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Impartial Selection with Additive Guarantees via Iterated Deletion

Javier Cembrano, Felix Fischer, David Hannon, Max Klimm

TL;DR

It is shown that for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.

Abstract

Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. For this problem, we give a deterministic mechanism with an additive performance guarantee of $O(n^{(1+κ)/2})$ in a setting with $n$ individuals where each individual casts $O(n^κ)$ nominations, where $κ\in[0,1]$. For $κ=0$, i.e., when each individual casts at most a constant number of nominations, this bound is $O(\sqrt{n})$. This matches the best-known guarantee for randomized mechanisms and a single nomination. For $κ=1$ the bound is $O(n)$. This is trivial, as even a mechanism that never selects provides an additive guarantee of $n-1$. We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.

Impartial Selection with Additive Guarantees via Iterated Deletion

TL;DR

It is shown that for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.

Abstract

Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. For this problem, we give a deterministic mechanism with an additive performance guarantee of in a setting with individuals where each individual casts nominations, where . For , i.e., when each individual casts at most a constant number of nominations, this bound is . This matches the best-known guarantee for randomized mechanisms and a single nomination. For the bound is . This is trivial, as even a mechanism that never selects provides an additive guarantee of . We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.
Paper Structure (10 sections, 10 theorems, 120 equations, 12 figures, 3 algorithms)

This paper contains 10 sections, 10 theorems, 120 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Iterated Deletion of Nominations
  6. A Tight Impossibility Result for Approval
  7. Proof of Lemma \ref{['lem:indegree-changes']}
  8. Proof of Lemma \ref{['lem:indegree-changes-two-graphs']}
  9. Proof of Lemma \ref{['lem:odd-graphs']}
  10. Proof of Lemma \ref{['lem:symmetry-axiom']}

Key Result

Theorem 1

For every $n\in\mathbb{N}$, $\kappa\in[0,1]$, and $k=O(n^{\kappa})$, there exists an impartial and $O(n^{\frac{1+\kappa}{2}})$-additive mechanism on $\mathcal{G}_n(k)$. Specifically, for every $n\in\mathbb{N}$, there exists an impartial and $\sqrt{7.25n}$-additive mechanism on $\mathcal{G}_n(1)$.

Figures (12)

  • Figure 1: Values of $\alpha_n$, for certain values of $n$, such that the supermajority rule with threshold $\lfloor n/2\rfloor+1$ (SR), the two contenders mechanism (TC), and an optimal twin threshold mechanism (TT$^*$) are $\alpha_n$-additive on $\mathcal{G}_n(1)$, together with a lower bound (LB) on $\alpha_n$ for any impartial $\alpha_n$-additive mechanism. Each bound for TT$^*$ in the table is accompanied by values of the thresholds $t$ and $T$ that achieve the bound, which may not be the unique such values.
  • Figure 2: Illustration of the edge deletion process in \ref{['alg:TTM']}. Here and in other figures, we arrange the vertices above $t$ vertically according to their indegree, with larger indegrees above, and horizontally according to their index, with greater indices to the left. Vertices below $t$, as well as edges with one or both endpoints below $t$, are omitted for simplicity.
  • Figure 3: Illustration of \ref{['lem:indegree-changes']} for $r=3$. If the indegree of $v$ drops as shown by the dashed arrow, there must be a vertex with an edge to $v$ in $A$, another vertex with an edge to $v$ in $A\cup B$, and a third vertex with an edge to $v$ in $A\cup B\cup C$. Note that this exact condition is satisfied for the example of \ref{['fig:example-mechanism']}.
  • Figure 4: By changing its outgoing edge, $\tilde{v}$ is able to affect whether it is selected by the mechanism or not. \ref{['lem:impartiality-additive-ub']} gives a condition over $T,\ t$, and $k$ such that this cannot happen. In this example, the sum of the indegrees of the vertices of $G_1$ that are shown in the figure is $5(t+1)$, so we need that $5(t+1)\leq kn$. On the other hand, when $T=t+2$\ref{['lem:impartiality-additive-ub']} states that impartiality is guaranteed as long as $4t+3>kn+\min\{k,2\}$. The graphs of this example clearly violate this condition.
  • Figure 5: Graphs in $\mathcal{G}^T_2$ and $\mathcal{G}^T_3$.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof : Proof of \ref{['lem:impartiality-additive-ub']}
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • ...and 14 more