Abstract
This paper considers the question of whether it is profitable for a weaker player to be closely linked to a strong/powerful player—our conjecture is ‘Yes’—and whether it is more beneficial to a strong/powerful player to be closely linked with a weak player than being linked with a strong player. Our understanding of power is based on the Public Good Index. We will demonstrate that in this sense power is a non-local concept indicating that strong players form a ‘hot-region’ about the strongest player. To obtain this result, we present an easy to perform algorithm for the computer-based determination of the Public Good Index on networks that equips us with instruments for studying the voting power in small networks.
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Notes
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Though, it can be that no winning coalitions exist. For instance, assume a set of players who are not connected with each other. If none has a weight larger or equal to the quorum then no winning coalition can be formed.
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Hotelling (1929) applied such a model to discuss competition in a one-dimensional spatial duoply market. However, he also applied his model to discuss spatial competition between the Democrats and the Republicans in the USA. The Public Choice literature by and large ignored Hotelling’s work and results. See the pioneering work by Anthony Downs (1957).
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Acknowledgements
The Authors would like to thank Luciano Andreozzi, Nicola Friederike Maaser and Aurelien Mekuko for their very helpful comments.
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Appendix: Essential Source Code for the Computation of the PGI in Voting Games on Networks
Appendix: Essential Source Code for the Computation of the PGI in Voting Games on Networks
As outlined, the programming environment of our choice is R, and for the following computation of power indices, the commands require the packages igraph and CoopGame to be installed and loaded. The first step is to define the underlying voting game:
Next, we pave the ground by discussing the simple voting game to gain information about its MWCs:
As a preparation for the PGI on the network structure, we generate an auxiliary Matrix \(M_2\), such that each row of \(M_2\) contains the numbers of the elements that constitute the MWC described in the row, together with the dummy element 0. Note, we assume that the grand coalition (containing all elements) is not a MWC, i.e., each row of \(M_2\) contains at least one dummy element.
Now, we turn to the network structure, and check whether a MWC of the simple voting game is feasible within the network structure:
Finally, we compute the total number of MWC a vertex belongs to and thus the PGI on the network structure:
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Holler, M.J., Rupp, F. (2023). The Power of Closeness in a Network. In: Kurz, S., Maaser, N., Mayer, A. (eds) Advances in Collective Decision Making. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-031-21696-1_15
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