A Probabilistic Model for Positional Voting - Spectrum Games -
Sammanfattning: This thesis considers a class of cooperative n-person games (voting games) in which the voters are spread across an ideological scale. The two most commonly used measures of individual voting power in voting games are the Shapley-Shubik index and the Banzhaf-Coleman index. In both indices, the actors are assumed anomymous and treatad symmetrically. That is, the indices do not take into account that there might be communicative problems among the actors. This fact suggests that the indices should be suitably modified to better reflect the true individual strength of the actors in such voting situations. Several attempts to solve this problem have been made and we focus on the approach suggested by Edelman. The idea is to use the Shapley-Shubik index only on the connected coalitions, i.e. those coalitions where there cannot be any non-members who are ideologically in an intermediate position between any two coalition members. This approach, however, gives some counter-intuitive results, and one part of the thesis provides a reason for this phenomenon. The Edelman approach is then extended by combining connected coalitions with a probabilistic model for these coalitions. Paper 1 uses Edelman's extension of the Shapley-Shubik index to determine the voting power distribution in a number of common voting situations. Paper 2 extends the ideological scale to admit each position to contain more than one voter and suggests several ways to generalize the concept connected coalitions. Using a simple binary probabilistic process, Paper 3 introduces the Markov-Pólya index as a parametrized family of power indices which has Edelman's model as a special case. Paper 4 investigates the Markov-Pólya index further, and, finally, in Paper 5 the Markov-Pólya index is compared with other power indices suggested for ideological voting.
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