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On the distortion of single winner elections with aligned candidates

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Abstract

We study the problem of selecting a single element from a set of candidates on which a group of agents has some spatial preferences. The exact distances between agent and candidate locations are unknown but we know how agents rank the candidates from the closest to the farthest. Whether it is desirable or undesirable, the winning candidate should either minimize or maximize its aggregate distance to the agents. The goal is to understand the optimal distortion, which evaluates how good an algorithm that determines the winner based only on the agent rankings performs against the optimal solution. We give a characterization of the distortion in the case of latent Euclidean distances such that the candidates are aligned, but the agent locations are not constrained. This setting generalizes the well-studied setting where both agents and candidates are located on the real line. Our bounds on the distortion are expressed with a parameter which relates, for every agent, the distance to her best candidate to the distance to any other alternative.

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Notes

  1. We shall see that these properties hold under the mild assumption that no agent is equidistant from two distinct candidates.

  2. There is no incentive for a single agent or a group of agents to misreport their true rankings.

  3. In the present work, the location of the agents and the candidates are private.

  4. Note that the location of the candidates are public in [6, 15].

  5. Some previous works do not explicitly specify how to deal with ties probably because the input must contain strict preferences and ties are thus implicitly excluded. However, in [2, 19], the authors clearly state that no agent is equidistant from two candidates. In [24], the authors mention that candidates that are equidistant to an agent can be ranked arbitrarily by the agent.

  6. Technically, we consider that the distortion is 1 when both its numerator and denominator are 0. The distortion is infinite when its numerator is positive and its denominator is 0.

  7. Again, we consider that the distortion is 1 when both its numerator and denominator are 0; it is infinite if the denominator is 0 but the numerator is positive.

  8. The two candidates can even be co-located

  9. Under Assumption 2, there is a unique leftmost candidate and a unique rightmost candidate on the candidate line. The least preferred candidate of every agent (i.e., the farthest) must be one of them.

  10. See, for example, [16] and references therein for a similar result on a real line or a path.

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Acknowledgements

We thank anonymous reviewers for their valuable comments on the preliminary version of this work. Laurent Gourvès is supported by Agence Nationale de la Recherche (ANR), project THEMIS ANR-20-CE23-0018.

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Dimitris Fotakis

  2. CNRS, LAMSADE, Université Paris Dauphine-PSL, 75016, Paris, France

    Laurent Gourvès

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  1. Dimitris Fotakis
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Appendix

Appendix

1.1 About randomization

In the following result, the \({\bar{\alpha }}\)-distortion of a randomized algorithm \(\mathcal{A}\) under preference profile \(\succ \) is the worst case value that \(\frac{\sum _{i \in \mathcal {N}} d(i,\mathsf{opt}(\succ ))}{{\mathbb {E}}_{f \sim \mathcal {A}(\succ )}[\sum _{i \in \mathcal {N}} d(i,f)]}\) takes.

Proposition 2

When \(m=2\), any randomized algorithm has \({\bar{\alpha }}\)-distortion at least \(\frac{1+2 {\bar{\alpha }}}{1+ {\bar{\alpha }}}\).

Proof

Suppose there are two candidates \(f_1\) and \(f_2\), and two agents. Agent 1 has preference order \(f_1 \succ _1 f_2\) and agent 2’s preference order is \(f_2 \succ _2 f_1\). Suppose \(f_1\) and \(f_2\) are output with probability p and \(1-p\), respectively, with \(p \le 1-p\) (the case \(p > 1-p\) is symmetric).

Consider an instance where the candidates and the agents are on a line. The location of \(f_1\), \(f_2\), agent 1 and agent 2, are 0, \({\bar{\alpha }} +1\), \({\bar{\alpha }}\), and \({\bar{\alpha }} +1\), respectively.

The instance is consistent with the preference profile and \(\bar{\alpha }\)-decisive. The distortion is \(\frac{1+2{\bar{\alpha }}}{p(1+2 {\bar{\alpha }}) + 1-p} = \frac{1+2{\bar{\alpha }}}{1+ 2{\bar{\alpha }} p}\). The largest value that p can take is 1/2, giving a lower bound of \(\frac{1+2 {\bar{\alpha }}}{1+ {\bar{\alpha }}}\). \(\square \)

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Fotakis, D., Gourvès, L. On the distortion of single winner elections with aligned candidates. Auton Agent Multi-Agent Syst 36, 37 (2022). https://doi.org/10.1007/s10458-022-09567-5

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