The Complexity of Testing Properties of Simple Games

The Complexit y of T esting Prop erties of Simple Games J. F reixas 1 , X. Molinero 2 , M. Olsen 3 , and M. Serna 4 1 Universitat P olit ` ecnica de Catalun ya. DMA3 and EPSEM. E-08240 Manresa, Spain. josep.f reixas@upc .edu 2 Universitat P olit ` ecnica de Cataluny a. LSI and EPSEM. E-08240 Manresa, Spain. molinero@lsi.upc .edu 3 MADALGO ⋆ . Department of Computer Science. U niversi ty of Aarhus. Aab ogade 34, DK 8200 Aarh us N, Denmark. mo@madalgo.au .dk 4 Universitat P olit` ecnica de Cataluny a. LSI. E-08034 Barcelona, Spain. mjserna@lsi.upc.ed u Octob er 29, 2018 Abstract. Simple games cov er voting systems in which a single alterna- tive, such as a bill or an amendment, is pitted against the status quo. A simple game or a y es–no voting system is a set of rules that sp ecifies ex- actly which collections of “yea” votes yield passage of the issue at hand. A collection of “y ea” voters forms a winning coalition. W e are interested on p erforming a complexit y analysis of problems on such games dep end- ing on the game represen tation. W e consider fo ur natural explicit rep- resen tations, winning, lo osing, minimal winn in g, and maximal lo osing. W e first analyze the computational complexity of obtaining a particu- lar representation of a simple game from a different one. W e show that some cases this transformation can b e done in p olynomial time while th e others require ex p onential time. The second qu estion is classifying the complexity for testing whether a game is simple or wei ghted. W e show that for the four types of representati on b oth problem can b e solved in p olynomial time. Finally , we provide results on the complexity of testing whether a simple game or a weigh ted game is of a sp ecial type. In this w a y , we analyze strongness, prop erness, d ecisiv eness and homogeneity , whic h are desirable p roperties to b e fulfilled for a simple game. Keywords S imple/W eig hted/Majority Games, NP- Completeness. 1 In tro duction Simple game theory is a very dynamic a nd expa nding field. T a ylo r and Zwick er [17] po int ed out that “ few structu r es arise in mor e c ont exts and lend themselves to mor e diverse interpr etations than do simple games ”. Indeed, simple ga mes c ov er voting sy s tems in which a single alternative, such as a bill or an amendment, is ⋆ Cen ter for Massiv e Data A lgorithmics, a Center of th e Danish National Research F oundation. 2 J. F reixas, X . Molinero, M. Olsen, and M. Serna pitted against the status q uo . In these systems , each voter resp onds with a v ote of “yea” or “nay”. A simple g ame or a yes–no voting sys tem is a set o f rules that sp ecifies exactly which collections o f “yea” votes yield pass age of the is sue at hand. A collectio n of “yea” voters for ms a winning coa litio n. Demo c ratic s o cieties and international org anizations us e a wide v ariety of complex rules to reach decisions. Ex amples, where it is not alwa ys easy to under- stand the cons equences of the way voting is done, include the Elector al College to elec t the President o f the United States , the United Nations Security Co uncil, the gov ernance s tr ucture of the W orld Bank, the In ternational Monetar y F und, the Europ ea n Union Co uncil of Ministers, the national gov ernments of many countries, the councils in several counties, and the system to elect the ma jor in cities or villa ges of many countries. Another source o f examples comes from economic en terprises who se owners ar e share holders of the so ciety and divide profits or losse s prop or tionally to the num ber s of sto cks they p osses, but make decisions following sp ecific rules which a re voted and ea ch shareholder ha s a vote prop ortiona l to the num ber of share s tha t hav e in the so ciety . One natural wa y to constr uct a simple ga me is to a ssign a (p ositive) real num- ber weigh t to each voter, and declare a coa lition to b e winning pr ecisely when its total weight meets or exceeds s ome pr edetermined q uota. Simple games de s crib ed in this wa y ar e said to b e weigh ted. Y et not every simple g ame is weighted, every simple game can b e decomp osed as a n in tersection o f so me weigh ted games . It is related with the notion of dimensio n co nsidered by T aylor and Zwick er [16,17]. The computational effort to weigh up the dimension of a game was determined by De ˘ ıneko and W o eg ing er [2]: computing the dimensio n of a simple game is a NP - hard pro ble m. Related work with the co mplexity theory in game theory app ears in [14], where Pras ad a nd Kelly provide exa mples o f NP -completeness on deter mining pr op erties o f weigh ted ma jor ity voting games. F or instance, they show that computing standard po litical p ow er indices, such a s absolute Ba nzhaf, Banzhaf–Colema n and Shapley-Shubik, ar e all NP - hard pro blems. In a ddition, Elkind et al. [3] deal with complexit y analys is using representations for w eighted games. All those results relate es sentially to weight ed games, how ever there a re als o several alter native ways to introduce a simple ga me; the most natural is by giving the list of winning coalitions , then the complementary set is the set of losing c o alitions and the simple game is fully describ ed. A co nsiderably r e duction in intro ducing a simple game can b e obtained by considering only the list of minimal winning c o alitions, i.e. winning coa litions which are minimal b y the inclusion o pe r ation. Coa litio ns containing minimal winning co alitions ar e also winning. Analogous ly , o ne may present a simple g ame by using either the set of losing coalitions or the set of maximal losing coalitions . W e are interested o n per forming a complexity analysis depending on the g ame representation. A first ob jective to analyze in this work is: given a particular representation of a s imple ga me how difficult is it, from the computationa l view p o in t, to obtain the other three sets of co alitions? In other w ords, we a re in terested in classifying the complexity o f passing from one a llowable representation to another. The Complexity of T esting Prop erties of Simple Games 3 The se c ond g oal co nsists in cla ssifying the c o mplexity fo r testing w he ther a game is w eighted a s suming that the simple g ame is presented by one of the four sets of coalitio ns des crib ed ab ov e. The third aim conce r ns classifying the co mplexity o f testing whether a sim- ple g a me is o f a sp ecial type. Apart from weigh ted ga mes there are some other sub c lasses of simple games which are very s ig nificant in the litera ture of vot- ing systems. Strongnes s, pr op erness, decisiveness a nd homo g eneity are, among others, desira ble prop erties to b e fulfilled for a simple game. Our results a re summar ized o n T ables 1 a nd 2. T able 1 s hows the c o mplexity to pa ss from a given form to another one. All ex plicit for ms ar e repres ent ed b y a pair ( N , C ) in which N = { 1 , . . . , n } for some p ositive in teger n , and C is the set o f winning, minima l winning, los ing and ma ximal losing coalitions. Note that it is p ossible to pass from winning (or losing) coa litions to minima l winning (or maximal losing) coalitions in p oly nomial time, but the other s waps requiere ex- po nential time. On the o ther hand, given a game in a sp ecific form, T able 2 shows the complexity on determining whether it is simple, s trong, prop er, weighted, homogeneous , decisive o r ma jority . Here ( q ; w ) denotes an int e ger r epr esentation of a w eighted ga me where q is the quo ta and w are the w eights. How ev er, there are some proble ms that still r emain op en. Input → ( N , W ) ( N , L ) ( N , W m ) ( N , L M ) Output ↓ ( N , W ) – EXP EXP EXP ( N , L ) EXP – EXP EXP ( N , W m ) P P – EXP ( N , L M ) P P EXP – T able 1. Complexity of c hanging th e representation form of a simple game. Input → ( N , W ) ( N , W m ) ( N , L ) ( N , L M ) ( q ; w ) IsSimple P P P P – IsStr ong P co-NPC P P co -NPC IsPr oper P P P co-NPC co-NPC IsWeighted P P P P – IsHomogeneous P ? P ? ? IsDecisive P ? P ? co-NPC IsMajority P ? P ? co-NPC T able 2. Complexit y on p roblems on simple games. 4 J. F reixas, X . Molinero, M. Olsen, and M. Serna Up till now we deal with ex plicit or e x tensive forms (fro m viewp oint of giving an exhaustive list of coalitions which defines the given ga me), how e ver we also consider succinct forms (a bo olean formula which defines the given g ame). W e refer the reader to Papap dimitriou [12] for the definitions of the complex- it y classes P , NP , co-NP , and their sub clas s es o f co mplete pro blems NPC a nd co-NPC , and the counting class #P . 2 Recognizing simple games W e star t b y giving some basic definitions on simple games (we r e fer the in terested reader to [17] for a thorough presentation). Simple games ca n b e viewed as mo dels of voting systems in which a single alternative, such as a bill or an amendment, is pitted ag ainst the status quo. Definition 1. A simple game Γ is a p air ( N , W ) in which N = { 1 , . . . , n } for some p ositive inte ger n , and W is a c ol le ction of subsets of N that satisfi es N ∈ W , ∅ / ∈ W , and the monotonicity pr op erty: S ∈ W and S ⊆ R ⊆ N ⇒ R ∈ W . An y set of voters is called a c o alition , the set N is called the gr and c o alition , and the empty set ∅ is ca lled the nul l c o alition . Mem bers of N are called players or voters , and the subsets o f N that a re in W are called winning c o alitions . The int uition here is that a set S is a winning coalition iff the bill or amendment passes when the play ers in S are prec is ely the one s who voted for it. A subset of N that is not in W is called a losing c o alitio n . The collection of lo os ing c o alitions is denoted b y L . The set of minimal winn ing c o alitions ( maximal losing c o alitions ) is deno ted by W m ( L M ), where a minimal winning coa lition (a max imal losing coalition) is a winning (losing ) coalition all of whose prop er subse ts (sup ersets) are losing (winning). Becaus e of monotonicity , any s imple game is completely determined by its set of minima l winning co alitions. A voter i is null if i / ∈ S for all S ∈ W m . ¿F rom a computatio nal p oint o f view a simple game can b e given under different r epresentations. In this pap er we consider the following options: – Explicit or Extensive winni ng fo rm: the game is given as ( N , W ) b y providing a listing of the colle c tion of subsets W . – Explicit or Exten sive minimal winning form: the game is given a s ( N , W m ) by providing a listing of the family W m . Obser ve that this form requires less space than the ex plicit winning form whenever W 6 = { N } . When we consider descr iptions of a game in terms of winning coa litions in this pap er, we a lso cons ide r the corr esp onding r epresentations for los ing coalitio ns, replacing minimal by maximal. Thus, in addition we co nsider the explicit or extensive losi ng , and explicit or extensive maximal l osing forms. W e a nalyse fir st the computatio nal complexity of obtaining a r epresentation of a ga me in a g iven form when a repr esentation in another form is given. F or doing so we r e quire some additional res ult on families of sets. The Complexity of T esting Prop erties of Simple Games 5 Lemma 1. Given a family of subsets C of a set N , we c an che ck whether it is close d un der ⊆ or ⊇ in p olynomial time. Pr o of. W e have to chec k for any set S ∈ C whether a ll their sup ers ets (or subsets) a re included in C . That is, for any S ∈ C we ha ve to chec k whether for all s et D s uch that S ⊂ D ⊆ N (o r D ⊂ S ) then D ∈ C . This can be done in po lynomial time in | C | (see [12]). ⊓ ⊔ Lemma 2. Given a family of su bsets C of a set N , and let C b e the closur e of C under ⊆ , and C b e the closur e of C u nder ⊇ , then the families C m and C M c an b e obtaine d in p olynomial time. Pr o of. O bserve that, for any set S in C we hav e to check whether there is a subset (sup erset) of S that fo r ms part of C , and keep tho se S that do not verify the prop erty . Therefore, the complete co mputation can b e done in po lynomial time on the input leng th of C . ⊓ ⊔ Definition 2. Given a family of subsets C of a set N , we say that it is minimal if C = C m and m aximal iff C = C M . As a consequence of Lemma 2 we ha ve the following result. Lemma 3. Given a family of subsets C of a set N , we c an che ck whether it is maximal or minimal in p olynomial time. Now we can s ta te our first res ult for simple ga mes given in explicit winning or losing form. Lemma 4. Given a game Γ in explicit winning (or losing) form, the re pr esen- tation of Γ in explicit minimal winning or maximal losing ( m aximal losing or minimal winning) form c an b e obtaine d in p olynomial time. Pr o of. Given a game Γ = ( N , W ), consider the set R = | N | [ i =1 W − i where W − i = { S \ { i } : i ∈ S ∈ W } . Obser ve that all the sets in R \ W are losing coalitions, R \ W ⊆ L . W e claim that ( R \ W ) M = L M . W e are g oing to prov e that in tw o steps: – ( R \ W ) M ⊆ L M : Now supp ose that T ∈ ( R \ W ) M and that T / ∈ L M . Consequently we hav e that T ∈ L and that T ∪ { i } ∈ W for some i ∈ N . W e also hav e that T ⊂ U for some U ∈ L . Due to the mono to nicity we conclude that U ∪ { i } ∈ W . This means that U ∈ R \ W which contradicts that T is maximal in R \ W . – L M ⊆ ( R \ W ) M : W e will show this inclusion in tw o s teps : a. L M ⊆ R \ W : If T ∈ L M then T ∪ { i } ∈ W for any i / ∈ T . Thus T can be obtained from a winning coa lition ( T ∪ { i } ) from r emoving an element ( i ). This means that T ∈ R \ W since T is a losing co alition. 6 J. F reixas, X . Molinero, M. Olsen, and M. Serna b. Ma ximal elements in a set will also be max ima l in a ny subset they app e ar in. F rom L M ⊆ R \ W ⊆ L we conclude that L M ⊆ ( R \ W ) M . F or the co st o f the algorithm, o bserve that, given ( N , W ), the set R has cardinality at most | N | · | W | , and thus R can b e obtained in p olynomia l time. Using Lemma 2, from W and R \ W , we can compute W m and L M in p olynomial time. Analogously , when the ga me is given by the family of losing coalitio ns a symmetric ar gument provides the pro of for explicit maximal los ing o r minimal winning form. ⊓ ⊔ Now we fo cus on simple ga mes g iven in explicit minimal winning or explicit maximal losing form. Lemma 5. Given a game Γ in explicit minimal winning (maximal losing) form, c omputing the r epr esentation of Γ in explicit maximal losing (minimal winning) form r e quir es exp onential time. Pr o of. T he following tw o examples s how tha t the s ize of the c omputed family can b e exp onential in the size of the given one. Therefo re, any algorithm that solves the pro blem requires exp onential time. Consider N = { 1 , . . . , 2 n } and coalitions S i = { 2 i − 1 , 2 i } for all i = 1 , . . . , n . Then, (i) The simple g ame defined b y W m = S n i =1 { S i } has L M = { T ⊆ N : | T ∩ S i | = 1 , for all i = 1 , . . . , n } . Therefore, | W m | = n and | L M | = 2 n . (ii) The simple g ame defined b y W m = { T ⊆ N : | T ∩ S i | = 1 , for all i = 1 , . . . , n } has L M = S n i =1 { N \ S i } . Therefore, | W m | = 2 n and | L M | = n . ⊓ ⊔ As a consequence of Lemmata 4 a nd 5 we have, Lemma 6. Given a game Γ in explicit minimal winning (maximal losing) form, c omputing the r epr esentation of Γ in ex plicit losing (winning) form re quir es ex- p onential t ime. The remaining cases are ag ain computationally hard. Lemma 7. Given a game Γ in explicit winning (losing) form, c omput ing t he r epr esent ation of Γ in explicit losing (winning) form r e quir es exp onential time. Pr o of. W e pre sent t w o examples where the size o f the computed family is expo - nent ial in the size of the g iven one. Let ( N , W ) be the game, wher e W = { N } , then | W | = 1 and | L | = 2 | N | − 1. Similarly , let be the game ( N , W ), where L = {∅} , then | W | = 2 | N | − 1 a nd | L | = 1. ⊓ ⊔ The Complexity of T esting Prop erties of Simple Games 7 Lemmatta (1)–(7) give us all results pres ent ed in T able 1 . Note that most of the sw aps need exp onential time. Now we analyse the co mputational complexity of the following pr oblems: Name: IsSimpleE Input: ( N , C ) Question: Is ( N , C ) a corr ect e xplicit repres ent ation of a simple game? W e have in total fo ur differ ent problems dep ending on the test for winning , minimal winning, losing and maximal losing. How ever, the pro blem b ecomes po lynomial time so lv able when the game r epresentation provides an explicit description of the winning, minimal winning, losing, or max imal losing coa litions. This is a direct conseq uence o f Lemmata 2, 1, and 3 , stating that whether the family is monoto nic 5 or minimal/maximal c an b e tested in p olyno mial time. This result establishe s the first row of T able 2. Theorem 1 . The I sSimpleE p r oblem b elongs to P for any explicit form F : winning, minimal winning, losing, or m aximal losing. 3 Problems on simple games In this section we consider a set of decis ion problems related to prop erties that define some s pe c ial type s o f simple ga mes (a gain w e re fer the rea der to [17]). In general we will state a prop erty P for simple games a nd consider the asso ciated decision problem which has the form: Name: IsP Input: A simple ga me Γ Question: Do es Γ sa tisfy prop erty P ? F urther co nsiderations on the complexity of such problems will b e sta ted in terms of the input re presentation. 3.1 Recognizing strong and prop er games Definition 3. A simple game ( N , W ) is stro ng if S / ∈ W implies N \ S ∈ W . A simple game that is n ot st ro ng is c al le d weak . Int uitively sp eaking, if a g ame is weak it ha s to o few winning coalitions, bec ause adding sufficiently many winning coa litions would make the game strong. Note that the a ddition of winning coalitions ca n never destroy stro ngness. Definition 4. A simple game ( N , W ) is prop er if S ∈ W implies N \ S / ∈ W . A simple game that is n ot pr op er is c al le d improp er . 5 W e sa y that a family of sets is monotonic iff it satisfies the monotonicit y prop erty . 8 J. F reixas, X . Molinero, M. Olsen, and M. Serna An impr op er game has to o many winning coa litions, in the sense that deleting sufficiently many winning coalitions would ma ke the ga me pro p e r. Note that the deletion of winning co alitions can never destroy prop erness. When a game is bo th pro pe r and strong, a coalition wins iff its complement loses. Therefor e, in this case we ha ve | W | = | L | = 2 n − 1 . Theorem 2 . The IsStrong pr oblem, when the input game is given in explicit losing or m aximal losing form, and the IsProper pr oblem, when t he game is given in explicit winn ing or minimal winning form, c an b e solve d in p olynomial time. Pr o of. Fir st observe tha t, given a family of subsets F , we can chec k , for any set in F , w he ther its co mplement is not in F in p olyno mia l time. Ther efore, taking into account the definitions, we have that the IsStrong problem, w he n the input is given in explicit loo sing for m, and the I sProper problem, when the input is given in explicit winning form, are p olyno mial time solv able . Thu s, taking into acc o unt that – A simple ga me is weak iff ∃ S ⊆ N : S ∈ L ∧ N \ S ∈ L which is equiv alen t to ∃ S ⊆ N : ∃ L 1 , L 2 ∈ L M : S ⊆ L 1 ∧ N \ S ⊆ L 2 The last assertion is eq uiv alent to the fact that ther e are tw o maximal lo o sing coalitions L 1 and L 2 such that L 1 ∪ L 2 = N . – A simple ga me is impr op er iff ∃ S ⊆ N : S ∈ W ∧ N \ S ∈ W which is equiv alen t to ∃ S ⊆ N : ∃ W 1 , W 2 ∈ W m : W 1 ⊆ S ∧ W 2 ⊆ N \ S. This la st ass ertion is equiv alent to the fac t tat there are tw o minimal winning coalitions W 1 and W 2 such that W 1 ∩ W 2 = ∅ . Observe that, given a family of subsets F , c hec king whether an y o ne of the t wo conditions hold ca n be done in p olynomial time. Thus the theore m holds a lso when the set of max imal lo osing (or minimal winning ) coalitions is given. ⊓ ⊔ As a consequence of Lemma 4 a nd the previo us theorem we ha ve Corollary 1. The IsStrong pr oblem, when the input game is given in explicit winning form, and the IsProper pr oblem, when the game is given in explicit losing form, c an b e s olve d in p olynomial time. Our next result states the co mplexity o f the I sStrong problem when the game is given in the remaining for ms. The Complexity of T esting Prop erties of Simple Games 9 Theorem 3 . The IsStrong pr oblem is co -NP -c omplete when t he input game is given in explicit minimal winn ing form. Pr o of. ¿F rom the definitio n of stro ng game, it is straig htforward to show that the problem belo ngs to co-NP . W e show her e that the complementary problem, the IsWeak problem, when the input game is g iven in extensive winning form, is NP -har d. This will settle the claimed result. W e provide a p o lynomial time r eduction from the set splitting pr o blem w hich is known to b e NP - c omplete [5]. An insta nce of the set s plitting pr oblem is a collection C o f subsets of a finite set N . The question is whether it is p ossible to partition N into tw o subse ts P and N \ P suc h that no subset in C is entirely contained in either P or N \ P . In other words we have to decide whether P ⊆ N exists such that ∀ S ∈ C : S 6⊆ P ∧ S 6⊆ N \ P (1) W e transform a set splitting ins tance ( N , C ) into the simple game in explicit minimal winning for m ( N , C m ). This trans formation ca n b e computed in p oly- nomial time according to Lemma 2. W e will now show that ( N , C ) has a set splitting iff ( N , C m ) is a weak g ame: – Now assume that P ⊆ N satisfying (1 ) exists. This means that P a nd N \ P are lo osing coalitio ns in the game ( N , C m ). – Let P and N \ P b e lo osing coalitions in the game ( N , C m ). As a conse q uence we hav e that S 6⊆ P and S 6⊆ N \ P for a ny S ∈ C m . This implies tha t S 6⊆ P and S 6⊆ N \ P holds for any S ∈ C s ince a ny se t in C contains a se t in C m . ⊓ ⊔ Theorem 4 . The IsProper pr oblem is co-NP - c omplete when the input game is given in extensive maximal losing form. Pr o of. ¿F rom Definition 4 , a g ame is impr op er if and only if ∃ S ⊆ N : S 6∈ L ∧ N \ S 6∈ L ⇐ ⇒ ∃ S ⊆ N : ∀ T 1 , T 2 ∈ L M : S 6⊆ T 1 ∧ N \ S 6⊆ T 2 Therefore the pr o blem I sImproper b elongs to NP , and the problem I sProper belo ngs to co-NP . T o show that the problem is also co-NP -ha r d we provide a reduction from the IsStrong pr oblem for games given in extensive minimal winning form. First obser ve that, if a fa mily C of subs ets o f N is minimal then the family { N \ L : L ∈ C } is maximal. Given a g ame Γ = ( N , W m ), in minimal winning form, let us co nsider its dual game Γ ′ = ( N , { N \ L : L ∈ W m } ) given in maximal losing form. O f cour se Γ ′ can b e obtaine d from Γ in p olynomia l time. Thus Γ is weak iff ∃ S ⊆ N : S ∈ L ( Γ ) ∧ N \ S ∈ L ( Γ ) which is equiv alen t to ∃ S ⊆ N : N \ S ∈ W ( Γ ′ ) ∧ S ∈ W ( Γ ′ ) iff Γ ′ is improp er ⊓ ⊔ 10 J. F reixas, X . Molinero, M. Olsen, and M. Serna 3.2 Recognizing w eigh ted games Definition 5. A simple game ( N , W ) is weigh ted if t her e ex ist a “ quota ” q ∈ R and a “ weigh t function ” w : N → R such t hat e ach c o alition S is winning exactly when the sum of weights of S m e ets or exc e e ds q . Note that, fr o m the definition of simple game, we ha ve 0 < q ≤ w ( N ). W eighted g ames are proba bly the most imp ortant k ind of simple g ames. Any sp e- cific e x ample of a weigh t function w and quo ta q is said to r e alize G a s a weigh ted game. A particula r rea lization of a weight ed game is denoted ( q ; w 1 , . . . , w n ), or briefly ( q ; w ). By w ( S ) we denote P i ∈ S w i . Observe also that, fr o m the m onotonicity pr op erty , it is obvious that a simple game ( N , W ) is weighte d iff there exist a “ quota ” q ∈ R and a “ weight funct ion ” w : N → R such that w ( S ) ≥ q ∀ S ∈ W m w ( S ) < q ∀ S ∈ L M . On the other ha nd, although a simple game can fail to be pr op er and fail to be strong, this canno t happen with weigh ted g ames. Prop ositi o n 1. Any weighte d game is either pr op er or str ong. ¿F rom Prop osition 1, it fo llows that there ar e thre e kind of weigh ted games: prop er but not s trong, strong but not prop er , and both strong and pro p er . It is w ell–known that any weigh ted g ame admits a n integer r ealization (see for instance [1 ]), that is, a weigh t function with no nnegative in teger v alues, and a p ositive integer a s quota. Integer rea lizations naturally aris e; just consider the seats distributed amo ng p o litical parties in any voting s ystem. Theorem 5 . The IsW eighted pr oblem c an b e solve d in p olynomial time when the input game is given in explicit winning or losing form. Pr o of. W e provide a p olyno mial time re duction from the IsWeighted pro ble m to the Line ar Pr o gr amming pro blem, which is known to b e solv able in po lynomial time [7,8]. T aking into acco unt Lemma 2, in b oth cases we can o btain W m and L M in po lynomial time. Once this is done we ca n write, aga in in p olynomia l time, the following Line ar Pr o gr amming instance Π : min q subjec t to w ( S ) ≥ q if S ∈ W m w ( S ) < q if S ∈ L M 0 ≤ w i for all 1 ≤ i ≤ n 0 ≤ q As ( N , W ) is weigh ted iff Π has a s olution, the prop osed co nstruction is a p oly- nomial time reduction. ⊓ ⊔ The Complexity of T esting Prop erties of Simple Games 1 1 Theorem 6 . The IsW eighted pr oblem c an b e solve d in p olynomial time when the input game is given in explicit minimal winning or maximal losing form. Pr o of. Given ( N , W m ), w e are going to prove tha t we can decide in poly nomial time whether a simple game is weigh ted. F or C ⊆ N we let x C ∈ { 0 , 1 } n denote the vector with the i ’th co ordinate equal to 1 if a nd only if i ∈ C . In po lynomial time we transform W m int o the bo olean function Φ W m given by the dnf : Φ W m ( x ) = _ S ∈ W m ( ∧ i ∈ S x i ) By construction we have the following: Φ W m ( x C ) = 1 ⇔ C is winning in the g a me given by ( N , W m ) (2) Note that Φ W m is a threshold function if and only if the g ame given by ( N , W m ) is weigh ted: – only if ( ⇒ ): Assume that Φ W m is a thr eshold function. Let w ∈ R n be the weigh ts and q ∈ R the threshold v alue. Thus we hav e that Φ W m ( x C ) = 1 ⇔ h w , x C i ≥ q where h· , ·i denotes the usual inner pro duct. By using (2) we conclude that the game given b y ( N , W m ) is weigh ted. – if ( ⇐ ): Now assume that the ga me given by ( N , W m ) is weigh ted and that ( q ; w ) is a r ealization of such g ame. In this case we have the following: C is winning in the ga me given by ( N , W m ) ⇔ h w , x C i ≥ q Again we us e (2) and conclude that Φ W m is a threshold function. The b o olean function Φ W m is mono tonic (i.e. p ositive ) so acco rding to the pap ers [6,13] (pages 211 and 59, resp ectively) we ca n in p oly nomial time de- cide whether Φ W m is a threshold function. Conseq uent ly we ca n also decide in po lynomial time whether the game given by ( N , W m ) is weigh ted. On the other hand, we can prove a similar result given ( N , L M ) just taking int o acount that a game Γ is weigh ted iff its dual g ame Γ ′ is weigh ted. Then, we can use the same technique from the pro of of Theorem 4. ⊓ ⊔ Note that now w e can offer an alternative to the linear progr a mming appro a ch to dec ide whether the game is w eighted g iven ( N , W ) (first part of Theo r em 5 ): W e compute ( N , W m ) in p olynomia l time (Lemma 4) a nd use the pro ce dure describ ed in Theor em 6. How ever, this approa ch do es not provide a wa y to determine the complexity of the IsH omogeneous problem given a game in explicit winning o r losing form. 12 J. F reixas, X . Molinero, M. Olsen, and M. Serna It is importa nt to remark that it is known that “ a s imple game is weighte d iff it is t r ade r obust ” [3,17]. Thus, according to Theorems 5 and 6, chec king whether a simple g ame is trade ro bust can b e solved in p olynomial time. W e might also note her e that this result is different from the one obtained by De ˘ ınek o a nd W o eginger [2] who pr ov ed that deter minig the dimension of a simple g ame is a NP -hard problem. Corollary 2. The IsTradeRobustness pr oblem c an b e solve d in p olynomia l time when the input game is given in ex plicit winning, minimal winning, losing or maximal losing form. This subsection prov es the fourth row o f T able 2 (problems o n p olyno mial). 3.3 Recognizing homogeneous , decisive and ma jority games In this se c tion w e define the homo geneous, decisive and ma jority games and, afterwards, we analyse the complexity of the IsHomogeneous , IsD ecisive and IsMajority problems. Definition 6. A s imple game ( N , W ) is homogeneous if ther e exist a (weighte d) r e alization ( q ; w ) such that q = w ( S ) for al l S ∈ W m . That is, a weigh ted g ame is ho mogeneous iff the sum o f the weigh ts of a ny minimal winning co alition is equal to the quota. Theorem 7 . The IsHomogeneous pr oblem c an b e solve d in p olynomial time when the input game is given in ex plicit winning or losing form. Pr o of. T he p olynomial time r eduction fro m the I sHomogeneous pro blem to the Line ar Pr o gr amming pro blem is done in the same vein a s we did to pr o of Theorem 5, but consider ing the instance Π ′ obtained replacing w ( S ) ≥ q , in the first set of inequalities o f Π , by w ( s ) = q . It is immediate to see that the game is homog e neous iff Π ′ has a solution. ⊓ ⊔ Definition 7. A simple game is decisive (or s elf–dual , or co nstant sum ) if it is pr op er and str ong. A simple game is indecisive if it is n ot de cisive. A related concept with decisiveness is the dualityness. Definition 8. Given a simple game ( N , W ) , its dual game is ( N , W ∗ ) , wher e S ∈ W ∗ if and only if N \ S / ∈ W . That is, winning coalitio ns in the dua l g ame ar e just the “ blo cking” c oalitions in the or iginal g a me. Note that ( N , W ) is prop er iff ( N , W ∗ ) is strong , and ( N , W ) is stro ng iff ( N , W ∗ ) is prop er. As a conseq uence, we hav e that a simple game ( N , W ) is decisive iff W = W ∗ . On the other hand, ( N , W ) is closed under ⊆ or ⊇ iff ( N , W ∗ ) is clos ed under ⊆ or ⊇ , resp ectively . The Complexity of T esting Prop erties of Simple Games 1 3 In the s eminal work on game theory b y v on Neumann and Morgenstern [10] only decisive simple games were considere d. Now adays, ma ny gov ernmen tal in- stitutions made their decisions through voting rules that are in fact decisive games. If abstention is no t allowed (see [4] for voting games with abstention) ties are not p oss ible in decisive ga mes. Another in teresting subfamily o f simple games are the so–called ma jority games: Definition 9. A simple game is a ma jority game if it is weighte d and de cisive. Theorem 8 . The I sMajority and the I sDecisive pr oblems c an b e s olve d in p olynomial time when the input game is given in explicit winning or losing form. Pr o of. W e know tha t a g a me is b o th prop er a nd strong requir es that | W | = | L | = 2 n − 1 , a nd this test can b e p e rformed in polyno mial time when W or L is g iven. F urthermor e , under b oth forms, we can chec k whether the game is weighted in po lynomial time using Theor em 5. ⊓ ⊔ Unfortunately , now we just susp ect the following claims without any proo f. Conje ctur e 1. The IsDecisive problem is co-NP - complete when the input game is given in ex plicit minimal winning o r maximal losing form. Conje ctur e 2. The I sMajority proble m is co-NP -complete when the input game is given in ex plicit minimal winning o r maximal losing form. W e have studied the firs t four columns of T able 2. Henceforth we will only consider weigh ted games giv en b y an in teger repres e n tation ( q ; w ). 4 Problems on w eigh ted games In this section we cons ider weighted g ames which are describ ed by an integer realization ( q ; w ). Obser ve that for this case ma jority a nd decisive a r e just the same pro pe rty , so we will consider only the ma jority games. W e analyse the complexity of problems of the type: Name: IsP Input: An integer r ealization ( q ; w ) of a weigh ted ga me Γ . Question: Do es Γ sa tisfy P? W e ar e interested in such proble ms a sso ciated to the prop erties o f b eing strong, prop er, homog eneous, and ma jority . ¿F rom now on so me of the pro ofs are based on reductions fr o m the NP - complete problem P ar tition [5], which is defined as: Name: P ar tition Input: n integer v alues, x 1 , . . . , x n Question: Is there S ⊆ { 1 , . . . , n } for which P i ∈ S x i = P i / ∈ S x i . 14 J. F reixas, X . Molinero, M. Olsen, and M. Serna Observe that, for any instance of the P ar tition problem in which the sum of the n input num b er s is o dd, the a nswer must be no . Theorem 9 . The IsStrong , IsProper and IsMajority (her e, e quivalent to IsD ecisive ) pr oblems, when the input describ es a inte ger r e alization of a weighte d game ( q ; w ) , ar e co- NP -c omplete. Pr o of. ¿F rom the definitions of strong, prop er and ma jo r ity g ames, it is str a ight- forward to show that the three problems b elong to co-NP . Observe that the w eigh ted game with integer repr e sentation (2; 1 , 1 , 1) is prop er and strong , and th us decisive. W e trans fo rm an instance x = ( x 1 , . . . , x n ) of P ar tition problem in to a realization of a weight ed game according to the following s chema f ( x ) = ( ( q ( x ); x ) when x 1 + · · · + x n is even, (2; 1 , 1 , 1) otherwise. F unction f c a n b e co mputed in p olynomial time provided q do es, and we will use a different q for each problem. Nevertheless, indepe nden tly o f q , when x 1 + · · · + x n is o dd , x is a no input for partition, but f ( x ) is a yes instance o f IsStrong , IsProper , a nd IsMajority , and th us a no insta nce of the complementary pr oblems. Therefore, we hav e to tak e care only of the ca se in which x 1 + · · · + x n is even . Assume tha t this is the case and let s = ( x 1 + · · · + x n ) / 2 and N = { 1 , . . . , n } . W e will provide the pr o of that f reduces P ar tition to the complementary pro blem. a) IsStrong pr oblem. F or the ca s e o f strong games, taking q ( x ) = s + 1 , we hav e: – If there is a S ⊂ N such that P i ∈ S x i = s , then P i / ∈ S x i = s , thus b oth S and N \ S are losing coa litio ns and f ( x ) is weak. – Now assume that S a nd N \ S a re b o th losing coalitions in f ( x ) If P i ∈ S x i < s then P i / ∈ S x i ≥ s + 1 , whic h cont radicts that N \ S is los ing. T hus w e hav e that P i ∈ S x i = P i 6∈ S x i = s , a nd there exists a partition of x . Therefore, f is a p olytime r eduction from P ar tition to IsWeak b) IsProper pr oblem. F or the ca se of prop er games we take q ( x ) = s . Then, if ther e is a S ⊂ N such that P i ∈ S x i = s , then P i / ∈ S x i = s , thus bo th S and N \ S are winning coalitions and f ( x ) is impr o p er. When f ( x ) is improp er ∃ S ⊆ N : X i ∈ S x i ≥ s ∧ X i / ∈ S x i ≥ s , and thus P i ∈ S x i = s . Thus, we ha ve a p olytime r eduction fr om P a r tition to IsImproper . The Complexity of T esting Prop erties of Simple Games 1 5 c) IsMajority pr oblem. F or the ca se o f ma jority ga mes we take again q ( x ) = s . Observe that f ( x ) cannot b e weak, as in such a case ther e must b e some S ⊆ N for which, X i ∈ S x i < s ∧ X i / ∈ S x i < s , contradicting the fac t that s = ( x 1 + · · · + x n ) / 2. Therefor e, the game is not ma jority iff it is improp er, and the cla im follows. ⊓ ⊔ In the s ame vein we hav e no res ults for the IsHo mogeneous pr oblem, but befo re finishing this s ection we introduce the following related proble m: Name: IsHomogeneousRealiza tion Input: An integer r ealization ( q ; w ) of a weigh ted ga me Γ . Question: Is ( q ; w ) a n homogeneous realiza tio n? Theorem 1 0. The IsHomogeneousRealiza tion pr oblem c an b e solve d in p olynomial time. Pr o of. Given the w eights w , Rosenm ¨ uller [15] presents an algorithm that com- putes all q such tha t ( q ; w ) is a homogeneo us realiza tion. The a nalysis on the complexity of the algor ithm is omitted. It is not hard to construct a dyna mic pro gramming algo r ithm – bas ed on Lemma 1 .1 (also ca lled the Basic L emma ) in [15] – that runs in po lynomial time and chec ks whether the integer realiza tion ( q ; w ) is a ho mogeneous realiza tion. ⊓ ⊔ Note that, given an integer r ealization ( q ; w ) of a weigh ted game, we hav e not proven that this ga me is ho mogeneous, we hav e just pr ov en whether such realization is homo geneous. The IsHomogeneous problem still r emains op en. 5 Succinct represen tations Besides extensive repr esentations there ar e o ther repr esentations to pr esent sim- ple games. F or instance, using a Bo o lean function, there is the s o -called succinct representation [9]. ¿F rom a computational p oint of view, a simple game can also b e given under a succinct repr e sentation: – Succinct wi nning form: the g ame is given by ( N , Φ ) where Φ is a b o olean formula on | N | v ariables pr oviding a c o mpact de s cription of the sets in W . W e hav e Φ : { 0 , 1 } | N | → { 0 , 1 } such that Φ ( x 1 , . . . , x n ) = 1 ⇐ ⇒ ∃ S ∈ W ; S = [ x i =1 { i } . 16 J. F reixas, X . Molinero, M. Olsen, and M. Serna In this wa y W is identified with the truth assignments that satisfy the for- m ula Φ . – Succinct minima l winning form: the ga me is given by ( N , Φ ) but now Φ de- scrib es the family W m . Observe aga in that this form might re q uire less space than the previo us one whenever W 6 = { N } . The ha rdness o f the problems asso ciated to the succinct form ca me from reductions from problems o n b o ole a n formula. Theorem 1 1. The IsS impleS-F pr oblem is co-NP c omplete for any suc cinct form F : winning, minimal winning, losing, or m aximal losing. Pr o of. O bserve that a s et W ( L ) is not monoto nic iff there are tw o se ts S 1 and S 2 such that S 1 ⊆ S 2 but S 1 ∈ W and S 2 / ∈ W ( S 1 / ∈ L and S 2 ∈ L ). When the game is given in succinct winning or losing form, thes e tests can b e do ne by guessing t wo truth as s ignments x 1 and x 2 and checking that x 1 < x 2 , Φ W ( x 1 ) = 1 and Φ W ( x 2 ) = 0 ( Φ L ( x 1 ) = 0 and Φ W ( x 2 ) = 1). Both pr op erties can b e chec ked in po lynomial time once S 1 and S 2 are given. In the case that Φ repr esents W m ( L M ) we hav e to chec k that the repre - sented set is minimal (maximal). O bserve that Φ do es not represe n t a minimal (maximal) se t if there are α, β ∈ { 0 , 1 } n with α < β such that Φ ( β ) = 1 a nd Φ ( α ) = 1. Therefore, all the problems of recognizing succinct forms b elong to co-NP . The c anonic al or der on the set of truth assignments of a set o f n v a riables is defined as follows: x ≤ y iff x i ≤ y i for all i ; and, x < y iff x ≤ y and x i 6 = y i for some i . A b o o lean formula is monotonic if for any pair o f truth a ssignments x, y , such that x < y in canonica l order , we have that Φ ( x ) < Φ ( y ) (ass uming that false < true ). The later problem is co-NP -co mplete (even for dnf , disjunctive normal form, formulas) [9]. Observe that, if Φ W is a b o olea n formula representing W , then w e hav e that ( N , W ) is s imple iff Φ W is mo notonic. F urthermor e, if Φ L is a bo olean formula representing L , then we have that ( N , L ) is simple iff ¬ Φ L is monotonic. Thus both a sso ciated problems are co -NP -hard. Recall that the sa t problem a sks whether a given b o olean formula has a satisfying assignment. sa t is a well known NP - c omplete pr oblem. W e cons ider here the v ariatio n of sa t in which the input is a b o olean formula Φ such that Φ (0 n ) = 0, of course this v ar iation is also NP -complete. Given a b o olean formula Φ on n v ariables with Φ (0 n ) = 0, w e co nstruct a new formula on n + 1 v a riables defined as follows: Ψ ( αa ) =      Φ ( α ) if a = 1 0 if a = 0 and | α | 1 > 1 0 if a = 1 and | α | 1 = 1 Observe that Φ has a satisfying assignment iff Ψ r epresents a non minimal set (or a non maxima l set). Thus the re maining problems are co-NP - hard. ⊓ ⊔ The Complexity of T esting Prop erties of Simple Games 1 7 6 Conclusions Given a simple game, we have studied the complexity to pas s from an explicit form to another one. All explicit forms that w e ha ve considered are represented by a pa ir ( N , C ) in which N = { 1 , . . . , n } for some p ositive integer n , a nd C is the set of winning, minimal winning, losing and maximal losing coalitions. Given a simple g ame in an explicit form ( N , C ), we hav e studied the com- plexity to decide whether it is stro ng, pro p er , weigthed, homogene o us, decisive or ma jority . In the same vein, given a weighted g ame in an integer repr esen- tation ( q ; w ), we have a ls o considered the complexity to decide whether it is strong, prop er, homog eneous o r ma jor it y (here to be ma jo rity is equiv alent to be decisive). There ar e s o me interesting op en pr oblems in which we are working on. F or instance, given a game in explicit minimal winning o r maximal los ing form, we conjecture that to decide whether it is a decisive (or a ma jority) g ame is co- NP - complete. W e would also like to remar k that our study can b e enlarge d by co nsider- ing new forms to present a simple game. F or example, blo cking co alitions and minimal blo cking coa litions provide a n a lternative way to fully describ e a simple game. P r ecisely , a blo cking coalition wins whenever its complementary loses. Of course, there ar e other pr esentations for a simple g ame, which can b e imple- men ted using a Bo olean function. Pr obably the most natural w ay to do so is b y means of the multilinear extension of a simple g ame [11]. How ev er, w e leav e this part for future res e arch. Ac kno wledgemen ts Josep F reix a s was pa rtially s uppo rted by Grant MTM 2 006– 06064 of “Ministerio de Ciencia y T ecnolog ´ ıa y el F ondo Europ eo de Des a rrollo Regiona l” and SGRC 2005- 00651 of “ Generalitat de Cataluny a”. Xavier Mo linero was partially supp orted by pro ject TIN2005- 05446 (ALINEX) of “Ministerio de Educa cion y Ciencia ” and SGRC 2005- 00516 o f “Genera litat de Cataluny a”. Maria Ser na was partially supp orted by by FET pr o-active Int egrated Pro ject 15964 (AEOLUS), the spanish pro jects MEC TIN2005- 0919 8 -C02-0 2 (ASCE) and TIN2005-2 5859 -E, and SGR C 2005- 0051 6 of “ Generalitat de Cataluny a”. W e would a lso like to aknowledge Gerth S. B ro dal fro m University of Aar hus for v aluable comments and constructive criticism. References 1. F. Carreras and J. F reixas. 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