial functions are permitted, and polynomial otherwise; for MinMax the easiness results extend to arbitrary-arity logical ORs and ANDs taken in isolation, but not beyond. For the classic model of delegating to individual agents, we give asymptotically near-tight algorithms for carrying out the two procedures and efficient algorithms for finding optimal unravellings with the highest vote count for a given alternative. These algorithms inspire novel tie-breaking rules for the setup of voting to change a status quo. We then introduce a new axiom, which can be viewed as a variant of the participation axiom, and use algorithmic techniques developed earlier in the paper to show that it is satisfied by MinSum and a lexicographic refinement of MinMax (but not MinMax itself).

Unravelling Expressive Delegations: Complexity and Normative Analysis

TL;DR

The paper analyzes binary decisions under an expressive liquid democracy where agents can submit monotone Boolean delegation functions. It establishes complete complexity dichotomies for the MinSum (minimizing total certificate rank) and MinMax (minimizing the maximum rank) unravellings, showing NP-hardness when nontrivial functions are allowed and tractability only under restricted classes, with nuanced results for the classic model of delegation to individuals. It provides near-optimal algorithms for the classic model, efficient methods to realize unravellings that maximize the chance of a given alternative winning, and tie-breaking rules that can bias outcomes toward a status quo when desirable. A new cast-monotonicity axiom is introduced, showing MinSum and a lexicographicMinMax refinement satisfy it, while MinMax itself does not, offering a normative lens on these rules. Overall, the work blends algorithmic, complexity-theoretic, and normative analyses to advance understanding of expressive delegation in liquid democracy and to inform practical tie-breaking and policy-design choices.

Abstract

We consider binary group decision-making under a rich model of liquid democracy recently proposed by Colley, Grandi, and Novaro (2022): agents submit ranked delegation options, where each option may be a function of multiple agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such ballots are unravelled into a profile of direct votes by selecting one entry from each ballot so as not to introduce cyclic dependencies. We study delegation via monotonic Boolean functions, and two unravelling procedures: MinSum, which minimises the sum of the ranks of the chosen entries, and its egalitarian counterpart, MinMax. We provide complete computational dichotomies: MinSum is hard to compute (and approximate) as soon as any non-trivial functions are permitted, and polynomial otherwise; for MinMax the easiness results extend to arbitrary-arity logical ORs and ANDs taken in isolation, but not beyond. For the classic model of delegating to individual agents, we give asymptotically near-tight algorithms for carrying out the two procedures and efficient algorithms for finding optimal unravellings with the highest vote count for a given alternative. These algorithms inspire novel tie-breaking rules for the setup of voting to change a status quo. We then introduce a new axiom, which can be viewed as a variant of the participation axiom, and use algorithmic techniques developed earlier in the paper to show that it is satisfied by MinSum and a lexicographic refinement of MinMax (but not MinMax itself).
Paper Structure (25 sections, 44 theorems, 5 equations, 3 figures, 1 algorithm)

This paper contains 25 sections, 44 theorems, 5 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Classic Liquid Democracy
  6. Enhanced Liquid Democracy
  7. Complexity of the Enhanced Model
  8. Complexity of $\textsc{MinSum}\xspace_\mathcal{F}$.
  9. Complexity of $\textsc{MinMax}\xspace_\mathcal{F}$.
  10. Computational Dichotomies.
  11. Complexity of the Classic Model
  12. Efficient Computation of MinSum and MinMax Certificates
  13. Control for MinSum and MinMax
  14. Axiomatic Results
  15. Summary and Future Work
  16. ...and 10 more sections

Key Result

Lemma 2

In the classic model, a certificate $\mathbf{c}$ is consistent iff there exists a linear order $\triangleleft$ on $N$ such that for all agents $a \in N$ it holds that either $B_a(c_a) \in D$ or $B_a(c_a) \triangleleft a.$

Figures (3)

  • Figure 1: Preference profile in \ref{['example:full-model']}. Solid lines indicate first preferences, dashed lines indicate second preferences, and dotted lines indicate third preferences. To avoid clutter we have removed the third preferences of $a, b$ and $c$.
  • Figure 2: Gadget of clause $x_a \lor x_b$ for $\textsc{MinSum}\xspace_{\textsc{Or}_2}.$
  • Figure 3: Gadget of clause $x_i \lor x_j \lor x_k$ for $\textsc{MinMax}\xspace_{\textsc{OrAnd}_2}.$

Theorems & Definitions (74)

  • Example 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • proof
  • ...and 64 more