Abstract
Isbell (1956) introduced a class of homogeneous weighted majority games based on the Fibonacci sequence. In our paper, we generalize this approach to other homogeneous representations of weighted majority games in a suitable Fibonacci framework. We provide some properties of such representations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
That is w(N) is the smallest among all equivalent integer representations of the game.
- 2.
As usual \(\lfloor x\rfloor \) denotes the floor(x). In particular, for \(m-h-2\) integer even (odd), \(\lfloor \frac{(m-h-2)}{2}\rfloor =\frac{(m-h-2)}{2}\) (or \(\frac{(m-h-3)}{2})\).
References
Chandra P., Weisstein E.W.: Fibonacci Number, MathWorld, A Wolfram Web Resource (2011). http://mathworld.wolfram.com/FibonacciNumber.html
Fibonacci (Leonardo Pisano), Liber Abbaci (1202)
Freixas, J., Kurz, S.: The golden number and Fibonacci sequences in the design of voting structures. Eur. J. Oper. Res. 226, 246–257 (2013)
Gambarelli, G.: Weighted majority game. In: Dowding, K. (ed.) Encyclopedia of Power, pp. 709–710. SAGE, Thousand Oaks (2011)
Isbell, R.: A class of majority games. Q. J. Math. 7, 183–187 (1956)
Ostmann, A.: On the minimal representation of homogeneous games. Int. J. Game Theor. 16, 69–81 (1987)
Peters, H.: Homogeneous weighted majority games. In: Dowding, K. (ed.) Encyclopedia of Power, p. 324. SAGE, Thousand Oaks (2011)
Pressacco, F., Ziani, L.: A Fibonacci approach to weighted majority games. J. Game Theor. 4, 36–44 (2015)
Rosenmüller, J.: Weighted majority games and the matrix of homogeneity. Oper. Res. 28, 123–141 (1984)
Rosenmüller, J.: Homogeneous games: recursive structure and computation. Math. Oper. Res. 12–2, 309–330 (1987)
Rosenmüller, J., Sudhölter, P.: The nucleolus of homogeneous games with steps. Discrete Appl. Math. 50, 53–76 (1994)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, Princeton (1947)
Acknowledgements
On November 4, 1977 two promising young researchers, Flavio Pressacco and Gianfranco Gambarelli, presented two works on Cooperative Games at the first Italian meeting of AMASES (Associazione per la Matematica Applicata alle Scienze Economiche e Sociali), held in Pisa. They met again in Pisa on September 15, 2011, with some white hair for the 35-th meeting of the same association, for which, in the meanwhile, both of them played the role of members of the Scientific Committee and Flavio Pressacco also the charge of President. This paper was thought as a tribute to the first meeting with implicit thanks to the AMASES for the opportunities of meeting each other and of enjoying an academic life. In the following, Nicola Gnocchi, after his graduation in Bergamo, joined in the part of the research that required a fresh mind, with the thanks and wishes of the other authors. Finally, also Vito Fragnelli and Laura Ziani entered, with valuable contributions, in the author’s team. This work is sponsored by MIUR.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fragnelli, V., Gambarelli, G., Gnocchi, N., Pressacco, F., Ziani, L. (2016). Fibonacci Representations of Homogeneous Weighted Majority Games. In: Nguyen, N., Kowalczyk, R., Mercik, J. (eds) Transactions on Computational Collective Intelligence XXIII. Lecture Notes in Computer Science(), vol 9760. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52886-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-662-52886-0_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-52885-3
Online ISBN: 978-3-662-52886-0
eBook Packages: Computer ScienceComputer Science (R0)