The conditional Shapley–Shubik measure for ternary voting games

Friedman, Jane; Parker, Cameron

doi:10.1016/j.geb.2018.03.014

Cited by 3 publications

(7 citation statements)

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“…6 However, this is an artifact for j = 2 and for j > 2 different probabilities for the input levels, as well as more complicated probability distributions on vote vectors, lead to different results in the roll call model. The case (j, k) = (3, 2) was studied in more detail in [14], where the authors defined a conditional Shapley-Shubik index given some fixed probability of the voters to abstain.…”

Section: The Roll Call Modelmentioning

confidence: 99%

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An Axiomatization of the Shapley-Shubik Index for Interval Decisions

2019

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The Shapley-Shubik index was designed to evaluate the power distribution in committee systems drawing binary decisions and is one of the most established power indices. It was generalized to decisions with more than two levels of approval in the input and output. In the limit we have a continuum of options. For these games with interval decisions we prove an axiomatization of a power measure and show that the Shapley-Shubik index for simple games, as well as for (j, k) simple games, occurs as a special discretization. This relation and the closeness of the stated axiomatization to the classical case suggests to speak of the Shapley-Shubik index for games with interval decisions, that can also be generalized to a value.

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Section: The Roll Call Modelmentioning

confidence: 99%

“…This changes if voters have at least a third option. So given some probability that voters do not abstain, Friedman and Parker consider a conditional Shapley-Shubik power index for (3,2) simple games [14]. For general (j, k) simple games a Shapley-Shubik power index was introduced in [9].…”

Section: Introductionmentioning

confidence: 99%

An Axiomatization of the Shapley-Shubik Index for Interval Decisions

2019

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“…However, also other variants have been introduced in the literature, see e.g. [2,8,10]. Here, we will only consider the variant from [5].…”

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confidence: 99%

Axiomatizations for the Shapley-Shubik Power Index for Games With Several Levels of Approval in the Input and Output

2020

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The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called (j, k) simple games. Here we present a new axiomatization for the Shapley-Shubik index for (j, k) simple games as well as for a continuous variant, which may be considered as the limit case.

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“…Using (8), one gets Φ a (u b ) = (0, 1, 0, • • • , 0). It then follows from equations (15) and (16) that Φ a does not satisfy (AC).…”

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confidence: 99%

“…The proof for (C) and (L) goes along the same lines as the proof of Proposition 3.1. Also the generalization of the power index to a parametric class can be done just as the one for (j, k) simple games in Equation(8).…”

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confidence: 99%