Generalized roll-call model for the Shapley-Shubik index

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Kurz, Sascha:
Generalized roll-call model for the Shapley-Shubik index.
2018
Event: 5. Workshop on Cooperative Game Theory in Business Practice - Financial Networks , 14.-15.06.2018 , Leipzig.
(Conference item: Workshop , Speech )

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In 1996 D. Felsenthal and M. Machover considered the following model. An assembly consisting of n voters exercises roll-call. All n! possible orders in which the voters may be called are assumed to be equiprobable. The votes of each voter are independent with expectation 0<p<1 for an individual vote yea. For a given decision rule v the pivotal voter in a roll-call is the one whose vote finally decides the aggregated outcome. It turned out that the probability to be pivotal is equivalent to the Shapley-Shubik index. Here we give an easy combinatorial proof of this coincidence, further weaken the assumptions of the underlying model, and study generalizations to the case of more than two alternatives.