NDLI: A Special Case of Penrose’s Limit Theorem When Abstention is Allowed

Content Provider	Springer Nature Link
Author	Lindner, Ines
Copyright Year	2007
Abstract	In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: ‘yes’ or ‘no’. Felsenthal and Machover [Felsenthal, D.S. and Machover, M. (1997), International Journal of Game Theory 26, 335–351.] introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley–Shubik and Banzhaf indices in TVGs. Braham and Steffen [Braham, M. and Steffen, F. (2002), in Holler, et al. (eds.), Power and Fairness, Jahrbuch für Neue Politische Ökonomie 20, Mohr Siebeck, pp. 333–348.] argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between ‘yes’ or ‘no’. The purpose of this paper is two-fold. First, we compare the two approaches and show how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose’s Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.
Starting Page	495
Ending Page	518
Page Count	24
File Format	PDF
ISSN	00405833
Journal	Theory and Decision
Volume Number	64
Issue Number	4
e-ISSN	15737187
Language	English
Publisher	Springer US
Publisher Date	2007-08-11
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	limit theorems ternary voting games voting power weighted voting games Operations Research/Decision Theory Methodology of the Social Sciences Game Theory, Economics, Social and Behav. Sciences Game Theory/Mathematical Methods Economic Theory
Content Type	Text
Resource Type	Article
Subject	Arts and Humanities Economics, Econometrics and Finance Developmental and Educational Psychology Social Sciences Decision Sciences Applied Psychology Computer Science Applications

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