Abstract
The Isbell desirability relation (I), the Shapley–Shubik index (SS) and the Banzhaf–Coleman index (BC) are power theories that grasp the notion of individual influence in a yes–no voting rule. Also, a yes–no voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes–no voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.
Similar content being viewed by others
References
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Diffo Lambo L, Moulen J (2000) Quel pouvoir mesure-t-on dans un jeu de vote? Math Sci Hum. 38e année, no 152, pp 27–47
Diffo Lambo L, Moulen J (2002) Ordinal equivalence of power notions in voting games. Theory Decis 53: 313–325
Diffo Lambo L, Tchantcho B, Mulen J (2004) Pouvoir mesuré et capacité à à influencer le résultat du vote. Math Sci Hum 166: 5–24
Felsenthal DS, Machover M (1997) Voting games with abstention. Intl J Game Theory 26: 335–351
Felsenthal DS, Machover M (1998) The measurement of voting power. Theory and practice, problems and paradoxes. Edward Elgar, London
Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21: 399–431
Isbell JR (1975) A class of simple games. Duke Math J 25: 423–439
Moulen J, Diffo Lambo L (2001) Théorie du vote. HERMES Science, Paris
Pongou R, Tchantcho B, Diffo Lambo L (2011) Political Influence in multi-choice institutions: cyclicity, anonymity and transitivity. Theory Decis 70(2): 157–178
Shapley LS, Shubik M (1954) A model for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792
Taylor AT (1995) Mathematics and politics. Springer, New York
Taylor AT, Zwicker W (1992) A characterization of weighted voting. In: Proceedings of the American Mathematical Society, Providence, vol 115, No 4, August 1992
Taylor AT, Zwicker W (1993) Weighted voting, multicameral representation and power. Games Econ Behav 5: 170–181
Tchantcho B, Diffo Lambo L, Pongou R, Mbama Engoulou B (2008) Voters’ power in voting games with abstention: influence relation and ordinal equivalence of power theories. Games Econ Behav 64: 335–350
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Diffo Lambo, L., Tchantcho, B. & Moulen, J. Comparing influence theories in voting games under locally generated measures of dissatisfaction. Int J Game Theory 41, 719–731 (2012). https://doi.org/10.1007/s00182-012-0342-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-012-0342-x