The Complexity of Power-Index Comparison

The Complexit y of P o w er-Index Compariso n ∗ Piotr F aliszewsk i † Departmen t o f Computer Science Univ ersit y of Ro ches ter Ro ch ester, NY 14627 Lane A. Hem aspaandra ‡ Departmen t o f Computer Science Univ ersit y of Ro c hester Ro ch ester, NY 14627 Jan ua ry 29, 2008 Abstract W e study the c omplexity of the following problem: Given t wo w eighted voting games G ′ and G ′′ that each contain a play er p , in which o f these games is p ’s p ow er index v alue higher? W e study this problem with resp ect to b oth the Sha pley-Shubik pow er index [SS54 ] and the Banzhaf power index [Ban65, DS79]. Our main result is that for b oth of these power indices the problem is complete for probabilistic p oly no mial time (i.e., is PP-complete). W e apply our results to partially r esolve s ome recently prop osed problems regarding the complexity of weigh ted v oting g ames. W e also study the complexity of the raw Shapley-Shubik p ow er index. Deng and P apadimitriou [DP94] sho wed that the r aw Shapley-Sh ubik p ow er index is #P - metric- complete. W e strengthen this b y sho wing tha t the ra w Shapley -Shubik p ow er index is many-one complete f or #P. And our streng thening cannot pos sibly b e further improved to par simonious completeness, since we observe that, in contrast with the raw Banzhaf p ower index, the raw Shapley-Shubik power index is not #P-parsimonious- complete. 1 In tro duction In an abs tract, direct demo cracy , eac h mem b er in a certain sense has equal p otentia l for impact on the deci sions that the so ciet y mak es. Ho wev er, in many practical decision-making scenarios it is reasonable to give up this n oble idea and consider we igh ted v oting ins tead. Here are a few motiv ating e xamples. In a coun try divided in to districts it mak es sense to giv e eac h d istrict v oting p o w er prop ortional to its p opulation (consider, e.g., the US House of Representat iv es or v arious decision making p ro cesses within the Europ ean Union). In fact, the p o wer that v arious app ortion- men t metho ds giv e to the US states in its House of Represen tativ es has b een s tu died in terms of ho w well it is prop ortional to th e sizes of the states [HRSZ98]. In a b u siness setting, stockholders in a compan y might h op e to hav e vot ing p o wer prop ortional to the amount of stock they o wn. Within computer science, Dw ork et al. [DKNS01] suggested building a meta searc h engine for the w eb via treating other searc h engines as v oters in an election. It wo uld only b e n atural to w eigh ∗ Also appears as URCS T R-2008-929. † W ork done in part while visiting Heinric h- Heine-Universit¨ at D ¨ usseldorf, German y . Supp orted in part by grant NSF-CCF-0426761. ‡ Supp orted in part by grant NSF- CCF-04267 61, a T ransCo op gran t, and a F riedrich Wilhelm Bessel Researc h Awa rd. 1 the participating searc h engines with their (quan tified in some wa y) qualit y . Naturally , one can pro vide man y other examples. The f o cus of this pap er is on the compu tational complexity of the follo win g issue: Given an individual and t wo w eigh ted vo ting scenarios (in eac h of them ou r individual m igh t ha v e differen t w eigh t and ea c h scenario migh t inv olv e d ifferen t sets of v oters with differen t w eigh ts), in which one of th em is our individual more in fluentia l? (W e pro vid e a formal definition of this prob lem in Section 1.1.) T h is prob lem has a v ery n atural motiv ation. F or example, consider a compan y that wishes to join so me business consortium and has a c hoice among sev eral consortia (e.g., consider an airline deciding w hic h airline alliance to j oin). It is natural to assum e that within eac h consortium companies mak e decisions via wei gh ted v oting, with companies we igh ted, e.g., via their size o r rev en u e or some com b ination thereof. In a political con text, mem b ers of the Europ ean Union sometimes try to promote new sc hemes of distributing vot e weigh ts among EU member s . I t is imp ortant for the countries in v olv ed to see whic h scheme is b etter for them. One can easily giv e man y other applications of the issue we study . F ormally , we mo del the ab o ve problem via comparing the v alues of p o w er in dex fu nctions— in our case those of S hapley and Sh ubik [SS54] and of Banzhaf ([Ban65], see also [DS79])—of a particular pla y er within tw o give n weig h ted vo ting games. Ou r main result is that this p roblem is PP-complete for b oth the Sh apley-Sh ubik p o wer index and the Banzhaf p o wer index. Let u s now define our problem formally . 1.1 The Po wer-Index Comparison Problem W e mo del w eigh ted v oting via so-called w eigh ted v oting ga mes. An n -pla yer weigh ted v oting game is a sequence of n nonnegativ e in teger weig h ts, w 1 , . . . , w n , together with a quota q . W e d enote it as ( w 1 , . . . , w n ; q ). W e refer to the pla yer with w eigh t w i as the i ’th play er. W eigh ted vo ting games m o del the f ollo wing scenario: The play ers a re given a y es/no question (e.g., should w e lo w er the taxes? should w e bu y out our comp etitors?) and eac h pla y er either agrees (answ ers ye s ) or disagrees (answ ers no ). If the total weigh t of the v oters who agree is at least as h igh as the quota then the result of the game is yes and otherwise it is no . Let G b e a vo ting game ( w 1 , . . . , w n ; q ). Any s u bset of { 1 , . . . , n } is a coalition in G . W e sa y that a coalition S is successful if P i ∈ S w i ≥ q . W e define succ G ( S ) to b e 1 if S is a s u ccessful coaliti on for G and to b e 0 otherwise. In terestingly , the rela tion b et w een the effec tiv e p o w er o f a pla ye r within a v oting game and his or her weigh t is not as simp le as one migh t think. Consider game G = (8 , 7 , 2; 9), i.e., a game w ith quota q = 9 and th ree pla yers with weigh ts 8, 7, and 2, resp ectiv ely . It is easy to see th at in this game an y coalition of at least tw o p la yers is successful. In effect, eac h of the play ers can influ ence the final r esult of the game to exactly the same d egree, regardless of the fact that their we igh ts differ significan tly . Th u s when analyz ing w eigh ted v oting games it is standard to measure pla y ers’ p ow er using, e.g., the Sh ap ley-Shubik p o wer index [SS54] or the Banzhaf p o w er index [Ban65, DS79]. In essence , these p o wer indices measure the pr obabilit y that, assuming some coalitio n formation mo del, our designated p la yer is critical for th e f orming coalition. By critic al w e mean here that the coalition is successful with our designated p la yer but is not successful without him or her. Let G = ( w 1 , . . . , w n ; q ) b e a vot ing game, let i b e a pla y er in this g ame, and le t N = { 1 , . . . , n } b e the set of all p la yers of G . The v alue of th e Banzhaf p ow er index of i in G is defined as 2 Banzhaf ( G, i ) = Banzhaf ∗ ( G,i ) 2 n − 1 , where Banzhaf ∗ ( G, i ) is the r aw version of the index, Banzhaf ∗ ( G, i ) = X S ⊆ N −{ i } (succ G ( S ∪ { i } ) − succ G ( S )) . The Shapley-Shubik p o wer index of pla ye r i in game G is d efined as SS( G, i ) = SS ∗ ( G,i ) n ! , w here SS ∗ ( G, i ) is the raw v ersion of th e index, SS ∗ ( G, i ) = X S ⊆ N −{ i } k S k !( n − k S k − 1)!(succ G ( S ∪ { i } ) − succ G ( S )) . In tuitiv ely , Banzhaf ( G, i ) giv es the probability that a rand omly c hosen coalition of pla y ers in N − { i } is not successful bu t wo uld b ecome su ccessful had pla yer i joined in. The intuition for the S h apley-Sh ubik index is that we coun t the prop ortion of p er mutations for wh ic h a giv en pla y er is pivota l. Giv en a p ermutatio n π of { 1 , . . . , n } , the π ( i )’th play er is piv otal if it holds that the coaliti on { π (1) , π (2) , . . . , π ( i ) } is succe ssful and the coalit ion { π (1) , π (2) , . . . , π ( i − 1) } is not. This p ermutatio n-based intuitio n is motiv ated by the view of the su ccessful-coalit ion formation as th e pro cess of pla y ers joining in in random order. Naturally , the fir st pla yer that makes th e coalition successful is crucial and so the i dea is to measure p o wer via counting h o w often o ur pla y er-of-in terest is piv otal. The fo cus of this pap er is o n the computational complexit y analysis of the follo wing p roblem. Definition 1.1 L et f b e either the Shap ley-Shubik or the Banzhaf p ower index. By Po w erC ompare f we me an the pr oblem wher e the input ( G ′ , G ′′ , i ) c ontains two weighte d voting games, G ′ = ( w ′ 1 , . . . , w ′ n , q ′ ) and G ′′ = ( w ′′ 1 , . . . , w ′′ n , q ′′ ) , and an inte ger i , 1 ≤ i ≤ n , and wher e we ask whether f ( G ′ , i ) > f ( G ′′ , i ) . Note that in the ab ov e definition w e assume that b oth g ames hav e the same num b er of p la yers. A t first this migh t seem to b e a w eakness but it is easy to see that giv en t wo games with differen t n um b ers of pla y ers w e can easily pad the s m aller one with weig h t-0 pla yers. On the other hand, the assumption that b oth games ha ve th e same n um b er of pla yers allo ws us to solv e the prob lem via c omparing the raw v alues of the index: Th e scaling factor for b oth game s is the same and thus it do es not affect the result of the comparison. 1.2 Computational Complexit y W e briefly review some notions and notations. W e fix the alphab et Σ = { 0 , 1 } , and we assume that all the p roblems we consider are encod ed in a natural, efficien t manner o v er Σ. By | · | we mean the length fun ction. W e assume h· , ·i to b e a standard, natural pairing function such that |h x, y i| = 2( | x | + | y | ) + 2. The main result of this pap er, Theorem 2.1, sa ys that the p o wer-index comparison problem is PP -complete b oth for the Sh ap ley-Shubik p ow er index and for th e Banzhaf p o wer index. The class PP, probabilistic p olynomial time, w as defined b y Simon [Sim75] and Gill [G il77]. A language L ⊆ Σ ∗ b elongs to PP if and only if there exists a p olynomial p and a p olynomial-time computable relation R su c h that x ∈ L ⇐ ⇒ k{ w ∈ Σ p ( | x | ) | R ( x, w ) holds }k > 2 p ( | x | ) − 1 . PP captures the set of languages ha ving a prob ab ilistic T uring mac hine that on precisely the elemen ts of the set has s tr ictly more than 50% probability of acceptance. Let us men tion th at PP is a v ery p o w erf ul 3 class. F or example, it is w ell-kno w n that NP is a sub set of PP (as are e v en v arious larger classes). Via T o d a’s Theorem [T o d 91], we kno w that PH ⊆ P PP . Th at is, PP is at least as p ow erful as p olynomial-time hierarc hy , giv e or tak e the flexibilit y of p olynomial-time T u r ing reductions. Ma n y other prop erties of PP ha ve b een established in the literature. Let us no w recall the definition of the cl ass #P [V al79 ]. F or eac h NP mac hin e N (i.e., for eac h nondeterministic p olynomial-time mac hine N ), b y #acc N ( x ) we mean th e num b er of accepting computation paths of N running with input x . A fu nction f , f : Σ ∗ → N , b elongs to #P if and only if there is an NP mac hine N such that ( ∀ x ∈ Σ ∗ )[ f ( x ) = #acc N ( x )]. #P is, in a very loose sense, a functional coun terpart of P P. F or example, P #P = P PP [BBS86]. More typical ly , #P is describ ed as the coun ting analogue of NP. As is usual, we sa y that a language L is hard for a complexit y cla ss C if every language in C p olynomial-time man y-one redu ces to L . If in addition L b elongs to C then we sa y th at L is C -complete. A language A p olynomial-time man y-one reduces to a language B if th er e exists a p olynomial-time computable function f suc h that for eac h strin g x ∈ Σ ∗ it holds that x ∈ A ⇐ ⇒ f ( x ) ∈ B . On the other hand, there is no one agreed-up on n otion of completeness for function classes. F or example, V alian t [V al79] in his seminal pap er used T urin g redu ctions bu t other p eople ha ve p r eferred n otions s uc h as Krentel’ s metric reductions [Kr e88], Zank´ o’s many-one redu ctions (for functions) [Zan91], and Simon’s [Sim75] parsim on ious r eductions. In the context of p o wer ind ex functions, Prasad and Kelly [PK 90] (implicitly) s ho wed that the (raw) Banzhaf p o w er index is #P-parsimonious-complete and Deng a nd Papadimitriou [DP94 ] established th at the (ra w) Shapley-Sh ubik p ow er index is #P-metric-complete (reg arding th e com- plexit y analysis of p o w er indices, we also ment ion the pap er of Matsui and Matsui [MM01]). W e no w review parsimonious and metric r eductions, as those underp in the notions of parsimon ious - completeness and metric-completeness. Definition 1.2 1. [Kr e88] A function f : Σ ∗ → N metric r e duc es to a fu nction g : Σ ∗ → N if ther e exist two p olynomial-time c omputable functions, ϕ and ψ , such that ( ∀ x ∈ Σ ∗ )[ f ( x ) = ψ ( x, g ( ϕ ( x )))] . 2. [Zan91] A function f : Σ ∗ → N many-one r e duc es to a function g : Σ ∗ → N if ther e exists two p olynomia l-time c omputable functions, ϕ and ψ , such that ( ∀ x ∈ Σ ∗ )[ f ( x ) = ψ ( g ( ϕ ( x )))] . 1 3. [Sim75] f p arsimoniously r e duc es to g i f ther e is a p olynomial-time c omputa ble function ϕ such that ( ∀ x ∈ Σ ∗ )[ f ( x ) = g ( ϕ ( x ))] . Note th at (a) if f parsimoniously reduces to g , then f many-one reduces to g , and (b ) if f many-one reduces to g , th en f m etric reduces to g . Giv en a function class C , w e sa y that a fu nction f is C -parsimonious-complete if f ∈ C and e ac h fun ction in C parsimonious reduces to f . C -metric-complete ness and C -many-o ne-completeness are defined analogously . T ypically , parsimonious-complete functions are easier to work with th an functions that are merely metric- complete or man y-one-complete. In particular, our pro of of T h eorem 2.11 is more in v olv ed than o ur pro of of Theorem 2.4 b ecause, as we note, the ra w Sh apley-Sh ubik p o wer index is not parsimoniously complete. 1 Note that Za nk´ o’s man y- one reduction i s a a nalogue for functions of th e standard many-one reduction notion for sets. T o avo id confusion, we mention to t h e reader that the term “functional man y- one redu ction” (whic h w e do not use here) is sometimes used in the literature [V ol94] as a synon ym for “parsimo nious reductions.” 4 2 Main Results Our main result, Theorem 2.1, says that the p ow er in d ex comparison problem is PP-co mplete. This section is devot ed to building the infrastru cture for Theorem 2.1’s pro of and giving that pro of. W e also sho w that the r a w Shapley-Shubik p o wer index is #P-man y-one-complete but not #P-parsimonious-complete. Theorem 2.1 L et f b e either the Banzhaf or the Shapley-Shubik p ower index. The pr oblem P o werCompare f is PP -c omplete. W e start via showing PP-mem b ership of a problem closely r elated to our Po w erCompare Banzhaf and P ow erCompare SS problems. L et f b e a # P f unction and let Compare f b e the language {h x, y i | x, y ∈ Σ ∗ ∧ f ( x ) > f ( y ) } . (Po w erCompare Banzhaf and P o werCompare SS are essen tially , up to a minor definitional issue, incarnations of Compare f for appropriate functions f .) Lemma 2.2 L et f b e a #P function. Th e language Compare f is in PP . Pro of. Let f b e an arbitrary #P function and let N b e an NP machine suc h that f = #acc N . Without th e loss of generalit y , w e assu me that there is a p olynomial q su c h that for eac h inpu t x ∈ Σ ∗ all computation p aths of N make exactly q ( | x | ) binary nond eterministic c hoices. Thus eac h computation path of N on inp ut x can b e represented as a string w in Σ q ( | x | ) . In order to sho w that Compare f is in P P w e need to pro vide a p olynomial-time computable relation R and a polynomial p such that for e ac h string z = h x, y i it holds that: z ∈ Compare f ⇐ ⇒ k{ w ∈ Σ p ( | z | ) | R ( z , w ) holds }k > 2 p ( | z | ) − 1 . W e now defin e suc h R and p . Let us fix t wo strings, x and y , and le t z = h x, y i and n = | z | . W e define p ( n ) = q ( n ) + 1 and, for eac h s tring w = w 0 w 1 . . . w p ( n ) − 1 ∈ Σ p ( n ) , w e define R ( z , w ) as follo ws: Case 1. If w 0 = 0 th en R ( z , w ) is true exactly if the strin g w 1 , . . . , w q ( | x | ) denotes an accepting computation path of N on x and the symb ols w q ( | x | )+1 through w p ( n ) − 1 are all 0. R ( z , w ) is false otherwise. Case 2. If w 0 = 1 then R ( z , w ) is false exac tly if the string w 1 , . . . , w q ( | y | ) denotes an accepting computation path of N on y and th e symbols w q ( | x | )+1 through w p ( n ) − 1 are all 0. R ( z , w ) is true otherwise. Via analyzi ng the abov e t wo cases it is easy to see that there are exactly f ( x ) + (2 p ( n ) − 1 − f ( y )) = f ( x ) − f ( y ) + 2 p ( n ) − 1 strings w ∈ Σ p ( n ) for w hic h R ( z , w ) is true. T h is v alue is greater than 2 p ( n ) − 1 if and only if f ( x ) > f ( y ). Th us the relation R a nd the p olynomial p join tly witness that Compare f b elongs to PP. ❑ Lemma 2.2 giv es an upp er b oun d on the complexit y of Compare f (assuming that f ∈ #P). W e n o w prov e a matc hing lo w er b ound, PP-completeness, for th e case that f is #P-parsimonious- complete. Lemma 2.3 L et f b e a #P -p arsimonio us-c omplete function. The language Compare f is PP - c omplete. 5 Pro of. Let f b e a #P-parsimonious-complete function. Via Lemma 2.2 we kn ow that Compare f is in PP and thus to sh o w PP-completeness it remains to sho w PP-hardn ess. W e do so via reducing an arbitrary PP language L to Compare f . Let L b e an arbitrary PP language. By d efinition, there exists a p olynomial-time relation R and a p olynomial p such that for eac h s tr ing x ∈ Σ ∗ it holds that x ∈ L ⇐ ⇒ k{ y ∈ Σ p ( | x | ) | R ( x, y ) holds }k > 2 p ( | x | ) − 1 . W e defin e t wo fu nctions, g 1 and g 2 , suc h that g 1 ( x ) = k{ y ∈ Σ p ( | x | ) | R ( x, y ) h olds }k and g 2 ( x ) = 2 p ( | x | ) − 1 . It is easy to see that b oth g 1 and g 2 are in #P. g 1 can b e computed via a an NP mac h in e that on input x guesses a binary string y of length p ( | x | ) and accepts if and only if R ( x, y ) holds. g 2 can b e computed via a mac hin e that on input x guesses a binary string of length 2 p ( | x | ) − 1 and then accepts. Na turally , x ∈ L i f and only if g 1 ( x ) > g 2 ( x ). Since f is #P-parsim on ious -complete, b oth g 1 and g 2 parsimoniously r ed uce to f . Let ϕ 1 b e the r ed uction fu nction for g 1 and let ϕ 2 b e the redu ction function for g 2 . W e hav e that for eac h string x it holds that g 1 ( x ) = f ( ϕ 1 ( x )) and g 2 ( x ) = f ( ϕ 2 ( x )). Our reduction from L to C ompare f w orks as follo w s. On input x w e ou tp ut the strin g z = h ϕ 1 ( x ) , ϕ 2 ( x ) i . Clearly , this can b e done in p olynomial time. T o sho w co rrectness it is enough to recall that x ∈ L if and on ly if g 1 ( x ) > g 2 ( x ), whic h is equiv alen t to testing w hether z is in Compare f . S ince L w as c hosen as an arbitrary PP language, this pro v es P P-completeness. ❑ W e a re almost r eady to sh o w th at Po w erCompare Banzhaf is PP-complete. Ho wev er, in order to do so, w e need to justify the claim that the r a w v ersion of the Ba nzhaf p ow er index is #P - parsimonious-complete. (This wa s shown imp licitly in th e w ork of Prasad and K elly [PK90], but w e feel that it is imp ortant to explici tly outline the pr o of.) One of our imp ortant to ols here (and later on) is the function #X3C. The inp ut to the X3C problem is a set B = { b 1 , . . . , b 3 k } and a f amily S = { S 1 , . . . , S n } of 3-elemen t sub sets of B . The X3C problem asks whether there exists a collection of exactly k sets in S whose u nion is B . #X3C( B , S ) is the n um b er of solutions of the X3C instance ( B , S ). Hun t et al . [HMRS98] sho wed that # X3C is parsimonious complete for #P. Th is is very useful for us as the standard reduction fr om #X3C to #Sub setSum (see, e.g., [P ap94, Theorem 9.10]; #Subs etSu m is the function that accepts as input a ve ctor of nonnegativ e inte gers ( s 1 , . . . , s n ; q ) and retur ns the n um b er of sub s ets of { s 1 , . . . , s n } that sum up to q ) is p arsimonious and Prasad and Kelly’s r eduction fr om #S ubsetSum to Banzhaf ∗ (the ra w version of Banzhaf ’s p o wer in dex) is parsimonious as well. Since Banzhaf ∗ is in # P, Banzhaf ∗ is #P-parsimonious-complete. Thus the follo win g theorem is, essenti ally , a direct consequence of Lemma 2.3. Theorem 2.4 Po w erCompare Banzhaf is PP -c omplete. Pro of. The raw v ersion of the Banzhaf p o wer index is #P-parsimonious-complete and so, via Lemma 2.3, Compare Banzhaf ∗ is PP-complete. Via a sligh t misuse of notation, w e can say that Compare Banzhaf ∗ accepts as input t wo we igh ted v oting games, G ′ and G ′′ , and t w o pla y ers, p ′ and p ′′ , suc h that p ′ participates in G ′ and p ′′ participates in G ′′ and accepts if and only if Banzhaf ∗ ( G ′ , p ′ ) > Banzhaf ∗ ( G ′ , p ′′ ). W e giv e a red u ction from Compare Banzhaf ∗ to Po w erC ompare Banzhaf . Let G ′ , p ′ and G ′′ , p ′′ b e our input to the Compare Banzhaf ∗ problem. W e can assume that G ′ and G ′′ ha ve the same n um b er of p la yers. If G ′ and G ′′ do not ha v e the same n um b er of p la yers then it is easy to see that the game w ith few er play ers can b e padd ed w ith pla y ers wh ose weigh t is equal to th is game’s quota v alue. Su c h a padding lea ves the r a w Banzhaf p o wer index v alues of the ga me’s original pla ye rs un c h anged. (The reason for this is that an y coaliti on th at includes an y 6 of the padding cand idates is already winning and so n one of the original p la yer’s is critical to the success of the coalitio n, and so the coalitio n d o es not con trib ute to original p la yers’ p o wer index v alues.) W e form t w o games, K ′ and K ′′ , that are iden tical to games G ′ and G ′′ , resp ectiv ely , except that K ′ lists pla yer p ′ as first and G ′′ lists pla yer p ′′ as first. Our reduction’s output is ( K ′ , K ′′ , 1). Naturally , Banzhaf ( K ′ , 1) > Banzhaf ( K ′′ , 1) if and only if Banzhaf ∗ ( G ′ , p ′ ) > Ba nzhaf ∗ ( G ′′ , p ′′ ). Also, clearly , K ′ and K ′′ can b e computed in p olynomial time. Thus w e ha v e successfully reduced Compare Banzhaf ∗ to P o w erCompare Banzhaf . This sho ws PP-hardness of Po w erCompare Banzhaf . PP- mem b ership of P ow erCompare Banzhaf is, essen tially , a simp le consequence of Lemma 2. 2. T his completes the pro of. ❑ Let us no w focus on the computatio nal complexit y of t he pow er index comparison problem for the case of Shapley-Sh u bik. It w ould b e nice if t he ra w Shapley-Shubik p o wer index w ere #P-parsimonious-complete. I f that were the case then w e could establish PP-completeness of P o werCompare SS in essen tially th e same wa y as w e did for P o werCompare Banzhaf . Thus it is natural to ask wh ether the Shapley-Shubik p o wer in d ex (i.e ., its ra w version) is #P-parsimonious - complete. P rasad and Kelly [PK90 ] at th e end of their pap er, after—in effect—sho wing #P- parsimonious-completeness of t he ra w Banzhaf p o w er ind ex (their Theorem 4) , w rite: “S u c h a straigh tforward approac h do es not seem p ossible with the Shapley-Shubik [pow er index].” W e reinforce their intuition by now pro ving th at the ra w Shapley-Shubik p o wer in d ex in fact is not #P-parsimonious-complete. Theorem 2.5 The r aw Shapley-Shubik p ower index (i.e., S S ∗ ) is not #P - p arsimonious-c omplete. Pro of. F or the s ak e of contradict ion, let us assume that SS ∗ is #P -parsimonious-complete. Thus for eac h natural num b er k there is a w eigh ted v oting game G and a p lay er i within G suc h that SS ∗ ( G, i ) = k . This is the case b ecause th e fun ction f ( x ) = x b elongs to #P (w e assu me that the “output x ” is an in teger obtained via a standard bijection b et w een Σ ∗ and N ) and if SS ∗ is #P-parsimonious-complete then there has to b e a parsimonious reduction from f to S S ∗ . Let G b e an arb itrary voting game with n ≥ 4 pla ye rs and let i b e a pla y er in G . By definition, SS ∗ ( G, i ) is a sum of terms of the form k !( n − k − 1 )!, where k is some v alue in { 0 , . . . , n − 1 } . Since n ≥ 4, eac h such term is eve n and th u s SS ∗ ( G, i ) is even. The ra w Shapley-Shubik pow er ind ex of an y pla yer in a game with at most 3 pla ye rs is at most 3! = 6 and th us there is n o input on whic h SS ∗ yields the v alue 7. This con tradicts the assumption that SS ∗ is #P-parsimonious-complete and completes the pro of. ❑ So the we ll-kno wn result of Deng and P apadimitriou [DP94] that the raw S hapley-Shubik p o w er index is #P-metric-complete cann ot b e strengthened to #P -p arsimonious-completeness. Theorem 2.5 preven ts us fr om directly using Lemma 2.3 to s ho w that P o werCompare SS is PP-complete. Nonetheless, via the f ollo wing set of r esults n ot only do we establish that P o werCompare SS is PP-complete, b ut w e also strengthen the r esu lt of Deng and P apadimitriou via sho wing that the ra w Shapley-Shubik pow er index is #P-man y-one-complete (i.e., is #P-c omplete w.r.t. Zank´ o’s man y-one red u ctions [Zan91]). T o establish our resu lts we need to b e able to bu ild X3C in stances that satisfy c ertain prop erties. F act 2.6 b elo w lists three basic transformations that we use to enforce these prop erties. 7 F act 2.6 L et ( B , S ) b e an instanc e of X3C and let b 1 , b 2 , . . . , b 6 b e elements that do not b elong to B . L et B 1 = { b 1 , b 2 , b 3 } , B 2 = { b 4 , b 5 , b 6 } , B 3 = { b 1 , b 4 , b 5 } and B 4 = { b 1 , b 4 , b 6 } . The fol lowing tr ansforma tions pr eserve the numb e r of solutions of the input instanc e : 1. g ( B , S ) = ( B ∪ B 1 , S ∪ { B 1 } ) , 2. h ′ ( B , S ) = ( B ∪ B 1 ∪ B 2 , S ∪ { B 1 , B 2 , B 3 } ) , 3. h ′′ ( B , S ) = ( B ∪ B 1 ∪ B 2 , S ∪ { B 1 , B 2 , B 3 , B 4 } ) , In the follo wing lemma w e use these transformations to, in some sense, norm alize X3C instances. Lemma 2.7 Ther e is a p olynomial-time algorithm that given an X3 C inst anc e X = ( B , S ) outputs instanc e X ′′ = ( B ′′ , S ′′ ) such that #X3C( X ′′ ) = #X3C( X ) and 1 3 k B ′′ k kS ′′ k = 2 3 . Pro of. L et X = ( B , S ) b e ou r inp u t X3C instance and let 3 k = k B k and m = kS k . Let g and h ′′ b e the transform ations as in F act 2.6. The idea of our algorithm is to app ly transformation g to X so man y times as to ac hieve th e 2 3 ratio. Let t b e some nonnegativ e in teger and let ( B t , S t ) = g ( t ) ( B , S ). W e observe that 1 3 k B t k kS t k = k + t m + t and that if t = 2 m − 3 k (assumin g this v alue is nonnegativ e) then k + t m + t = 2 3 . Our algorithm works as follo ws. First, we form instance X ′ = ( B ′ , S ′ ) such that 2 kS ′ k − 3 · 1 3 k B ′ k ≥ 0. If 2 m − 3 k ≥ 0 then w e set X ′ = X and otherwise we rep eatedly apply trans- formation h ′′ , until this condition is met. (It is easy to see that ⌈ 3 k − 2 m 2 ⌉ ap p lications are suffi- cien t.) Then we derive the instance X ′′ from X ′ via 2 kS ′ k − 3 · 1 3 k B ′ k applicatio ns of g . T hat is, X ′′ = g (2 kS ′ k− 3 · 1 3 k B ′ k ) ( X ′ ). Naturally , the algo rithm runs in p olynomial t ime. The corr ectness follo ws via the observ ation in the firs t p aragraph and the fact that tr ansformations g and h ′′ preserve the num b er of solutions. ❑ Finally , w e are ready to sho w that the ra w Shapley-Shubik p ow er index is #P-man y-one- complete. Theorem 2.8 The r aw Shapley-Shubik p ower index (i.e., S S ∗ ) is #P -many-one-c omplete. Pro of. The ra w Shapley-Shubik p o w er index is in #P and th u s it r emains to sho w that it is #P-man y-one-hard. T o do so , w e giv e a man y-one redu ction from #X3C ′ to SS ∗ . #X3C ′ is a restriction of #X3C to in stances X = ( B , S ) suc h that: (1) 1 3 k B k kS k = 2 3 . (2) If n is a n onnegativ e in teger suc h that 1 3 k B k = 2 n and kS k = 3 n then there is a n on n egativ e inte ger t suc h that n = 4 t . T o see that th e th us restricted # X3C function is #P -parsimonious-complete it is e nough to consider Lemma 2.7 and transformation h ′ from F act 2.6. Let ϕ s b e the standard, parsimonious redu ction from #X3C to #SubsetSum (see, e.g., [P ap94, Theorem 9.10]) . ϕ s has th e pr op ert y that giv en an instance ( B , S ), where k B k = 3 k and k S k = m , ϕ s ( B , S ) is an instance ( s 1 , . . . , s m ; q ) of SubsetSu m su ch that every subset of { s 1 , . . . , s m } that sums u p to q h as exactly k elemen ts. Giv en su c h an instance ( s 1 , . . . , s m ; q ), Deng and P apadimitriou [DP94, Th eorem 9] obs erv e that the ra w S hapley-Shubik p o wer index of the fi rst pla yer in game (1 , s 1 , . . . , s m ; q + 1) is exactly ( m − k )! k ! · #Su bsetSum( s 1 , . . . , s n ; q ). Since ϕ s is parsimonious, this v alue is equal to ( n − m )! m ! · #X3C( B , S ). 8 W e now pro vide fu nctions ϕ and ψ that constitute a man y-one r eduction from #X3C ′ to SS ∗ . W e need to ensure that for ea c h #X3C ′ instance X 2 it holds that #X3C ′ ( X ) = ψ (SS ∗ ( ϕ ( X ))). W e first describ e h o w to compute ϕ and ψ and then explain why th ey ha v e this prop er ty . Giv en #X3C ′ instance X , w e compute ϕ ( X ) as follo ws: W e compute Su bsetSum instance ϕ s ( X ) = ( s 1 , . . . , s n ; q ) and output g ame (1 , s 1 , . . . , s n ; q + 1). F unction ψ is a li ttle more in volv ed. Define r 1 ( n ) = n !(2 n )! and r 2 ( n ) = n !(2 n )!2 3 n . Giv en a nonn egativ e int eger x , w e compute ψ ( x ) using the follo wing algorithm. If x = 0 then r eturn 0. Otherwise, find the smallest nonnegativ e in teger t suc h that r 1 (4 t ) ≤ x ≤ r 2 (4 t ) and output ⌊ x r 1 (4 t ) ⌋ . I f there is no su ch t then return 0. F u nction ψ ( x ) can b e computed in p olynomial time via computing r 1 (4 t ) and r 2 (4 t ) for successiv e v alues of t . It is easy to see that w e only n eed to try O (log x ) many t ’s and th u s ψ is computable in p olynomial time with resp ect to the binary represent ation of x . Let us no w sh o w that ind eed for any #X3C ′ instance X it holds that #X3C ′ ( X ) = ψ (SS ∗ ( ϕ ( X ))). L et X = ( B , S ) b e an arbitrary #X3C ′ instance and let n b e a nonnegativ e in teger suc h that 1 3 k B k = 2 n and kS k = 3 n . (The existence of suc h an n is guarante ed via th e fact that in an y #X3C ′ instance 1 3 k B k kS k = 2 3 .) Via the prop erties of ϕ s and ϕ w e see that SS ∗ ( ϕ ( X )) = n !(2 n )!#X3C ′ ( X ) = r 1 ( n )#X3C ′ ( X ) . It is easy to see that #X3C ′ ( X ) ≤ 2 3 n and th u s , assuming that #X3C ′ ( X ) ≥ 1, w e ha ve that r 1 ( n ) ≤ S S ∗ ( ϕ ( X )) ≤ r 2 ( n ) . Via routine calculation we see that f or an y p ositiv e int eger n it holds that r 1 (4 n ) > r 2 ( n ). Th us the inte rv als [ r 1 (4 t ) , r 2 (4 t )] are disjoin t and given SS ∗ ( ϕ ( X )) as inp u t, the function ψ correctly iden tifies the r 1 ( n ) factor and outputs the answer #X3C ′ ( X ). Clearly , ψ also w orks correctly wh en SS ∗ ( ϕ ( X )) = 0. ❑ Lemma 2.9 Ther e is a p olynomial-time algorithm that g iven two X3C insta nc es X = ( B x , S x ) and Y = ( B y , S y ) outputs two X 3C instan c es X ′′ = ( B ′′ y , S ′′ y ) and Y ′′ = ( B ′′ y , S ′′ y ) such that k B ′′ x k = k B ′′ y k , kS ′′ x k = kS ′′ y k , #X3C( X ) = #X3C( X ′′ ) , and #X3C( Y ) = #X3C( Y ′′ ) . Pro of. W e fir st use the algorithm from Lemma 2.7 to deriv e instances X ′ = ( B ′ x , S ′ X ) and Y ′ = ( B ′ x , S ′ X ) suc h that #X3C( X ) = #X3C( X ′′ ), #X3C( Y ) = #X3C( Y ′′ ), 1 3 k B ′ x k kS k = 2 3 , and 1 3 k B ′ x k kS k = 2 3 . Without the loss o f generalit y w e can assume that k B ′ x k ≤ k B ′ y k . W e set Y ′′ = Y ′ and deriv e X ′′ via rep eatedly a pplying transf ormation h ′ from F act 2 .6 to X ′ , until the condition of the theorem is met. ❑ In the next lemma and theorem we pro v e the PP-completeness of Po w erCompare SS . Lemma 2.10 L et f and g b e two arbitr ary #P func tions. Ther e exists a p olynomial-time c omputable function cmp f ,g ( x, y ) such tha t ( ∀ x, y ∈ Σ ∗ )[ f ( x ) > g ( y ) ⇐ ⇒ cmp f ,g ( x, y ) ∈ P o werCompare SS ] . 2 W e assume that th e inputs t o ϕ satisfy the requ irements of being #X3C ′ instances. W e implicitly replace an y instance that does not fulfill this requiremen t wi th a fixed instance that does satisfy it and that h as no solutions. 9 Pro of. L et f and g b e as in the lemma and let x and y b e tw o arbitrary strin gs. Since b oth f and g are in #P and # X3C is #P-parsimonious-complete, there exist functions ϕ f and ϕ g that compute parsimonious reductions fr om f to #X3C and from g to #X3C, resp ectiv ely . 3 Let ( B x , S x ) = ϕ f ( x ) and ( B y , S y ) = ϕ g ( y ). Via Lemma 2.9 (and through a sligh t abuse of notation) w e ensure that k B x k = k B y k = 3 k and that kS x k = kS y k = r , where r and k are tw o nonnegativ e intege rs. Let ϕ b e the redu ction fun ction from the pr o of of Theorem 2.8. (Note that in the pro of of Theorem 2.8 we restricted ϕ to w ork only on instances of X3C that fulfi ll a sp ecial requirement . F or th e purp ose of th is pro of w e disregard this requirement.) W e n o w describ e our fun ction c mp f ,g . Giv en an instance ( B x , S x ) w e compute G x = ϕ ( X ) and G y = ϕ ( Y ). W e defi n e cmp f ,g ( x, y ) to outpu t ( G x , G y , 1). Vi a the prop erties of ϕ discussed in the pro of of Theorem 2.8, it holds that SS ∗ ( G x , 1) = ( r − k )! k ! · #X3C( B x , S x ) = ( r − k )! k ! f ( x ), and SS ∗ ( G y , 1) = ( r − k )! k ! · #X3C( B y , S y ) = ( r − k )! k ! g ( y ) . Th us f ( x ) > f ( y ) if and only if SS( G x , 1) > SS( G y , 1), and so it is clear that th e function cmp f ,g do es what the theorem claims. Na turally , cmp f ,g can b e computed in p olynomial time. ❑ Theorem 2.11 Po w erC ompare SS is PP - c omplete. Pro of. Via Lemma 2.2 it is easy to see th at Po w erCompare SS is in P P. Let h b e some #P- parsimonious-complete function. PP-hardness of P ow erCompare SS follo ws via a reduction from PP-complete problem C ompare h (see L emm a 2.3 ). As a r eduction we can use, e.g., the fun ction cmp h,h from Lemma 2.10. This completes the pro of. ❑ 3 Conclusions and Op en Pr oblems W e h a ve sho wn that the problem of deciding in wh ic h of the t w o giv en vo ting games our designated pla yer has a h igher p o wer in dex v alue is PP-c omplete for b oth the B anzhaf and the S h apley-Sh ubik p o w er ind ices. F or the case of Banzhaf, we hav e used the fact that the raw Banzhaf p o wer index is #P-parsimonious-complete. F or the ca se of Shapley-Sh ubik, w e ha ve sho wn that the ra w Shapley- Shubik p o wer index is #P-many-o ne-complete but not #P-parsimonious-complete. Nonetheless, using the index’s prop erties we were able to sho w th e PP-completeness of Po w erCompare SS . W e b eliev e that th ese resu lts are interesting and pr actica lly imp ortant. Belo w we mention one p articular application. In the con text of multiag en t systems, the S hapley-Shubik p o w er index is often u sed to distribute pla yers’ pa y offs, i.e., eac h pla y er’s pa y off is prop ortional to his or her p o w er in d ex v alue. Rec en tly Elkind [Elk07] ask ed ab out the exact complexit y of the follo w in g problem: Giv en a we igh ted v oting game G = ( w 1 , w 2 , . . . , w n ; q ), is it p rofitable f or play ers 1 and 2 to join? That is, if G ′ = ( w 1 + w 2 , w 3 , . . . , w n ; q ), is it the case that SS( G ′ , 1) > SS( G, 1 ) + S S( G, 2). Using Lemma 2.10 and the fact that #P is closed under addition w e can easily sho w that this p r oblem reduces to P o w erCompare SS 3 W e as sume that n eith er ϕ f nor ϕ g ever out puts a malformed instance of X3C. This prop erty is easy to enforce via the follow ing mod ification: When ever eith er ϕ f or ϕ g is a b out to outp ut a malformed instance, replace it with a fixed, correct one that has no solutions. 10 and thus is in PP. W e believ e that Elkind’s problem is, in f act, PP-complete and th at the tec h niques present ed in this p ap er will lead to the pro of of this fact. Ho we v er, at this p oin t the exact complexit y of the problem remains op en. Ac knowledgmen ts W e thank J¨ org R othe and Edith Elkind for helpful discussions on the topic of weigh ted v oting games and J¨ org Rothe for hosting a visit during which this work w as done in part. References [Ban65] J. Banzhaf. W eigh ted vo ting doesn’t w ork: A mat hematical analysis. R utgers L aw R eview , 19:317– 343, 1965. [BBS86] J. Balc´ azar, R. Book, and U. Sc h¨ oning. The p olynomial-time hierarc hy and sparse oracles. Journal of the ACM , 33(3):603–6 17, 1986. [DKNS01] C . Dw ork, R. Kumar, M. Naor, and D . Siv akumar. Rank ag gregation metho ds for the web. In Pr o c e e dings of the 10th International World Wide Web Confer enc e , pages 613–6 22. A CM Press, Marc h 2001. [DP94] X. Deng and C. P apadimitriou. On the complexit y of comparative solution concepts. Mathematics of Op er ations R ese ar ch , 19(2): 257–2 66, 1994. [DS79] P . Dubey and L. Sh apley . Mathematical prop erties of the Banzhaf p o w er index. Math- ematics of O p e r ations R ese ar ch , 4(2):99–13 1, Ma y 1979. [Elk07] E. Elkind, No vem b er 2007. Personal comm un ication. [Gil77] J. Gill. Computational complexit y of probabilistic Turing mac hines. SIAM J ournal on Computing , 6(4):6 75–69 5, 1977. [HMRS98] H. Hunt, M. Marathe, V. Radh akrishnan, and R. S tearns. The complexit y of p lanar coun ting problems. SIA M J ournal on Computing , 27(4):1142 –1167, 1998. [HRSZ98] L. Hemaspaandra, K. Ra jaseth u path y , P . S eth upath y , and M. Zimand. P o wer balance and app ortionment algorithms for t he United States Congress. ACM Journal of Exp erimental Algorithmics , 3(1), 1998. URL h ttp://www.jea.acm.o rg/1998 /HemaspaandraP ow er. [Kre88] M. Kr entel. The c omplexit y of optimization problems. Journal of Computer and System Scienc es , 36(3):4 90–50 9, 1988. [MM01] Y. Matsui and T. Matsui. NP-completeness f or calculating p ow er ind ices of w eigh ted ma jorit y games. The or etic al Computer Scienc e , 263( 1–2):3 05–310, 2001. [P ap94] C. P ap ad im itriou. Computatio nal Complexity . Add ison-W esley , 1994. [PK90] K. Pr asad and J. Kelly . NP-completeness of some problems concerning vo ting games. International Journal of Game The ory , 19(1):1–9 , 1990. 11 [Sim75] J. Simon. On Som e Centr al Pr oblems in Computational Complexity . PhD thesis, C ornell Univ ersit y , Ithaca, N.Y., J an uary 1975. Av ailable as Cornell Departmen t of Compu ter Science T ec hn ical Rep ort TR75-224. [SS54] L. Shapley and M. Sh ub ik. A metho d of ev aluating the distribu tion of p ow er in a committee system. Americ an Politic al Scienc e R ev iew , 48:787 –792, 1954. [T o d91] S. T o da. PP is as hard as the p olynomial-time hierarch y . SIA M Journa l o n Computing , 20(5): 865–8 77, 1991. [V al79] L. V alian t. The complexit y o f computing the p ermanent . The or etic al Computer Scienc e , 8(2):1 89–20 1, 1979. [V ol94] H. V ollmer. On differen t reducibilit y notions for function classes. In Pr o c e e dings of the 11th Annual Symp osium on The or etic al A sp e cts of Computer Scienc e , pages 449–460. Springer-V erlag L e ctur e Notes in Computer Scienc e #775 , F ebru ary 1994. [Zan91] V. Zank´ o. #P-completeness via many-one redu ctions. International Journal of F oun- dations of Computer Scienc e , 2(1):76–82 , 1991. 12