Power Indices and minimal winning Coalitions
The Penrose-Banzhaf index and the Shapley-Shubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in …
Authors: Werner Kirsch, Jessica Langner
P o w er indices and minimal winning coalitions W erner Kirsch and Jessica Langner Novemb er 11, 2018 Abstract The P enrose-Banzhaf index and the Shapley-Shubik index are the b est-kno wn and the most used to ols to measure p olitical pow er of voters in simple voting games. Most metho ds to calculate these pow er indices a re based on counting winning coalitions, in pa rticular tho s e coalitions a voter is decisive fo r. W e p resent a new combinato ri al fo rmula ho w to calculate b oth indices solely usi ng the set of minimal winning coalitions. 1 Intro duction The theo ry of p o we r indices is a systematic app ro ach to measure p olitical p ow er in voting systems (cp. [T a ylor1995], [F eMa199 8 ]). V o ting systems a re also kno wn as simple (voting) games in literature. The w ell-known Pe nro se-B anzhaf index [Penrose1946], [B anzhaf1965] and Shapley-Shubik index [ShSh1954] rely on the concept of decisiveness of voters. On the other hand, the Deegan-P a ck el index [DeP a1978] and the Holler-P acke l index [HoPa1983] a re based explicitly on the set of minimal winning coalitions ( M W C s). M W C s are those coalitions each vo ter is decisive fo r. P articula rly , a calculation of p ow er indices is easy to handle in w eighted voting systems . Here, v oting w eights a re assigned to each vo ter and a decision threshold is defined. A propo sal is accepted if the sum of the voting we ig hts in fa vo r meets or exceeds the given threshold. Usually , calculation metho ds a re based o n listing the set of winning coalitions. In this pa p er we develop a combinatorial approach ho w to determine p ow er indices solely using the M W C -set. F o r illustration w e use the examples of the Pe nrose-B a nzhaf index a nd the Shapley-Shubik i ndex . It is kno wn that each vo ting system (whether it is w eighted o r not) ha s got a M W C - set and it is completely defined b y it. More p recisely , each set of voting rules can b e quantified b y a M W C - set. Thus, our app roa ch makes it p ossible to calculate p o wer indices for each p otential M W C -set in a rather elegant w ay . F urthermore , we could systematically calculate each p otential constellation of voting p o wer fo r a given set o f voters. This might b e useful fo r a n optimization of existing v oting systems or to design scientifically based pr o p osals fo r further voting b o dies. P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 2 This pa p er is o rg a nized as follows. In first part w e pres ent basic definitions and concepts of the theory of voting p o wer. This section 2 is div ided in three subsections. In subsection 2 .1 voting systems will b e defined. The theo ry of influence on decision in a v oting b o dy will b e introduced in subsection 2.2. In this context minima l winning coalitions and its several properties will b e discussed. In the last subsection 2.3 w e p resent the b est-known metho ds fo r measuring the mentioned decisiveness of vo ters. The sec tio n 3 is the main pa rt of this pap er. Here, w e intro duce o ur approach of a combinato rial calculation of the p resented p o w er indices solely using the set of minima l winning coalitions. How these calcula tio n metho ds w ork will b e illustrated using the example of the Europ ean Economic Communit y of 1958-1 9 72 in section 4. The last section 5 of this pap er contains concluding rema rks. 2 Basic definitions and con cepts 2.1 V oti ng systems V oting systems consist o f a set o f voters and vo ting rules. The vo ting rules determine w ether a p rop osal is accepted o r not. The set of voters of a v oting system can b e rep resented by a finite non-empt y set W = { 1 , . . . , n } . W e call each element w ∈ W a v o ter . A collection of voters rep resented b y a subset A ⊆ W is called a coalition . Its ca rdinality # A is given by the numb er of voters in the coalition. The set of al l coalitions of W is denoted b y P ( W ) which is called the p o wer set of W . I ts cardinalit y is # P ( W ) = 2 n . Additionally , w e mention t w o imp ortant coa litions, the empt y coa lition ∅ and the grand coalition W . V oters decide ab out accepting o r rejecting a proposal b y a vote in fav o r o r against. Whenever w e talk ab out a coal itio n we mean the collection o f those voters who vote in fa vor of a given p rop o sal. V oting rules a re reflected in a split of P ( W ) in t wo disjunct pa rts: The first pa rt consists of those coalitions which can ma k e a p rop osal pass; the second pa rt consists o f those coalitions which can not make a p ro p osal pass. W e ca l l the first part the set of winning coa litions and it will b e denoted with G ⊂ P ( W ) . A coa lition A 6∈ G is called a losing coalition . W e will a l wa ys assume that the grand coalition is a winning coali tio n while the empt y coalition is a losing one. More o v er, we assume if A is a winning coalition and the coalition B comp rises A , then B should b e winning as w ell. This p rop ert y of G is called monotonicity . Definition 2.1. If W is a finite no n- empt y set of v oters and G is a monotone subset of P ( W ) with W ∈ G and ∅ 6∈ G a v o ting system is a pair W := ( W , G ) . P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 3 In many applications it is o bvious that either a coalition A or its complementa ry coalition ( W \ A ) is losing (or b oth a re losing) . A voting system with this p rop ert y is called a p rop er voting system. Otherwise, it is called imp rop er (see e.g. [F eMa1 9 95], [F eMa1998 ], [Fe M a1998b]). I n the following w e don’t hav e to distinguish b etw een p rop er and improper v oting systems as our results are valid in b oth situations. F requently , v oting systems consist of voting rules which assign voting weights to each voter a nd define a decision threshold. A p rop osal will b e passed if the sum of the w eights of the v oters, which vote in fa vor, meets or exceeds the giv en threshold. These a re the so-call ed weighted voting systems . Definition 2.2. A vo ting system W = ( W, G ) is said to b e we ig hted if a function g : W → [0 , ∞ ) and a numb er q ∈ [0 , ∞ ) exi st with X w ∈ A g ( w ) ≥ q holds for a l l A ∈ G and X w ∈ B g ( w ) < q holds for all B ∈ ( P ( W ) \ G ) . (1) g ( w ) is called the voting w eight o f w and q i s called the quota . 2.2 Decisiveness and m inimal winning coal itions An imp o rtant a sp ect of p ol itical sciences is p ol i tical p ow er which is also kno wn as voting p ow er . V oting p ow er is a mathematical concept which quantifies the influence of a voter o n election at a system. Its theo ry can b e traced back to w orks of P enrose [P enrose1946], Shapley and Shubik [ShSh1954] and Banzha f [Banzhaf1965]. If a voter can turn the v oting o utcome by changing his o r her v oting b ehav io r (from vote in favor to aga inst or v ice v ersa) then he or she has influence on the voting decision (cp. [T aylo r1995], [F eMa1 998] and [Kirsch2004]). This p rop erty is kno wn as decisiveness . Thus, i n a vo ting system W a voter w is decisive for a coalition A ∈ G i f w ∈ A and ( A \{ w } ) 6∈ G . Otherwise, w is said to b e non decisive for A . P articula rly , we consider those winning coalitions each voter in the coalition is decisive: A winning coalition V ∈ G is said to b e a minimal winning coalition ( M W C ) if V \{ i } is a losing coa l itio n for each voter i ∈ V . Definition 2.3. T he non-empt y subset M ( G ) with M ( G ) := { V ∈ G | V is a M W C } (2) is called the M W C - set of G (resp ectively W ). M W C - sets have v arious p rop erties w ell kno wn in set theory , combinato rics a nd discrete math- ematics: M W C - sets a re antichains in P ( W ) which are also kno wn as Sp erner families (cp. P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 4 [Anderson1987] a nd [Engel1997]) in literature. More precis ely , an antichain f M is a non-empt y set of subsets of W such that X * Y and Y * X holds fo r all X , Y ∈ f M . In addition, we observe that each vo ting system has a unique M W C -set due to monotonicit y . In [F eMa1998 ] the a utho rs F elsenthal and Machover rema rk ed that W is uniquely determined b y its assembly W a nd its M W C - set M ( G ) . T hus G = { A ∈ P ( W ) | ∃ V ∈ M ( G ) : V ⊆ A } . (3) This is due to the fact that minimal winning coalitions are just the minimal elements in G with resp ect to the partial o rder ⊆ . Also, each set G meets the conditions of an upset or filter (cp. [Anderson1987] and [Engel199 7]). Thus, a voting system is completely defined b y its M W C -set. W e call a M W C -set a basis as well. By a theorem of Sp erner [Sp erner1928] on the cardinalit y o f M ( G ) it is know n to satisfy: 1 ≤ # M ( G ) ≤ n ⌊ n 2 ⌋ . (4) F urthermo re, the numb er of different voting systems f or a given numb er of voters is equal to the co rresp onding Dedekind numb er [Dedekind1897 ] minus 2. Acco rding to the definition of G the t wo sets ∅ and {∅ } are not allo wed as M W C - set. The numb er of v o ting systems with up to eight voters is sho wn in ta bl e 1. T able 1: Numb er of antichains for a given set of # W v o ters. # W Numb er of antichains 1 1 2 4 3 18 4 166 5 7.579 6 7.828.352 7 2.414.682.0 40.996 8 56.130.437 .228.687.5 5 7.907.786 P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 5 2.3 P o w er i ndi ces V oting p ow er of each voter can b e measured by p ow er indices in terms of influence on decisions [F eMa1998 ]. Felse nthal a nd Ma chov er gave a g eneral axio matic definition of p ow er indices in their pap ers [Fe M a1995] and [F eMa1 9 98b]. In the follo wing w e survey the t wo most p o pula r p o wer indices. Definition 2.4 ( Penrose-Banzhaf index) . B S w := # { C ∈ G | w ∈ C , ( C \{ w } ) 6∈ G } (5) is called the Banzhaf score of a voter w and P B P w := B S w 2 n − 1 (6) is called the P enrose-Banzha f p ow er of w . Finally P B I w := B S w P n i =1 B S i (7) is called the P enrose-Banzha f index of w . It is easy to see that 0 ≤ P B I w ≤ 1 a nd P n i =1 P B I i = 1 . The Penrose-Banzhaf p ow er is equal to the p robabi li ty a voter is decisive for a coalition. The P enrose-Banzhaf index measures the a p riori voting p o w er of a voter. This means that the decisiveness of a voter will b e measured without any p revious kno wledge of the single voters. Therefo re it is natural to assume tha t all coalitions ar e equally lik ely . Definition 2.5 ( Sha pley-Shubik index) . S S I w := X S ∈ G with w is deci sive fo r S ( n − # S )!(# S − 1)! n ! (8) is called the Shapley-Shubik index of a voter w . As ab ove 0 ≤ S S I w ≤ 1 a nd P n i =1 S S I i = 1 . The Sha pl ey - Shubik index repres ents the fraction of o rderings of vo ters for which a voter is decisive. Both the Penr o se-Banzhaf index and the Shapley-Shubik index measure the influence of voters in different wa ys (cp. [T aylo r1995], [Fe M a1998] and [LaV a200 5]). In many cases they agree but imp ort a nt examples lik e the US federal system exist where they do not [T aylo r1995]. The right choice which index should b e used fo r ana lysing a v oting situation dep ends o n the assumption a b out P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 6 the voting b ehavior of the v oters. In situations in which the voters vote completely indep endently from each other the Penrose-Banzhaf index should b e used. Otherwise, if a common b elief has influence on the choice of all voters the Shapley-Shubik i ndex should b e used (cp. [Straffin1977], [LaV a20 05] and [Kirsch2007]). Both the Pe nro se-B anzhaf index and the Shapley-Shubik a re based on the decisiveness of voters. Mo reov er, further p ow er indices exist which a re based uniquely on the set of minimal winning coalitions, i.e. the Deegan-Pack el index [DePa1978] and the Holler-P acke l index [HoP a198 3 ]. In addition, the vecto r of the pla yer’s p ow er values can b e defined as p o we r profile concerning the p ow er index under consideration. W e a re interested in the set and val ues of p otential p o wer p rofiles of a giv en set of voters. It is a fact, that not every arbitra ry constellation of voting p o we r is p ossible (cp. [Kirsch2001]). Fo r example, in a v oting system consisting of tw o v oters only t w o p ow er distributions are p ossible: Either o ne voter has got the total p o we r and the other voter has no p o wer o r b oth ha ve the same (half ) part of p ow er. F rom section 2.2 w e know that the several M W C -sets o f a g i v en set of v oters define each p otential v oting system. T hus, w e a re able to calculate each p ossible Deegan-Pack el profile and Holler-P acke l profile . The definitions of the P enrose-Ba nzhaf index and the Shapley-Shubik index allo w us, in p rinciple, to calculate these p o wer indices b y insp ecting the list of all winning coalitions (cp. [T aylo r1995], [Leech2002] and [Leech2003]). W e have develop ed a new combinato ria l approach to calculate the P enrose-Banzhaf index and the Sha pley-Shubik index using simple terms which are solely based on the M W C -set. 3 Calculations Firstly , w e p resent a new calculation fo rmula fo r the Banzhaf score of a v oter. Out of this the P enrose-Banzhaf index can easily b e determined. In the second part of this section we pre sent a simila r calculation metho d fo r the Shapley- Shubik index. Theo rem 3.1. [ B S -direct-calculation form ula ] In a voting system W with M ( G ) = { V 1 , . . . , V m } and # M ( G ) = m w e have for each voter w B S w = m X r =1 ( − 1) r − 1 X 1 ≤ i 1 < ··· k w e gain n k = n − 1 k − 1 + n − 1 k . Hence, P ow er indices and minimal winning coalitions b y We rner Kirsch and Jessica Langner 11 S S I w = X A ∈ A 0 w ψ S S I ( A ) = X A ∈ ( S m i =1 B V i \ S m i =1 B ′ V i ) ψ S S I ( A ) = X A ∈ S m i =1 B V i ψ S S I ( A ) − X A ∈ S m i =1 B ′ V i ψ S S I ( A ) = m X r =1 ( − 1) r − 1 X 1 ≤ i 1 < ···
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment