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Unravelling multi-agent ranked delegations

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Abstract

We introduce a voting model with multi-agent ranked delegations. This model generalises liquid democracy in two aspects: first, an agent’s delegation can use the votes of multiple other agents to determine their own—for instance, an agent’s vote may correspond to the majority outcome of the votes of a trusted group of agents; second, agents can submit a ranking over multiple delegations, so that a backup delegation can be used when their preferred delegations are involved in cycles. The main focus of this paper is the study of unravelling procedures that transform the delegation ballots received from the agents into a profile of direct votes, from which a winning alternative can then be determined by using a standard voting rule. We propose and study six such unravelling procedures, two based on optimisation and four using a greedy approach. We study both algorithmic and axiomatic properties, as well as related computational complexity problems of our unravelling procedures for different restrictions on the types of ballots that the agents can submit.

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Notes

  1. Note that in order to use Boolean functions to express a delegation, the domain of the alternatives for the issue must be a Boolean algebra. We restrict this to a two-element Boolean algebra, namely \(\{0,1\}\).

  2. This notion, when restricted to ballots with single-agent delegations, corresponds to the definition of confluent sequence rules by Brill et al. [12].

  3. In the following, we will simply write Unravel(\(\#\)), for \(\# \in \{\mathbf{U}, \mathbf{DU}, \mathbf{RU}, \mathbf{DRU} \}\), to indicate the Unravel algorithm using Update procedure \(\#\).

  4. Unless otherwise specified, in case the condition in an if statement fails, our programs will skip to the next step. Recall also that \(Y_{\restriction S}\) denotes the restriction of vector Y to the elements in set S.

  5. Observe that a formula of propositional logic is a Boolean function.

  6. Note that in previous work [19], the language \(\textsc {Bool}\) was initially defined simply as the language of contingent propositional formulas in DNF, for which however the necessary winners cannot be computed in polynomial time. We are grateful to an anonymous reviewer for pointing this out.

  7. We previously showed [19] that checking if a ballot of contingent DNF formulas is valid is an NP-complete problem. Restricting formulas to contingent complete DNFs makes this problem tractable.

  8. The formulation by Karp [35] is on directed graphs G which allow for reflexive edges. However, our sub-problem is also NP-complete, since a reduction can be given where the constructed graph \(G'\) adds a dummy node \(a'\) for each node a that had a reflexive edge in G, as well as the edges \((a,a')\) and \((a',a)\).

  9. Also independently suggested by Chu [16] and Bock [7].

  10. For undirected graphs, the corresponding problem is that of finding a minimum spanning tree.

  11. Recall that since both \(\mathcal {D}\) and the possible sets of delegates are finite, and since all functions given in an agent’s valid ballot must differ, the possible number of functions must also be finite.

  12. Note that Definition 10 slightly differs from the one given in previous work [19], and thus Theorem 5 does not hold for \(\mathbf {RU}\) or \(\mathbf {DRU}\): a counterexample can be constructed exploiting the fact that an agent may prefer the outcome of one random iteration of the procedure to another.

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Acknowledgements

The authors acknowledge the support of the ANR JCJC project SCONE (ANR 18-CE23-0009-01).

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Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse (IRIT), University of Toulouse, Toulouse, France

    Rachael Colley & Umberto Grandi

  2. Centre d’Economie de la Sorbonne (CES), University of Paris 1 Panthéon-Sorbonne, Paris, France

    Arianna Novaro

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  1. Rachael Colley
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Correspondence to Rachael Colley.

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This paper revises and extends our previous work presented at IJCAI-2020 [19], which was developed from ideas discussed at the Dagstuhl Seminar 19381 on Application-Oriented Computational Social Choice in September 2019. We are grateful for the feedback received by the anonymous reviewers of IJCAI-2020 and JAAMAS, as well as the audience of MPREF-2020 and the COMSOC video seminar. Some of the work in this paper was performed while the third author was affiliated with the Institute for Logic, Language and Computation (ILLC) at the University of Amsterdam.

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Colley, R., Grandi, U. & Novaro, A. Unravelling multi-agent ranked delegations. Auton Agent Multi-Agent Syst 36, 9 (2022). https://doi.org/10.1007/s10458-021-09538-2

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