Individual Representation in Approval-Based Committee Voting

TL;DR

This work introduces individual representation (IR) for approval-based committee voting, formalizing each voter's fair share via $f_i = \max_{S \subseteq A_i} \{|S| : |N(S)| \ge |S| \cdot n/k\}$ and requiring $|W \cap A_i| \ge f_i$ for all voters. It shows IR is computationally hard to decide and that many standard ABC rules can fail to deliver IR even when possible, while drawing a sharp line between voter-interval and candidate-interval domains. The paper also proves NP-hardness for IR-related decision problems and provides a polynomial-time $(2,4)$-IR approximation under voter-interval restrictions, supported by experiments indicating IR is often attainable in realistic profiles but not guaranteed by common rules. Overall, IR enriches the landscape of proportionality concepts by emphasizing individual guarantees and guiding future algorithmic design for IR-consistent voting in structured domains.

Abstract

When selecting multiple candidates based on approval preferences of agents, the proportional representation of agents' opinions is an important and well-studied desideratum. Existing criteria for evaluating the representativeness of outcomes focus on groups of agents and demand that sufficiently large and cohesive groups are ''represented'' in the sense that candidates approved by some group members are selected. Crucially, these criteria say nothing about the representation of individual agents, even if these agents are members of groups that deserve representation. In this paper, we formalize the concept of individual representation (IR) and explore to which extent, and under which circumstances, it can be achieved. We show that checking whether an IR outcome exists is computationally intractable, and we verify that all common approval-based voting rules may fail to provide IR even in cases where this is possible. We then focus on domain restrictions and establish an interesting contrast between ''voter interval'' and ''candidate interval'' preferences. This contrast can also be observed in our experimental results, where we analyze the attainability of IR for realistic preference profiles.

Figures (7)

• Figure 1: Approval profile showing that IR committees do not always exist. Voters correspond to integers and approve all candidates placed above them. For $k=3$, we have $n/k=3$ and $f_i=1$ for each voter $i \in [9]$. Clearly, there is no $W\subseteq \{c_1,c_2,c_3,c_4\}$ of size $|W|\le 3$ that satisfies $|W\cap A_i|\ge 1$ for all $i$. This instance appears in the paper by ABC+16a as Example 7.
• Figure 2: Two profiles admitting IR committees that are not identified by common voting rules or proportionality axioms.
• Figure 3: Relationships between different notions of representation. An arrow from $X$ to $Y$ signifies that $X$ implies $Y$. A committee providing one of the shaded notions does not always exist (the case for core stability is an open problem). PR is defined in \ref{['app:rel_perfect_repr']}.
• Figure 4: Profile showing that semi-strong JR is incompatible with PJR, EJR, and corestability.
• Figure 5: The ratio of generated profiles that admit an IR committee.

Theorems & Definitions (60)

• Definition 1
• Definition 2: Individual Representation
• Definition 3: $(\alpha,\beta)$-IR
• Theorem 2
• proof
• Example 1
• Example 2
• Proposition 2
• Proposition 2
• proof