Constant-Factor Distortion Mechanisms for $k$-Committee Election
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| Format: | Preprint |
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2025
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| _version_ | 1866909479712325632 |
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| author | Pulyassary, Haripriya Swamy, Chaitanya |
| author_facet | Pulyassary, Haripriya Swamy, Chaitanya |
| contents | In the $k$-committee election problem, we wish to aggregate the preferences of $n$ agents over a set of alternatives and select a committee of $k$ alternatives that minimizes the cost incurred by the agents. While we typically assume that agent preferences are captured by a cardinal utility function, in many contexts we only have access to ordinal information, namely the agents' rankings over the outcomes. As preference rankings are not as expressive as cardinal utilities, a loss of efficiency is inevitable, and is quantified by the notion of \emph{distortion}.
We study the problem of electing a $k$-committee that minimizes the sum of the $\ell$-largest costs incurred by the agents, when agents and candidates are embedded in a metric space. This problem is called the $\ell$-centrum problem and captures both the utilitarian and egalitarian objectives. When $k \geq 2$, it is not possible to compute a bounded-distortion committee using purely ordinal information. We develop the first algorithms (that we call mechanisms) for the $\ell$-centrum problem (when $k \geq 2$), which achieve $O(1)$-distortion while eliciting only a very limited amount of cardinal information via value queries. We obtain two types of query-complexity guarantees: $O(\log k \log n)$ queries \emph{per agent}, and $O(k^2 \log^2 n)$ queries \emph{in total} (while achieving $O(1)$-distortion in both cases). En route, we give a simple adaptive-sampling algorithm for the $\ell$-centrum $k$-clustering problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constant-Factor Distortion Mechanisms for $k$-Committee Election Pulyassary, Haripriya Swamy, Chaitanya Computer Science and Game Theory Data Structures and Algorithms Multiagent Systems F.2.2; G.1.6; G.2; J.4 In the $k$-committee election problem, we wish to aggregate the preferences of $n$ agents over a set of alternatives and select a committee of $k$ alternatives that minimizes the cost incurred by the agents. While we typically assume that agent preferences are captured by a cardinal utility function, in many contexts we only have access to ordinal information, namely the agents' rankings over the outcomes. As preference rankings are not as expressive as cardinal utilities, a loss of efficiency is inevitable, and is quantified by the notion of \emph{distortion}. We study the problem of electing a $k$-committee that minimizes the sum of the $\ell$-largest costs incurred by the agents, when agents and candidates are embedded in a metric space. This problem is called the $\ell$-centrum problem and captures both the utilitarian and egalitarian objectives. When $k \geq 2$, it is not possible to compute a bounded-distortion committee using purely ordinal information. We develop the first algorithms (that we call mechanisms) for the $\ell$-centrum problem (when $k \geq 2$), which achieve $O(1)$-distortion while eliciting only a very limited amount of cardinal information via value queries. We obtain two types of query-complexity guarantees: $O(\log k \log n)$ queries \emph{per agent}, and $O(k^2 \log^2 n)$ queries \emph{in total} (while achieving $O(1)$-distortion in both cases). En route, we give a simple adaptive-sampling algorithm for the $\ell$-centrum $k$-clustering problem. |
| title | Constant-Factor Distortion Mechanisms for $k$-Committee Election |
| topic | Computer Science and Game Theory Data Structures and Algorithms Multiagent Systems F.2.2; G.1.6; G.2; J.4 |
| url | https://arxiv.org/abs/2501.19148 |