Impartial Selection with Predictions

TL;DR

This work investigates impartial selection mechanisms augmented with predictions of the most nominated agents. It introduces a tunable $\rho$-permutation mechanism family that trades off consistency, the quality of predictions, and robustness against prediction inaccuracy, and extends these ideas to more complex $k$-selection through a $\rho$-partition mechanism. The authors derive precise, often optimal, trade-offs between $\alpha$-consistency and $\beta$-robustness across single, double, and multi-selection settings, including special results for plurality voting. They also establish upper bounds showing inherent limits to what any impartial mechanism with predictions can guarantee, revealing that (asymptotically) near-optimal consistency can be achieved with only modest sacrifices in robustness. Overall, the paper demonstrates that predictions can substantially improve impartial selection performance while preserving strong fairness and strategic-robust guarantees, and it maps the landscape of achievable trade-offs for future work and applications.

Abstract

We study the selection of agents based on mutual nominations, a theoretical problem with many applications from committee selection to AI alignment. As agents both select and are selected, they may be incentivized to misrepresent their true opinion about the eligibility of others to influence their own chances of selection. Impartial mechanisms circumvent this issue by guaranteeing that the selection of an agent is independent of the nominations cast by that agent. Previous research has established strong bounds on the performance of impartial mechanisms, measured by their ability to approximate the number of nominations for the most highly nominated agents. We study to what extent the performance of impartial mechanisms can be improved if they are given a prediction of a set of agents receiving a maximum number of nominations. Specifically, we provide bounds on the consistency and robustness of such mechanisms, where consistency measures the performance of the mechanisms when the prediction is accurate and robustness its performance when the prediction is inaccurate. For the general setting where up to $k$ agents are to be selected and agents nominate any number of other agents, we give a mechanism with consistency $1-O\big(\frac{1}{k}\big)$ and robustness $1-\frac{1}{e}-O\big(\frac{1}{k}\big)$. For the special case of selecting a single agent based on a single nomination per agent, we prove that $1$-consistency can be achieved while guaranteeing $\frac{1}{2}$-robustness. A close comparison with previous results shows that (asymptotically) optimal consistency can be achieved with little to no sacrifice in terms of robustness.

Figures (7)

• Figure 1: Trade-off between $\alpha$-consistency and $\beta$-robustness of impartial $k$-selection mechanisms. Orange dots are the best mechanisms from previous work; orange areas are the whole ranges of possible consistency--robustness combinations implied by them. Green dots are the trivial mechanisms always selecting the predicted set; green areas are new ranges of consistency--robustness combinations implied by lotteries of them with previous work. Blue dots and blue rounded rectangles are new mechanisms introduced in this paper; blue areas are new ranges of consistency--robustness combinations implied by them or by lotteries of them with previous work. Gray areas are impossible consistency--robustness combinations as shown in \ref{['thm:ubs']}. Whether the combinations in the white areas are achievable by impartial mechanisms is left for future research.
• Figure 2: $\rho$-partition mechanism $\text{Pt}^\rho(\hat{S},G)$
• Figure 3: Impartial $1$-selection from $n$-vertex graphs. The predicted vertex is shown in white. Only $2$ vertices are shown; the remaining $n-2$ vertices do not have any incident edges.
• Figure 4: Impartial $1$-selection from $4$-vertex plurality graphs. The predicted vertex is shown in white.
• Figure 5: Impartial $2$-selection from $n$-vertex graphs. Predicted vertices are shown in white. Only $3$ vertices are shown; the remaining $n-3$ vertices do not have any incident edges.

Theorems & Definitions (26)

• Proposition 3.1
• Lemma 3.2: bjelde2017impartial
• Theorem 3.3
• Corollary 3.4
• Theorem 4.1
• Proposition 4.2
• Proposition 5.1
• Theorem 5.2
• Theorem 6.1
• Lemma A.1