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Gehrlein Stable Committee with Multi-modal Preferences

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Algorithmic Game Theory (SAGT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13584))

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Abstract

Inspired by Gehrlein stability in multiwinner election, in this paper, we define several notions of stability that are applicable in multiwinner elections with multimodal preferences, a model recently proposed by Jain and Talmon [ECAI, 2020]. In this paper we take a two-pronged approach to this study: we introduce several natural notions of stability that are applicable to multiwinner multimodal elections (MME) and show an array of hardness and algorithmic results.

In a multimodal election, we have a set of candidates, \(\mathcal{C}\), and a multi-set of \(\ell \) different preference profiles, where each profile contains a multi-set of strictly ordered lists over \(\mathcal {C}\). The goal is to find a committee of a given size, say k, that satisfies certain notions of stability. In this context, we define the following notions of stability: global-strongly (weakly) stable, individual-strongly (weakly) stable, and pairwise-strongly (weakly) stable. In general, finding any of these committees is an intractable problem, and hence motivates us to study them for restricted domains, namely single-peaked and single-crossing, and when the number of voters is odd. Besides showing that several of these variants remain computationally intractable, we present several efficient algorithms for certain parameters and restricted domains.

SG received funding from MATRICS Grant (MTR/2021/000869) and SERB-SUPRA Grant(SPR/2021/000860).

PJ received funding from Seed Grant (IITJ/R &D/2022-23/07) and SERB-SUPRA Grant(SPR/2021/000860).

SR is supported by the CTU Global postdoc fellowship program.

SS received funding from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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Notes

  1. 1.

    There are several other ways to submit a ballot.

  2. 2.

    For any \(x \in \mathbb {N}\), [x] denotes the set \(\{1,2,\ldots , x\}\).

  3. 3.

    In the graph-theoretic formulation, we will refer to the candidates as vertices.

  4. 4.

    GSCS is used in [25] as they only considered the weak stability notion.

  5. 5.

    Here QP denotes the complexity class quasi-polynomial.

  6. 6.

    In [2], the term “strict” is used instead of “strong” (Def. 1 and first para in Sec 5 of [2]).

  7. 7.

    The proofs marked by \(\spadesuit \) are deferred to the full version.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, HBNI, Chennai, India

    Sushmita Gupta & Saket Saurabh

  2. Indian Institute of Technology Jodhpur, Jodhpur, India

    Pallavi Jain

  3. University of California, Santa Barbara, Santa Barbara, USA

    Daniel Lokshtanov

  4. Czech Technical University in Prague, Prague, Czech Republic

    Sanjukta Roy

  5. University of Bergen, Bergen, Norway

    Saket Saurabh

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  1. Sushmita Gupta
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  5. Saket Saurabh
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Correspondence to Sanjukta Roy .

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Editors and Affiliations

  1. University of Essex, Colchester, UK

    Panagiotis Kanellopoulos

  2. University of Essex, Colchester, UK

    Maria Kyropoulou

  3. University of Essex, Colchester, UK

    Alexandros Voudouris

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Gupta, S., Jain, P., Lokshtanov, D., Roy, S., Saurabh, S. (2022). Gehrlein Stable Committee with Multi-modal Preferences. In: Kanellopoulos, P., Kyropoulou, M., Voudouris, A. (eds) Algorithmic Game Theory. SAGT 2022. Lecture Notes in Computer Science, vol 13584. Springer, Cham. https://doi.org/10.1007/978-3-031-15714-1_29

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  • DOI: https://doi.org/10.1007/978-3-031-15714-1_29

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