Abstract
In this paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the Hoede–Bakker index—the notion which computes the overall decisional ‘power’ of a player in a social network. The main drawback of the Hoede–Bakker index is that it hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banzhaf J. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19: 317–343
Barry, B. (1980). Is it better to be powerful or lucky?, Part I and Part II. Political Studies, 28, 183–194, 338–352.
Coleman J.S. (1971). Control of collectivities and the power of a collectivity to act. In: Lieberman, B. (eds) Social choice, pp 269–300. Gordon and Breach, New York
Coleman, J. S. (1986). Individual interests and collective action: Selected essays. Cambridge University Press.
Deegan J. and Packel E.W. (1978). A new index of power for simple n-person games. International Journal of Game Theory 7: 113–123
Dubey P. and Shapley L.S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4: 99–131
Feix M., Lepelley D., Merlin V. and Rouet J.-L. (2007). On the voting power of an alliance and the subsequent power of its members. Social Choice and Welfare 28: 181–207
Felsenthal D. and Machover M. (1998). The measurement of voting power: Theory and practice, problems and paradoxes. Edward Elgar Publishers, London
Felsenthal D. and Machover M. (2002). Annexations and alliances: When are blocs advantageous a priori?. Social Choice and Welfare 19: 295–312
Felsenthal D., Machover M. and Zwicker W. (1998). The bicameral postulates and indices of a priori voting power. Theory and Decision 44: 83–116
Grabisch M. and Lange F. (2007). Games on lattices, multichoice games and the Shapley value: A new approach. Mathematical Methods of Operations Research 65: 153–167
Hoede C. and Bakker R. (1982). A theory of decisional power. Journal of Mathematical Sociology 8: 309–322
Holler M.J. and Packel E.W. (1983). Power, luck and the right index. Journal of Economics 43: 21–29
Hu X. and Shapley L.S. (2003). On authority distributions in organizations: Equilibrium. Games and Economic Behavior 45: 132–152
Johnston R.J. (1978). On the measurement of power: Some reactions to Laver. Environment and Planning A 10: 907–914
Laruelle A. and Valenciano F. (2005). Assessing success and decisiveness in voting situations. Social Choice and Welfare 24: 171–197
Owen G. (1977). Values of games with a priori unions. In: Hein, R. and Moeschlin, O. (eds) Essays in mathematical economics and game theory, pp 77–88. Springer-Verlag, New York
Owen G. (1981). Modification of the Banzhaf–Coleman index for games with a priori unions. In: Holler, M.J. (eds) Power, voting and voting power, pp 232–238. Würzburg-Wien, Physica Verlag
Penrose L.S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society 109: 53–57
Rae D. (1969). Decision-rules and individual values in constitutional choice. American Political Science Review 63: 40–56
Rusinowska, A. (2008). On the not-preference-based Hoede–Bakker index. In L. A. Petrosjan & V. V. Mazalov (Eds.), Game theory and applications (Vol. XIII). New York: Nova Science Publishers, Inc. (forthcoming).
Rusinowska, A., & De Swart, H. (2006). Generalizing and modifying the Hoede–Bakker index. In H. De Swart, et al. (Eds.), Theory and applications of relational structures as knowledge instruments (pp. 60–88). Springer’s Lecture Notes in Artificial Intelligence, LNAI 4342, Springer, Heidelberg, Germany.
Shapley L.S. (1953). A value for n-person games. Annals of Mathematics Studies 28: 307–317
Shapley L.S. and Shubik M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review 48: 787–792
van den Brink R. and Borm P. (2002). Digraph competitions and cooperative games. Theory and Decision 53: 327–342
van den Brink, R., Borm, P., Hendrickx, R., & Owen, G. (2007). Characterizations of the β- and the degree network power measure. Theory and Decision (forthcoming).
van den Brink R. and Gilles R.P. (2000). Measuring domination in directed networks. Social Networks 22: 141–157
van den Brink R. and van der Laan G. (1998). Axiomatization of the normalized Banzhaf value and the Shapley value. Social Choice and Welfare 15: 567–582
van der Laan G. and van den Brink R. (1998). Axiomatization of a class of share functions for n-person games. Theory and Decision 44: 117–148
van der Laan G. and van den Brink R. (2002). A Banzhaf share function for cooperative games in coalition structure. Theory and Decision 53: 61–86
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grabisch, M., Rusinowska, A. A model of influence in a social network. Theory Decis 69, 69–96 (2010). https://doi.org/10.1007/s11238-008-9109-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-008-9109-z