Abstract
We introduce a voting rule for committee selection that captures positive correlation (synergy) between candidates. We argue that positive correlation can naturally happen in common scenarios that are related to committee selection. For example, in the movies selection problem, where prospective travelers are requested to choose the movies that will be available on their flight, it is reasonable to assume that they will tend to prefer voting for a movie in a series, only if they can watch also the former movies in that series. In elections to the parliament, it can be that two candidates are working extremely well together, so voters will benefit from being represented by both of them together.
In our model, the preferences of the candidates are represented by set functions, and we would like to maximize the total satisfaction of the voters. We show that although computing the best solution is \(\mathcal{NP}\)-hard, there exists an approximation algorithm with approximation guarantees that deteriorate gracefully with the amount of synergy between the candidates. This amount of synergy is measured by a natural extension of the supermodular degree [Feige and Izsak, ITCS 2013] that we introduce – the joint supermodular degree. To the best of our knowledge, our results represent the first voting rule that capture synergy between specific candidates.
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Notes
- 1.
A submodular set function is a function \(f:2^M \rightarrow \mathbb {R}^+\), such that for every \(S' \subseteq S \subseteq M\), and every \(j \in M\), \(f(j \mid S') \ge f(j \mid S)\), where \(f(j \mid S) = f(\{j\} \cup S) - f(S)\) is the marginal value of j with respect to S. That is, the marginal values are monotone non-increasing.
- 2.
NP-hardness is actually true also for submodular set functions, i.e. supermodular degree of 0.
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Acknowledgments
Work supported in part by the Israel Science Foundation (grant No. 1388/16). I would like to thank Uri Feige for many useful discussions and for his contributions to this paper. I would like to thank Nimrod Talmon for useful discussions and for directing me to the paper of Skowron, Faliszewski and Lang [21]. I would also like to thank Moshe Babaioff, Shahar Dobzinski and Moshe Tennenholtz for useful discussions.
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Izsak, R. (2017). Working Together: Committee Selection and the Supermodular Degree. In: Sukthankar, G., Rodriguez-Aguilar, J. (eds) Autonomous Agents and Multiagent Systems. AAMAS 2017. Lecture Notes in Computer Science(), vol 10642. Springer, Cham. https://doi.org/10.1007/978-3-319-71682-4_7
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