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Applications of Shapley-Owen Values and the Spatial Copeland Winner

Published online by Cambridge University Press:  04 January 2017

Joseph Godfrey ,
Bernard Grofman  and
Scott L. Feld
Joseph Godfrey*
Affiliation:
WinSet Group, LLC, 4031 University Drive, Suite 200, Fairfax, VA 22030
Bernard Grofman
Affiliation:
Department of Political Science and Institute for Mathematical Behavioral Sciences, University of California, Irvine, CA 92697-5100
Scott L. Feld
Affiliation:
Department of Sociology, Purdue University, Lafayette, IN 47907
*
e-mail: http://www.winset.com (corresponding author)
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Abstract

The Shapley-Owen value (SOV, Owen and Shapley 1989, Optimal location of candidates in ideological space. International Journal of Game Theory 125–42), a generalization of the Shapley-Shubik value applicable to spatial voting games, is an important concept in that it takes us away from a priori concepts of power to notions of power that are directly tied to the ideological proximity of actors. SOVs can also be used to locate the spatial analogue to the Copeland winner, the strong point, the point with smallest win-set, which is a plausible solution concept for games without cores. However, for spatial voting games with many voters, until recently, it was too computationally difficult to calculate SOVs, and thus, it was impossible to find the strong point analytically. After reviewing the properties of the SOV, such as the result proven by Shapley and Owen that size of win sets increases with the square of distance as we move away from the strong point along any ray, we offer a computer algorithm for computing SOVs that can readily find such values even for legislatures the size of the U.S. House of Representatives or the Russian Duma. We use these values to identify the strong point and show its location with respect to the uncovered set, for several of the U.S. congresses analyzed in Bianco, Jeliazkov, and Sened (2004, The limits of legislative actions: Determining the set of enactable outcomes given legislators preferences. Political Analysis 12:256–76) and for several sessions of the Russian Duma. We then look at many of the experimental committee voting games previously analyzed by Bianco et al. (2006, A theory waiting to be discovered and used: A reanalysis of canonical experiments on majority-rule decision making. Journal of Politics 68:838–51) and show how outcomes in these games tend to be points with small win sets located near to the strong point. We also consider how SOVs can be applied to a lobbying game in a committee of the U.S. Senate.

Type
Research Article
Information
Political Analysis , Volume 19 , Issue 3 , Summer 2011 , pp. 306 - 324
Copyright
Copyright © The Author 2011. Published by Oxford University Press on behalf of the Society for Political Methodology 

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Footnotes

Authors' note: The authors wish to thank Nicholas Miller for his encouragement and support; Guillermo Owen and Keith Dougherty for their helpful comments regarding an earlier version of this paper; Itai Sened and Michael Lynch for making available their compilation of data from experimental committee voting games; Fuad Aleskerov, Director, Departement of Higher Mathematics, Higher School of Economics, State University of Russia (Moscow) for making available data on party locations in the Russian Duma; and Sue Ludeman for bibliographic assistance. Supplementary materials for this article are available on the Political Analysis Web site.

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