GEOMETRIC APPROACH TO PARADOXES OF VOTING POWER Workshop on Voting Theory and Preference Modelling Sponsored by DIMACS and LAMSADE Paris, Universit´ e Paris Dauphine 28 October 2006 Michael A. Jones Department of Mathematical Sciences Montclair State University Montclair, NJ 07043, USA jonesm@mail.montclair.edu 1
Short History of the Literature on Power Indices Applications • Examining existent institutions: International Monetary Fund (Dreyer and Schotter, 1980; Leech, 2002b), the Electoral College (Mann and Shapley, 1964), the European Union Council of Ministers (Johnston, 1995; Leech, 2002a), and the Israeli Knesset (Laruelle, 2001). • Part of debate about the design of new institutions: new members in the EU (Turnovec, 1996; Widgr´ en, 1994). 2
Short History of the Literature on Power Indices Applications • Examining existent institutions: International Monetary Fund (Dreyer and Schotter, 1980; Leech, 2002b), the Electoral College (Mann and Shapley, 1964), the European Union Council of Ministers (Johnston, 1995; Leech, 2002a), and the Israeli Knesset (Laruelle, 2001). • Part of debate about the design of new institutions: new members in the EU (Turnovec, 1996; Widgr´ en, 1994). Counterintuitive results • paradox of redistribution (Dreyer and Schotter, 1980; Schotter, 1981), the donor and transfer paradoxes (Felsenthal and Machover, 1998), the paradox of quarreling members (Kilgour, 1974), the paradox of a new member (Brams, 1975; Brams and Affuso, 1976), and the paradox of large size (Brams, 1975; Shapley, 1953) 3
Short History of the Literature on Power Indices Applications • Examining existent institutions: International Monetary Fund (Dreyer and Schotter, 1980; Leech, 2002b), the Electoral College (Mann and Shapley, 1964), the European Union Council of Ministers (Johnston, 1995; Leech, 2002a), and the Israeli Knesset (Laruelle, 2001). • Part of debate about the design of new institutions: new members in the EU (Turnovec, 1996; Widgr´ en, 1994). Counterintuitive results • paradox of redistribution (Dreyer and Schotter, 1980; Schotter, 1981), the donor and transfer paradoxes (Felsenthal and Machover, 1998), the paradox of quarreling members (Kilgour, 1974), the paradox of a new member (Brams, 1975; Brams and Affuso, 1976), and the paradox of large size (Brams, 1975; Shapley, 1953) • Analyzing the domain and its geometry results in a classification of paradoxes of voting power 4
Voting, Simple Voting, and Simple Weighted-Voting Games Voter i ’s power in a voting game is defined by � p i = λ S [ v ( S ) − v ( S/i )] S ⊆ N where the set of parameters λ S indicates the specific measure of power (power index, semivalue, etc. ). • v ( S ) − v ( S/i ): value added to S by voter i • Simple Voting Game: v ( S ) − v ( S/i ) = 0 or 1 • Simple Weighted-Voting Game − Finite set of voters N = { 1 , 2 , . . ., n } − Voter i ’s vote carries weight w i such that w = � n i =1 w i − There exists a quota q ∗ where w 2 < q ∗ ≤ w − For S ⊆ N , � 0 if � i ∈ S w i < q ∗ , v ( S ) = 1 if � i ∈ S w i ≥ q ∗ . 5
Partitioning the Normalized Simplex (Domain) into Win- ning and Losing Coalitions w and q = q ∗ • Let x i = w i w . • ( x 1 , x 2 , . . . , x n ) can be viewed as a point on the simplex where x i ≥ 0 and � n i =1 x i = 1 6
Partitioning the Normalized Simplex (Domain) into Win- ning and Losing Coalitions w and q = q ∗ • Let x i = w i w . • ( x 1 , x 2 , . . . , x n ) can be viewed as a point on the simplex where x i ≥ 0 and � n i =1 x i = 1 • Winning and losing coalitions . . . � 0 if � i ∈ S x i < q, v ( S ) = 1 if � i ∈ S x i ≥ q. • What is the effect of this equation? − Equations of the form � i ∈ S x i = q partition the simplex into winning and losing coalitions 7
Power Indices and Generalized Power Indices • Many different power indices ( e.g. , measuring P -power or I -power) − Banzhaf (1965), Shapley-Shubik (1954), Coleman (1971), Penrose (1946), Deegan and Packel (1982), etc. • Geometry is independent of the particular power index − Even though certain power indices may not be susceptible to a specific paradox, when the paradox occurs we will understand what is happening − For concreteness, I will refer to specific power indices in examples but will be clear about what generalizes to all power indices 8
Power Indices and Generalized Power Indices • Many different power indices ( e.g. , measuring P -power or I -power) − Banzhaf (1965), Shapley-Shubik (1954), Coleman (1971), Penrose (1946), Deegan and Packel (1982), etc. • Geometry is independent of the particular power index − Even though certain power indices may not be susceptible to a specific paradox, when the paradox occurs we will understand what is happening − For concreteness, I will refer to specific power indices in examples but will be clear about what generalizes to all power indices − Geometry for 3 voters is enough to understand the geometry behind paradoxes of voting power 9
Power Indices and Generalized Power Indices • For a quota q , a power index is a map from the normalized simplex to R n : P q : S n − 1 → R n + where R + represents nonnegative real numbers. • Power Indices must satisfy the following conditions: 1. (Invariance) If σ is a permutation of the set of voters N , then voter i ’s power in [ q ; x 1 , . . ., x n ] should be the same as voter σ ( i ) = j in the permuted game [ q ; y 1 , . . . , y n ] where y j = x i . Equivalently, P q ( x 1 , x 2 , . . ., x n ) i = P q ( y 1 , y 2 , . . . , y n ) j where σ ( i ) = j and y j = x σ ( i ) for all i. 2. (Symmetry) If two voters are members of identical winning coalitions, then they have the same power. 3. (Dummy voter) If a voter a is never part of a minimal winning coalition, then voter a ’s power is 0. 10
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