The geometry of voting power: Weighted voting and hyper-ellipsoids

Suppose legislators represent districts of varying population, and their assembly's voting rule is intended to implement the principle of one person, one vote. How should legislators' voting weights appropriately reflect these population differences? An analysis requires an understanding of the relationship between voting weight and some measure of the influence that each legislator has over collective decisions. We provide three new characterizations of weighted voting that embody this relationship. Each is based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyper-ellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose–Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights.

[1]  L. Penrose.  The Elementary Statistics of Majority Voting , 1946 .

[2]  Avraham Brichta,et al.  The New Premier-Parliamentary System in Israel , 1998 .

[3]  Bernard Monjardet,et al.  The median procedure in cluster analysis and social choice theory , 1981, Math. Soc. Sci..

[4]  D. Felsenthal,et al.  The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes , 1998 .

[5]  Moshé Machover,et al.  L.S. Penrose's limit theorem: proof of some special cases , 2004, Math. Soc. Sci..

[6]  M. Hosli,et al.  Admission of European Free Trade Association states to the European Community: effects on voting power in the European Community Council of Ministers , 1993, International Organization.

[7]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[8]  Alan D. Taylor,et al.  Interval measures of power , 1997 .

[9]  Lancelot Hogben,et al.  On the Objective Study of Crowd Behaviour , 1952 .

[10]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

[11]  Dan S. Felsenthal,et al.  The Treaty of Nice and qualified majority voting , 2001, Soc. Choice Welf..

[12]  Dan S. Felsenthal,et al.  Voting power measurement: a story of misreinvention , 2005, Soc. Choice Welf..

[13]  Annick Laruelle,et al.  Voting and Collective Decision-Making - Bargaining and Power , 2008 .

[14]  Yair Tauman,et al.  Voting power and proportional representation of voters , 2014, Int. J. Game Theory.

[15]  A. Laruelle,et al.  Is the allocation of voting power among EU states fair? , 1998 .

[16]  Vincent C. H. Chua,et al.  L S Penrose's limit theorem: Tests by simulation , 2006, Math. Soc. Sci..

[17]  Moshé Machover,et al.  PENROSE'S SQUARE-ROOT RULE AND THE EU COUNCIL OF MINISTERS: SIGNIFICANCE OF THE QUOTA ** , 2007 .

[18]  Guillermo Owen,et al.  Cases where the Penrose limit theorem does not hold , 2007, Math. Soc. Sci..

[19]  Nikhil Srivastava,et al.  Voting with rubber bands, weights, and strings , 2012, Math. Soc. Sci..

[20]  Alan D. Taylor,et al.  A characterization of weighted voting , 1992 .

[21]  Arkadii M. Slinko,et al.  Roughly weighted hierarchical simple games , 2012, Int. J. Game Theory.

[22]  D. Leech.  Designing the Voting System for the Council of the European Union , 2002 .

[23]  P. Morriss,et al.  Power: A Philosophical Analysis , 1987 .

[24]  William S. Zwicker.  A characterization of the rational mean neat voting rules , 2008, Math. Comput. Model..

[25]  Tatiana Gvozdeva,et al.  Weighted and Roughly Weighted Simple Games , 2009, Math. Soc. Sci..

[26]  William S. Zwicker,et al.  Simple games - desirability relations, trading, pseudoweightings , 1999 .

[27]  Ron Johnston,et al.  The Conflict over Qualified Majority Voting in the European Union Council of Ministers: An Analysis of the UK Negotiating Stance Using Power Indices , 1995, British Journal of Political Science.

[28]  E. Lehrer,et al.  Regular simple games , 1989 .

[29]  DAVIDE P. CERVONE,et al.  CONVEX DECOMPOSITIONS , 2008 .

[30]  William S. Zwicker.  Consistency without neutrality in voting rules: When is a vote an average? , 2008, Math. Comput. Model..

[31]  Calvin C. Elgot,et al.  Truth functions realizable by single threshold organs , 1961, SWCT.