Deterministic Impartial Selection with Weights

TL;DR

This work advances the theory of impartial selection by providing the first deterministic, weighted-impartial mechanism for inexact $(n,k)$-selection with a nontrivial approximation ratio, achieved via robust partition systems and hypergraph duality. The Select$_k$ mechanism attains an explicit α-optimal bound $\alpha = \frac{k - k\bmod 2}{k \lceil \frac{2n}{k - k\bmod 2} \rceil}$ for the regime $k - k\bmod 2 \ge 2\sqrt{n}$, and an extension lemma broadens applicability to general $(n,k)$. The approach generalizes to impartial assignment across $m$ jobs, incurring a factor-$\tfrac12$ loss in the approximation. These results improve deterministic guarantees beyond the classic $1/k$ bound in the unweighted setting for large $k$, and hinge on a novel robust partition framework rooted in 2-regular, b-uniform, linear hypergraphs and their graph duals.

Abstract

In the impartial selection problem, a subset of agents up to a fixed size $k$ among a group of $n$ is to be chosen based on votes cast by the agents themselves. A selection mechanism is impartial if no agent can influence its own chance of being selected by changing its vote. It is $α$-optimal if, for every instance, the ratio between the votes received by the selected subset is at least a fraction of $α$ of the votes received by the subset of size $k$ with the highest number of votes. We study deterministic impartial mechanisms in a more general setting with arbitrarily weighted votes and provide the first approximation guarantee, roughly $1/\lceil 2n/k\rceil$. When the number of agents to select is large enough compared to the total number of agents, this yields an improvement on the previously best known approximation ratio of $1/k$ for the unweighted setting. We further show that our mechanism can be adapted to the impartial assignment problem, in which multiple sets of up to $k$ agents are to be selected, with a loss in the approximation ratio of $1/2$.

Figures (5)

• Figure 1: The construction of \ref{['lem:graph']} for $k = 8$ vertices and degree $b \in [4]$: (a) the $4P_2$ ($n = 4$ edges), (b) the cycle $C_8$ ($n = 8$), (c) the cube graph $Q_3$ ($n = 12$), and (d) the complete bipartite graph $K_{4, 4}$ ($n = 16$). Every edge represents an agent and every vertex corresponds to a partition. A vertex and an edge are incident if the corresponding agent is in the corresponding candidate set.
• Figure 2: The performance guarantee of \ref{['thm:main']} for permissible $n$ and $k$.
• Figure 3: Example of $\textsc{Select}_6(A)$ for $A\in \mathcal{A}_{9}$. The weight matrix $A$ is shown alongside its graph representation, where votes of weight 1 are in blue, weight 2 are in orange, weight 3 are in red, and votes of weight 0 are not included. The partition system is given below, where omitted votes are shown in gray. For each partition, the selected agent is highlighted in light blue. Observe that $\sigma(\textsc{Select}_6(A)) = 17$ and $\sigma(\textsc{Opt}_6(A)) = 27$; the multiplicative guarantee provided by \ref{['lem:bound-nice-n-k']} for this instance is $1/3$.
• Figure 4: Example of the construction of the proof of \ref{['thm:tightness']} for $n=9$ and $k=6$ with $3$ votes of weight $1$: agent $4$ votes for agent $1$, agent $5$ votes for agent $2$, and agent $6$ votes for agent $3$. All votes are only seen in the first partition. Since agents with positive scores have the smallest indices, they are not selected in their second candidate set.
• Figure 5: Example of $\textsc{Assign}_6(\mathbf{A})$ for $\mathbf{A} \in \mathcal{A}_9^2$. The votes carry unit weight and are shown in orange for job $1$ and in blue for job $2$. The partition system is depicted below, separately for each job, with omitted votes drawn in gray. For each partition and each job, the selected agent is highlighted in light orange or light blue, if it remains assigned to the corresponding job, and in gray, if it is unassigned due to a duplicate preliminary assignment of the agent to distinct jobs. This affects agents $3$, $5$, and $7$: Since agent $3$ has three votes for job $1$ and only one vote for job $2$, it is unassigned from job $2$. Agent $5$ has two votes for job $1$ and no votes for job $2$, so it is unassigned from job $2$. Likewise, agent $7$ has one vote for job $1$ and two votes for job $2$, and is thus unassigned from job $1$. It can be confirmed that $\sigma(\textsc{Assign}_6(\mathbf{A})) = 16$ and $\sigma(\textsc{Opt}_6(\mathbf{A})) = 18$ (reassign agents $1$ and $2$ to job $1$); the multiplicative guarantee provided by \ref{['lem:bound-nice-n-k-assignment']} for this instance is $1/6$.

Theorems & Definitions (20)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3