The power of small coalitions under two-tier majority on regular graphs

Pavel Chebotarev; David Peleg

doi:10.1016/j.dam.2023.07.011

The power of small coalitions under two-tier majority on regular graphs

Pavel Chebotarev, David Peleg

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network G, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex v has its own valuation of the proposal; we say that v is “happy” if its valuation is positive (i.e., it expects to gain from adopting the proposal) and “sad” if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex v is a proponent of the proposal if the majority of its neighbors are happy with it and an opponent in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class G_n|d|h of d-regular graphs of odd order n with all n loops and h happy vertices. We are interested in establishing necessary and sufficient conditions for the class G_n|d|h to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.

Original language	English
Pages (from-to)	239-258
Number of pages	20
Journal	Discrete Applied Mathematics
Volume	340
DOIs	https://doi.org/10.1016/j.dam.2023.07.011
State	Published - 15 Dec 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023 Elsevier B.V.

Funding

The work of P.C. was supported by the European Union (ERC, GENERALIZATION, 101039692 ). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. This paper was written while P. Chebotarev was visiting the Weizmann Institute of Science in May–July 2022. We thank Kieka Mynhardt for sharing several electronic copies of papers published in Ars Combinatoria in the 1990s, which are currently a bibliographic rarity. The work of P.C. was supported by the European Union (ERC, GENERALIZATION, 101039692). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Funders	Funder number
GENERALIZATION	101039692
Kieka Mynhardt
European Research Executive Agency
European Commission
European Commission

Keywords

Majority of majorities
Regular graphs
Small coalitions
Voting systems

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10.1016/j.dam.2023.07.011

Cite this

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The power of small coalitions under two-tier majority on regular graphs

Abstract

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Keywords

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