An issue based power index
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An issue based power index Qianqian Kong1,2 · Hans Peters2 Accepted: 6 September 2020 © The Author(s) 2020
Abstract An issue game is a combination of a monotonic simple game and an issue profile. An issue profile is a profile of linear orders on the player set, one for each issue within the set of issues: such a linear order is interpreted as the order in which the players will support the issue under consideration. A power index assigns to each player in an issue game a nonnegative number, where these numbers sum up to one. We consider a class of power indices, characterized by weight vectors on the set of issues. A power index in this class assigns to each player the weighted sum of the issues for which that player is pivotal. A player is pivotal for an issue if that player is a pivotal player in the coalition consisting of all players preceding that player in the linear order associated with that issue. We present several axiomatic characterizations of this class of power indices. The first characterization is based on two axioms: one says how power depends on the issues under consideration (Issue Dependence), and the other one concerns the consequences, for power, of splitting players into several new players (no advantageous splitting). The second characterization uses a stronger version of Issue Dependence, and an axiom about symmetric players (Invariance with respect to Symmetric Players). The third characterization is based on a variation on the transfer property for values of simple games (Equal Power Change), besides Invariance with respect to Symmetric Players and another version of Issue Dependence. Finally, we discuss how an issue profile may arise from preferences of players about issues. Keywords Power index · Simple game · Issue profile
Supported by the Innovation Foundation for Northwestern Polytechnical University (Grant Number CX201961) and by the China Scholarship Council (Grant Number 201906290180). We thank the reviewers and the associate editor for their comments.
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Hans Peters [email protected] Qianqian Kong [email protected]
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Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, People’s Republic of China
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Department of Quantitative Economics, Maastricht University, Maastricht, The Netherlands
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Q. Kong, H. Peters
JEL Classification C71 · D70 AMS Classification 91A12
1 Introduction 1.1 Background Power indices for simple games measure the power of players in such a game, independently of the issues at stake or the positions of players regarding these issues. For instance, a power index applied to a weighted majority game associated with a political parliament, typically considers how often a political party is needed to form a majority, without taking the issue at stake (for instance, a new law) into account. One may well argue that this is how it should be (for instance, Braham and Holler 2005), but one may also argue that this is a drawback (e.g., Napel and Widgrén 2005). For a relatively recent overview of power indi
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