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Controlling Delegations in Liquid Democracy

Shiri Alouf-Heffetz, Tanmay Inamdar, Pallavi Jain, Yash More, Nimrod Talmon

TL;DR

Here a certain kind of election control is formalized - in which an external agent may change certain delegation arcs - and the computational complexity of the corresponding combinatorial problem is studied.

Abstract

In liquid democracy, agents can either vote directly or delegate their vote to a different agent of their choice. This results in a power structure in which certain agents possess more voting weight than others. As a result, it opens up certain possibilities of vote manipulation, including control and bribery, that do not exist in standard voting scenarios of direct democracy. Here we formalize a certain kind of election control -- in which an external agent may change certain delegation arcs -- and study the computational complexity of the corresponding combinatorial problem.

Controlling Delegations in Liquid Democracy

TL;DR

Here a certain kind of election control is formalized - in which an external agent may change certain delegation arcs - and the computational complexity of the corresponding combinatorial problem is studied.

Abstract

In liquid democracy, agents can either vote directly or delegate their vote to a different agent of their choice. This results in a power structure in which certain agents possess more voting weight than others. As a result, it opens up certain possibilities of vote manipulation, including control and bribery, that do not exist in standard voting scenarios of direct democracy. Here we formalize a certain kind of election control -- in which an external agent may change certain delegation arcs -- and study the computational complexity of the corresponding combinatorial problem.
Paper Structure (11 sections, 8 theorems, 3 equations, 5 figures, 2 tables)

This paper contains 11 sections, 8 theorems, 3 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Control by Redirecting Arcs
  4. Liquid Democracy and Election Control
  5. Formal Model
  6. Hardness Results
  7. Algorithmic Results
  8. Preprocessing.
  9. Polynomial-Time Algorithm
  10. Parameterized Algorithms
  11. Outlook

Key Result

Theorem 1

CCRA is NP-hard when $\mathcal{R}$ is union function or approval function or GreedyMRC funtion even when Furthermore, it is W[2]-hard with respect to $k+\#{\texttt{redirections}}$ with all the above constraints except that in $1(a)$ out-degree is not bounded by a constant.

Figures (5)

  • Figure 1: (a) Input delegation graph, (b) Delegation graph after redirecting arc $(v_5,v_3)$ to $(v_5,v_6)$
  • Figure 2: Illustration of NP-hardness claimed in Theorem \ref{['thm:nph-general']} with constraints 1(a), 2, and 3. The dotted arcs are to illustrate that the out-degree of every $u'$ is three. Here, $e_1=u_1u_2$ is an edge in $G$. An approved candidate by an active voter is written in the blue color next to the voter name. The number on the top of an edge is the cost of redirection.
  • Figure 3: Modifications in Figure \ref{['fig:thm1-1']} in the vertex gadget, edge gadget, and dummy vertices for NP-hardness of Theorem \ref{['thm:nph-general']} with constraint 1(b). Here, $u$ is an endpoint of $e$ in $G$.
  • Figure 4: Illustration of NP-hardness claimed in Theorem \ref{['thm:nph-single-deleg']}. $E_u$ is the set of candidates corresponding to the edges incident to $u$ in $G$. The set of approved candidates by an active voter is written in the blue color next to the voter name. The number on an edge is the cost of redirection.
  • Figure 5: Modifications in Figure \ref{['fig:thm1-4']} for NP-hardness of Theorem \ref{['thm:nph-general']} with constraint 1(b).

Theorems & Definitions (20)

  • Example 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Claim 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 10 more