arXiv:2101.07621v2 [cs.GT] 29 May 2021Trading Transforms of Non-weighted Simple Games and Integer Weights of Weighted Simple Games∗ Akihiro Kawana†Tomomi Matsui‡ June 1, 2021 Abstract This study investigates simple games. A fundamental research question in this field is to determine necessaryand sufficient condition s for a simple game to be a weighted majority game. Taylor and Zwicker (1992) showed that a simple game is non-weighted if and only if there exists a trading transform of finite size. They also provided an uppe r bound on the size of such a trading transform, if it exists. Gvozdev a and Slinko (2011) improved that upper bound; their proof employed a property of linear inequalities demonstrated by Muroga (1971). In this study, we provide a new proof of the existence of a trading transf orm when a given simple game is non-weighted. Our proof employs Farkas’ lemma (1894), and yields an improved upper bound on the size of a trading transform. We also discuss an integer-weight representation of a weighted sim- ple game, improving the bounds obtained by Muroga (1971). We show that our bound on the quota is tight when the number of players is less than or equal to five, based on the computational results obt ained by Kurz (2012). Furthermore, we discuss the problem of finding an integer-weight representation under the assumption that we have minimal winning coalitions and maximal losing coalitions. In particular, we show a performance of a rounding method. Lastly, we address roughly weighted simple games. Gvozdeva and Slinko (2011) showed that a given simple game is not roughly weighted if and only if there exists a potent certificate of non-weightedness . ∗preliminary version of this paper was presented at Seventh I nternational Workshop on Computational Social Choice (COMSOC-2018), Rensselaer Polytechnic Institute, Troy, NY, USA, 25-27 June, 2018. †Graduate School of Engineering, Tokyo Institute of Technol ogy ‡Graduate School of Engineering, Tokyo Institute of Technol ogy 1We give an upper bound on the length of a potent certificate of non- weightedness. We also discuss an integer-weight representation o f a roughly weighted simple game. 1 Introduction A simple game consists of a pair G= (N,W),whereNis a finite set of players, and W ⊆2Nis an arbitrary collection of subsets of N. Throughout this paper, we denote |N|byn. Usually, the property (monotonicity): if S′⊇S∈ W,thenS′∈ W, (1) is assumed. Subsets in Ware called winning coalitions . We denote 2N\W byL, and subsets in Lare called losing coalitions . A simple game ( N,W) is said to be weighted if there exists a weight vector w∈RNandq∈R satisfying the following property: (weightedness): for any S⊆N,S∈ Wif and only if/summationdisplay i∈Swi≥q.(2) Previous research established thenecessary andsufficient c onditions that guarantee the weightedness of a simple. [Elgot, 1961] and [C how, 1961] in- vestigated the theory of threshold logic and showed the cond ition of the weightedness in terms of asummability . [Muroga, 1971] proved the suffi- ciency of asummability based on the theory of linear inequal ity systems and discussed some variations of their results in cases of a f ew variables. [Taylor and Zwicker, 1992,Taylor and Zwicker, 1999]obtain ednecessaryand sufficient conditions independently in terms of a trading transform . Atrad- ing transform ofsizejisacoalition sequence( X1,X2,...,X j;Y1,Y2,...,Y j), which may contain repetitions of coalitions, satisfying th e condition ∀p∈N, |{i|p∈Xi}|=|{i|p∈Yi}|. A simple game is called k-trade robust if there is no trading transform of size jsatisfying 1 ≤j≤k,X1,X2,...,X j∈ W, andY1,Y2,...,Y j∈ L. A simple game is called trade robust if it isk-trade robust for all positive integers k. Taylor and Zwicker showed that a given simple game Gwithnplayers is weightedifandonlyif Gis22n-traderobust. In2011, [Gvozdeva and Slinko, 2011] showed that agiven simplegame Gis weighted ifandonly if Gis (n+1)nn/2- trade robust. [Freixas and Molinero, 2009b] proposed a vari ant of trade ro- bustness, called invariant-trade robustness, whichdeter mineswhetherasim- ple game is weighted. The relations between the results in th reshold logic and simple games are clarified in [Freixas et al., 2016, Freix as et al., 2017]. 2In Section 2, we show that a given simple game Gis weighted if and only ifGisαn+1-trade robust, where αn+1denotes the maximal value of determinants of ( n+1)×(n+1) 0–1 matrices. It is well-known that αn+1≤ (n+2)n+2 2(1/2)(n+1). Our definition of a weighted simple game allows for an arbitra ry real number of weights. However, any weighted simple game can be r epresented by integer weights (e.g., see [Freixas and Molinero, 2009a] ). Aninteger- weight representation of a weighted simple game consists of an integer vec- torw∈ZNand some q∈Zsatisfying the weightedness property (2). [Isbell, 1956] found an example of a weighted simple game wit h 12 players withoutauniqueminimum-suminteger-weight representati on. Examplesfor 9, 10, or11playersaregivenin[Freixas and Molinero, 2009a ,Freixas and Molinero, 2010]. Inthefieldofthresholdlogic, examples of thresholdfuncti onsrequiringlarge weightsarediscussedby[Myhill and Kautz, 1961,Muroga, 19 71,H˚ astad, 1994]. Some previous studies enumerate (minimal) integer-weight representations of simple games with a small number of players (e.g., [Muroga et al., 1962, Winder, 1965,Muroga et al., 1970,Krohn and Sudh¨ olter, 199 5]). Inthecase ofn= 9 players, refer to [Kurz, 2012]. In general, [Muroga, 1971 ] (Proof of Theorem 9.3.2.1) showed that (under the monotonicity prope rty (1) and ∅ /\e}atio\slash∈ W ∋ N) every weighted simple game has an integer-weight repre- sentation satisfying 0 ≤wi≤αn≤(n+ 1)n+1 2(1/2)n(∀i∈N) and 0≤q≤nαn≤n(n+1)n+1 2(1/2)nsimultaneously. Here, αndenotesthemax- imal valueof determinantsof n×n0–1matrices. [Wang and Williams, 1991] discussed Boolean functions that require more general surf aces to sepa- rate their true vectors from false vectors. [Hansen and Podo lskii, 2015] in- vestigates the complexity of computing Boolean functions b y polynomial threshold functions. [Freixas, 2021] discusses a point-se t-additive pseudo- weighting for a simple game, which assigns weights directly to coalitions. In Section 3, we slightly improve Muroga’s result and show th at ev- ery weighted simple game (satisfying ∅ /\e}atio\slash∈ W ∋ N) has an integer-weight representation ( q;w⊤) satisfying |wi| ≤αn(∀i∈N),|q| ≤αn+1, and 1≤/summationtext i∈Nwi≤2αn+1−1 simultaneously. Based on the computational results of [Kurz, 2012], we also demonstrate the tightness o f our bound on the quota when n≤5. For a family of minimal winning coalitions, [Peled and Simeo ne, 1985] proposed a polynomial-time algorithm for checking the weig htedness of a given simple game. They also showed that for weighted simple games repre- sented by minimal winning coalitions, all maximal losing co alitions can be computed in polynomial time. When we have minimal winning co alitions 3and maximal losing coalitions, there exists a linear inequa lity system whose solution gives a weight vector w∈RNandq∈Rsatisfying property (2). However, it isless straightforward tofindaninteger-weigh t representation as the problem transforms from linear programming to integer p rogramming. In Section 4, we address the problem of finding an integer-wei ght rep- resentation under the assumption that we have minimal winni ng coalitions and maximal losing coalitions. We show that an integer-weig ht represen- tation is obtained by carefully rounding a solution of the li near inequality system multiplied by at most (2 −√ 2)n+(√ 2−1). A simple game G= (N,W) is called roughly weighted if there exist a non-negative vector w∈RN +and a real number q∈R, not all equal to zero ((q;w⊤)/\e}atio\slash=0⊤), such that for any S⊆Ncondition/summationtext i∈Swi< qim- pliesS/\e}atio\slash∈ W, and/summationtext i∈Swi> qimpliesS∈ W. We say that ( q;w⊤) is a rough voting representation forG. Roughly weighted simple games were ini- tially introduced by [Baugh, 1970]. [Muroga, 1971] (p. 208) studied them under the name of pseudothreshold functions. [Taylor and Zw icker, 1999] discussed roughly weighted simple games and constructed se veral examples. [Gvozdeva and Slinko, 2011] developed a theory of roughly we ighted simple games. A trading transform ( X1,X2,...,X j;Y1,Y2,...,Y j) with all coali- tionsX1,X2,...,X jwinningand Y1,Y2,...,Y jlosingiscalled a certificate of non-weightedness . This certificate is said to be potentif the grand coalition Nis among X1,X2,...,X jand the empty coalition is among Y1,Y2,...,Y j. [Gvozdeva and Slinko, 2011] showed that under the the monoto nicity prop- erty (1) and ∅ /\e}atio\slash∈ W ∋ N, a given simple game Gis not roughly weighted if and only if thereexists a potent certificate of non-weighted ness whose length islessthanorequalto( n+1)nn/2. Furtherresearchonroughlyweightedsim- plegamesappearsin[Gvozdeva et al., 2013,Freixas and Kurz , 2014,Hameed and Slinko, 2015]. In Section 5, we show that (under the the monotonicity proper ty (1) and ∅ /\e}atio\slash∈ W ∋ N) the length of a potent certificate of non-weightedness is le ss than or equal to 2 αn+1, if it exists. We also show that a roughly weighted simple game (satisfying ∅ /\e}atio\slash∈ W ∋ N) has an integer vector ( q;w⊤) of rough voting representation satisfying 0 ≤wi≤αn−1(∀i∈N), 0≤q≤αnand 0≤/summationtext i∈Nwi≤2αn. 2 TradingTransformsof Non-weighted Simple Games In this section, we discuss the size of a trading transform th at guarantees the non-weightedness of a given simple game. Throughout thi s section, we do not need to assume the monotonicity property (1). First, w e introduce a 4linear inequality system for determining the weightedness of a given simple game. For any nonempty family of player subsets ∅ /\e}atio\slash=N ⊆2N, we introduce a 0–1 matrix A(N) = (a(N)Si) whose rows are indexed by subsets in Nand columns are indexed by players in Ndefined by a(N)Si=/braceleftbigg1 (ifi∈S∈ N), 0 (otherwise) . A given simple game G= (N,W) is weighted if and only if the following linear inequality system is feasible: P1:/parenleftbiggA(W)1 0 −A(L)−1−1/parenrightbigg w −q ε ≥0, ε >0, where0(1) denotes a zero vector (all-one vector) of an appropriate di men- sion. Farkas’ Lemma [Farkas, 1902] states that P1 is infeasible if and only if the following system is feasible: D1: A(W)⊤−A(L)⊤ 1⊤−1⊤ 0⊤−1⊤ /parenleftbiggx y/parenrightbigg = 0 0 −1 , x≥0,y≥0. For simplicity, we denote D1 by A1z=c,z≥0,where A1= A(W)⊤−A(L)⊤ 1⊤−1⊤ 0⊤−1⊤ ,z=/parenleftbiggx y/parenrightbigg ,andc= 0 0 −1 . Subsequently, we assume that D1 is feasible. Let /tildewiderA1z=/tildewidecbe a linear equality system obtained from A1z=cby repeatedly removing redundant equalities. A column submatrix /hatwideBof/tildewiderA1is called a basis matrix if/hatwideBis a square invertible matrix. Variables corresponding to the c olumns of/hatwideBare calledbasic variables , andJ/hatwideBdenotes an index set of basic variables. A basic solution with respect to /hatwideBis a vector zdefined by zi=/braceleftbigg/hatwidezi(i∈J/hatwideB), 0 (i/\e}atio\slash∈J/hatwideB), 5where/hatwidezis a vector of basic variables satisfying /hatwidez=/hatwideB−1/tildewidec. It is well-known that if a linear inequality system D1 is feasible, then it has a basic feasible solution. Letz′be a basic feasible solution of D1 with respect to a basis matr ixB. By Cramer’s rule, z′ i= det(Bi)/det(B) for each i∈JB,whereBiis a matrix formed by replacing i-th column of Bby/tildewidec. Because Biis an integer matrix, det(B)z′ i= det(Bi) is an integer for any i∈JB. Let (x′⊤,y′⊤)⊤be a vector corresponding to z′,and (x∗⊤,y∗⊤) =|det(B)|(x′⊤,y′⊤). Cramer’s rule states that both x∗andy∗are integer vectors. The pair of vectors x∗and y∗satisfies the following conditions: A(W)⊤x∗−A(L)⊤y∗=|det(B)|(A(W)⊤x′−A(L)⊤y′) =|det(B)|0=0,/summationdisplay S∈Wx∗ S−/summationdisplay S∈Ly∗ S=|det(B)|(1⊤x′−1⊤y′) =|det(B)|0 = 0, /summationdisplay S∈Ly∗ S=|det(B)|1⊤y′=|det(B)|, x∗=|det(B)|x′≥0,andy∗=|det(B)|y′≥0. Next, we construct a trading transform corresponding to the pair ofx∗and y∗. LetX= (X1,X2,...,X|det(B)|) be a sequence of winning coalitions, where each winning coalition S∈ Wappears in Xx∗ S-times. Similarly, we introduce a sequence Y= (Y1,Y2,...,Y|det(B)|),where each losing coalition S∈ Lappears in Yy∗ S-times. The above equalities imply that ( X;Y) is a trading transform of size |det(B)|. Therefore, we have shown that if D1 is feasible, then a given simple game G= (N,W) is not|det(B)|-trade robust. Finally, weprovidean upperboundon |det(B)|. Letαnbethemaximum of the determinants of n×n0–1 matrices. For any n×n0–1 matrix M,it is easy to show that det( M)≥ −αnby swapping two rows of M(whenn≥2). If a column of Bis indexed by a component of x(i.e., indexed by a winning coalition), then each component of the column is either 0 or 1 . Otherwise, a column (of B) is indexed by a component of y(i.e., indexed by a losing coalition) whose components are either 0 or −1. Now, we apply elementary matrix operations to B(see Figure 1). For each column of Bindexed by a component of y, we multiply the column by ( −1). The resulting matrix, denoted by B′, is a 0–1 matrix satisfying |det(B)|=|det(B′)|. AsBis a submatrix of A1, the number of rows (columns) of B, denoted byn′, is less than or equal to n+ 2. When n′< n+ 2, we obtain the desired result: |det(B)|=|det(B′)| ≤αn′≤αn+1. Ifn′=n+ 2, then B has a row vector corresponding to equality 1⊤x−1⊤y= 0, which satisfies the condition that each component is either 1 or −1, and thus B′has an 60 0 1 1 0−1 0 1 0 1 0 0 1 0 0 1 0−1 1 1 1 0 −1−1 1 1 1 1 −1−1 0 0 0 0 −1−1 B0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 B′ Figure 1: Example of elementary matrix operations for D1. all-one row vector. Lemma 2.1 (c1) appearing below states th at|det(B)|= |det(B′)| ≤αn′−1≤αn+1. Lemma 2.1. LetMbe ann×n0–1 matrix, where n≥2. (c1)If a row (column) vector of Mis the all-one vector, then |det(M)| ≤αn−1. (c2)If a row (column) vector of Mis a 0–1 vector consisting of a unique 0-component and n−11-components, then |det(M)| ≤2αn−1. Proof of (c1). Assume that the first column of Mis the all-one vector. We apply the following elementary matrix operations to M(see Figure 2). For each column of Mexcept the first column, if the first component is equal to 1, then we multiply the column by ( −1) and add the all-one column vector. The obtained matrix, denoted by M′, is ann×n0–1 matrix satisfying |det(M)|=|det(M′)|,and the first row is a unit vector. Thus, it is obvious that|det(M′)| ≤αn−1. 11 0 1 0 11 1 1 0 10 1 0 0 11 1 0 1 10 0 1 1 M10 0 0 0 10 1 0 0 11 1 1 0 10 1 1 1 11 0 0 1 M′ Figure 2: Example of elementary matrix operations for (c1). Proof of (c2). Assume that the first column vector of M, denoted by a, contains exactly one 0-component. Obviously, e=1−ais a unit vector. LetM1andMebe a pair of matrices obtained from Mwith the first column 7replaced by 1ande, respectively. Then, it is easy to prove that |det(M)|=|det(M1)−det(Me)| ≤ |det(M1)|+|det(Me)| ≤2αn−1. QED From the above discussion, we obtain the following theorem ( without the assumption of the monotonicity property (1)). Theorem 2.2. A given simple game G= (N,W)withnplayers is weighted if and only if Gisαn+1-trade robust, where αn+1is the maximum of deter- minants of (n+1)×(n+1)0–1 matrices. Proof. If a given simple game is not αn+1-trade robust, then it is not trade robust and, thus, not weighted, as shown by [Taylor and Zwick er, 1992, Taylor and Zwicker, 1999]. We have discussed the inverse imp lication: if a given simple game Gis not weighted, then the linear inequality system P1 is infeasible. Farkas’ lemma [Farkas, 1902] implies that D1 is feasible. From the above discussion, we have a trading transform ( X1,...,X j;Y1,...Yj) satisfying j≤αn+1,X1,...,X j∈ W, andY1,...,Y j∈ L. QED Applying the Hadamard’s evaluation [Hadamard, 1893] of the determi- nant, we obtain Theorem 2.3. Theorem 2.3. For any positive integer n,αn≤(n+1)n+1 2(1/2)n. The exact values of αnfor small positive integers nappear in “The On- LineEncyclopediaof Integer Sequences (A003432)” [Sloane et al., 2018] and Table 1. 3 Integer Weights of Weighted Simple Games This section reviews the integer-weight representations o f weighted simple games. Throughoutthis section, we donot need to assume the m onotonicity property (1), except in Table 1. Theorem 3.1. Assume that a given simple game G= (N,W)satisfies ∅ /\e}atio\slash∈ W ∋ N. If a given simple game Gis weighted, then there exists an integer-weight representation (q;w⊤)ofGsatisfying |wi| ≤αn(∀i∈N), |q| ≤αn+1, and1≤/summationtext i∈Nwi≤2αn+1−1. 8Proof. It is easy to show that a given simple game G= (N,W) is weighted if and only if the following linear inequality system is feas ible: P2:A(W)w≥q1, A(L)w≤q1−1, 1⊤w≤u−1. We define A2= A(W)10 −A(L)−10 −1⊤0 1 ,v= w −q u ,d= 0 1 1 , and denote the inequality system P2 by A2v≥d. Subsequently, we assume that P2 is feasible. A non-singular submatrix /hatwideBofA2is called a basis matrix . Variables corresponding to columns of /hatwideB are called basic variables , andJ/hatwideBdenotes an index set of basic variables. Letd/hatwideBbe a subvector of dcorresponding to rows of /hatwideB. Abasic solution with respect to /hatwideBis a vector vdefined by vi=/braceleftbigg/hatwidevi(i∈J/hatwideB), 0 (i/\e}atio\slash∈J/hatwideB), where/hatwidevis a vector of basic variables satisfying /hatwidev=/hatwideB−1d/hatwideB. It is well-known that if a linear inequality system P2 is feasible, there exis ts a basic feasible solution. Let (w′⊤,−q′,u′)⊤be a basic feasible solution of P2 with respect to a basis matrix B. Assumption ∅ /\e}atio\slash∈ Wimplies that 0 ≤q′−1 and, thus, −q′/\e}atio\slash= 0. As N∈ W, we have inequalities u′−1≥1⊤w′≥q′≥1,which imply that u′/\e}atio\slash= 0. The definition of a basic solution implies that −qand uare basic variables with respect to the basis matrix B. Thus, Bhas columns corresponding to basic variables −qandu. A column of Bindexed byuis called the last column. As Bis invertible, the last column of Bis not the zero vector, and thus Bincludes a row corresponding to inequality 1⊤w≤u−1, which is called the last row (see Figure 3). Here, the numbe r of rows (columns) of B, denoted by n′, is less than or equal to n+2. For simplicity, we denote the basic feasible solution ( w′⊤,−q′,u′)⊤by v′. By Cramer’s rule, v′ i= det(Bi)/det(B) for each i∈JB,whereBiis obtained from Bwith a column correspondingto variable vireplaced by dB. Because Biis an integer matrix, det( B)v′ i= det(Bi) is an integer for any 9i∈JB. Cramer’s rule states that ( w∗⊤,−q∗,u∗) =|det(B)|(w′⊤,−q′,u′) is an integer vector satisfying the following conditions: A(W)w∗=|det(B)|A(W)w′≥ |det(B)|q′1=q∗1, A(L)w∗=|det(B)|A(L)w′≤ |det(B)|(q′1−1)≤q∗1−1,and 1⊤w∗=|det(B)|1⊤w′≤ |det(B)|(u′−1)≤u∗−1. From the above, ( q∗;w∗⊤) is an integer-weight representation of G. As N∈ W, we obtain 1⊤w∗≥q∗=|det(B)|q′≥1. w1w2w3w4−q u 1 1 1 0 10 0 1 1 1 10 0−1−1 0 −10 −1 0 0 −1−10 0−1 0 −1−10 −1−1−1−101 B w1w2w3w4−q u 1 1 1 0 00 0 1 1 1 00 0−1−1 0 10 −1 0 0 −110 0−1 0 −110 −1−1−1−111 Bqw1w2w3w4−q 1 1 1 0 0 0 1 1 1 0 0−1−1 0 1 −1 0 0 −11 0−1 0 −11 B′ qw1w2w3w4−q 1 1 1 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 1 B′′ q w1w2w3w4−q u 101 0 10 001 1 10 01−1 0 −10 −110−1−10 010−1−10 −11−1−101 B2w1w2w3w4−q 101 0 1 001 1 1 01−1 0 −1 −110−1−1 010−1−1 B′ 2w1w2w3w4−q 101 0 1 001 1 1 011 0 1 110 1 1 010 1 1 B′′ 2 w1w2w3w4−q u 1 1 1 0 10 0 1 1 1 10 0−1−1 0 −11 −1 0 0 −1−11 0−1 0 −1−11 −1−1−1−101 Buw1w2w3w4−q u 1 1 1 0 10 0 1 1 1 10 0 1 1 0 11 1 0 0 1 11 0 1 0 1 11 1 1 1 1 01 B′ u Figure 3: Examples of elementary matrix operations for P2. Now, we discuss the magnitude of |q∗|=|det(Bq)|,whereBqis obtained 10fromBwith a column corresponding to variable −qreplaced by dB. As the last column of Bqis a unit vector, we delete the last column and the last row fromBqand obtain a matrix B′ qsatisfying det( Bq) = det(B′ q). We apply the following elementary matrix operations to B′ q. First, we multiply the column corresponding to variable −q(which is equal to dB) by (−1). Next, we multiply the rows indexed by losing coalitions by ( −1). The resulting matrix, denoted by B′′ q, is 0–1 valued and satisfies the following condition: |q∗|=|det(Bq)|=|det(B′ q)|=|det(B′′ q)| ≤αn′−1≤αn+1. Next, we show that |w∗ i| ≤αn(i∈N). Ifw∗ i/\e}atio\slash= 0, then wiis a basic variable that satisfies |w∗ i|=|det(Bi)|,whereBiis obtained from Bbut the column corresponding to variable wiis replaced by dB. In a manner similar to that above, we delete the last column and the last r ow from Bi and obtain a matrix B′ isatisfying det( Bi) = det(B′ i). Next, we multiply a column corresponding to variable wiby (−1). We multiply rows indexed by losing coalitions by ( −1) and obtain a 0–1 matrix B′′ i. Matrix Bicontains a column corresponding to the original variable −q, which contains values 1 or−1. Thus, matrix B′′ icontains a column vector that is equal to an all-one vector. Lemma 2.1 (c1) implies that |w∗ i|=|det(Bi)|=|det(B′ i)|=|det(B′′ i)| ≤αn′−2≤αn. Lastly, we discuss the value of |u∗|=|det(Bu)|,whereBuis obtained fromBbut the last column (column indexed by variable u) is replaced by dB. In a manner similar to that above, we multiply the last colum n by (−1), multiply the rows indexed by losing coalitions by ( −1), and multiply the last row by ( −1). The resulting matrix, denoted by B′ u, is a 0–1 matrix in which the last row contains exactly one 0-component (inde xed by variable −q). Lemma 2.1 (c2) implies that |u∗|=|det(Bu)|=|det(B′ u)| ≤2αn′−1≤2αn+1, and thus 1⊤w∗≤u∗−1≤ |u∗|−1≤2αn+1−1. QED [Kurz, 2012] exhaustively generated all weighted voting ga mes satisfying the monotonicity property (1) for up to nine voters. Table 1 s hows max- ima of the exact values of minimal integer-weight represent ations obtained by [Kurz, 2012], Muroga’s boundsin [Muroga, 1971], and our u pperbounds. The table shows that our bound on the quota is tight when n≤5. 11Table 1: Exact values of integer weights representations. n 1 2 3 4 5 6 7 8 9 10 11 αn† 1 1 2 3 5 9 32 56 144 320 1458 max (N,W)min [q;w]max iwi‡1 1 2 3 5 9 18 42 110 Muroga’s bound (αn)•1 1 2 3 5 9 32 56 144 320 1458 max (N,W)min [q;w]q‡1 2 3 5 9 18 40 105 295 Our bound (αn+1)1 2 3 5 9 32 56 144 320 1458 Muroga’s bound (nαn)•1 2 6 12 25 54 224 448 1296 3200 16038 max (N,W)min [q;w]/summationtext iwi‡1 2 4 8 15 33 77 202 568 Our bound (2αn+1−1)1 3 5 9 17 63 111 287 639 2915 †[Sloane et al., 2018], ‡[Kurz, 2012], •[Muroga, 1971]. 4 Rounding Method This section addresses the problem of findinginteger-weigh t representations. In this section, we assume the monotonicity property (1). In addition, a weighted simple game is given by a triplet ( N,Wm,LM),whereWmand LMdenote the set of minimal winning coalitions and the set of ma ximal losing coalitions, respectively. We also assume that the em pty set is a losing coalition, Nis a winning coalition, and every player in Nis not a null player. Thus, there exists an integer-weight representati on in which q≥1 andwi≥1 (∀i∈N). We discuss a problem for findingan integer-weight represent ation, which is formulated by the following integer programming problem : Q: find a vector ( q;w) satisfying/summationdisplay i∈Swi≥q(∀S∈ Wm), (3) /summationdisplay i∈Swi≤q−1 (∀S∈ LM), (4) q≥1, wi≥1 (∀i∈N), (5) q∈Z, wi∈Z(∀i∈N). (6) A linear relaxation problem Q is obtained from Q by dropping the integer constraints (6). Let (q∗;w∗⊤) be a basic feasible solution of the linear inequality sys- temQ. Our proof in the previous section showed that |det(B∗)|(q∗;w∗⊤) 12gives a solution of Q (i.e., an integer-weight representati on), where B∗de- notes a corresponding basis matrix of Q. When |det(B∗)|> n, there ex- ists a simple method for generating a smaller integer-weigh t representation. For any weight vector w= (w1,w2,...,w n)⊤, we denote the integer vector (⌊w1⌋,⌊w2⌋,...,⌊wn⌋)⊤by⌊w⌋. Given a solution ( q∗;w∗⊤) ofQ, we intro- duce an integer vector w′=⌊nw∗⌋and an integer q′=⌊n(q∗−1)⌋+1. For any minimal winning coalition S∈ Wm, we have that /summationdisplay i∈Sw′ i>/summationdisplay i∈S(nw∗ i−1)≥n/summationdisplay i∈Sw∗ i−n≥nq∗−n=n(q∗−1)≥ ⌊n(q∗−1)⌋, /summationdisplay i∈Sw′ i≥ ⌊n(q∗−1)⌋+1 =q′. Each maximal losing coalition S∈ LMsatisfies /summationdisplay i∈Sw′ i≤/summationdisplay i∈Snw∗ i≤n(q∗−1), /summationdisplay i∈Sw′ i≤ ⌊n(q∗−1)⌋=q′−1. Thus, the pair of w′andq′gives an integer-weight representation satisfying (q′;w′⊤)≤n(q∗;w∗⊤). In the remainder of this section, we show that there exists an integer-weight representation (vector) that is l ess than or equal to ((2−√ 2)n+(√ 2−1))(q∗;w∗⊤)<(0.5858n+0.4143)(q∗;w∗⊤) for any solution ( q∗;w∗⊤) ofQ. Theorem 4.1. Let(q∗;w∗⊤)be a solution of Q. We define ℓ1= (2−√ 2)n− (√ 2−1)andu1= (2−√ 2)n+(√ 2−1). Then, there exists a real number λ•∈[ℓ1,u1]so that the pair Q=⌊λ•(q∗−1)⌋+1andW=⌊λ•w∗⌋gives a feasible solution of Q (i.e., an integer-weight representa tion). Proof. For any positive real λ, it is easy to see that each maximal losing coalition S∈ LMsatisfies /summationdisplay i∈S⌊λw∗ i⌋ ≤/summationdisplay i∈Sλw∗ i≤λ(q∗−1), /summationdisplay i∈S⌊λw∗ i⌋ ≤ ⌊λ(q∗−1)⌋. To discuss the weights of minimal winning coalitions, we int roduce a function g(λ) =λ−/summationtext i∈N(λw∗ i−⌊λw∗ i⌋). In thesecond part of this proof, we show that if we choose Λ ∈[ℓ1,u1] uniformly at random, then E[ g(Λ)]≥0. 13This implies that ∃λ•∈[ℓ1,u1] satisfying g(λ•)>0, because g(λ) is right- continuous, piecewise linear, and not a constant function. Wheng(λ•)>0, each minimal winning coalition S∈ Wmsatisfies λ•>/summationdisplay i∈N(λ•w∗ i−⌊λ•w∗ i⌋)≥/summationdisplay i∈S(λ•w∗ i−⌊λ•w∗ i⌋) =/summationdisplay i∈Sλ•w∗ i−/summationdisplay i∈S⌊λ•w∗ i⌋, (7) which implies /summationdisplay i∈S⌊λ•w∗ i⌋>/summationdisplay i∈Sλ•w∗ i−λ•=λ•/parenleftBigg/summationdisplay i∈Sw∗ i−1/parenrightBigg ≥λ•(q∗−1)≥ ⌊λ•(q∗−1)⌋, and thus /summationdisplay i∈S⌊λ•w∗ i⌋ ≥ ⌊λ•(q∗−1)⌋+1. Finally, we show that E[ g(Λ)]≥0 if we choose Λ ∈[ℓ1,u1] uniformly at random. It is obvious that E[g(Λ)] = E[Λ] −/summationdisplay i∈NE[(Λw∗ i−⌊Λw∗ i⌋)] =ℓ1+u1 2−/summationdisplay i∈N/integraldisplayu1 ℓ1(λw∗ i−⌊λw∗ i⌋)dλ u1−ℓ1 = (2−√ 2)n−/summationdisplay i∈N/integraldisplayu1 ℓ1(λw∗ i−⌊λw∗ i⌋)dλ u1−ℓ1. Let us discuss the last term appearing above. By substitutin gµforλw∗ i, we obtain /integraldisplayu1 ℓ1(λw∗ i−⌊λw∗ i⌋)dλ u1−ℓ1=/integraldisplayu1w∗ i ℓ1w∗ i(µ−⌊µ⌋)dµ w∗ i(u1−ℓ1) ≤/integraldisplay0 −w∗ i(u1−ℓ1)(µ−⌊µ⌋)dµ w∗ i(u1−ℓ1)=/integraldisplay0 −x(µ−⌊µ⌋)dµ x, where the last equality is obtained by setting x=w∗ i(u1−ℓ1). Asu1−ℓ1= 2(√ 2−1) andw∗ i≥1, it is clear that x=w∗ i(u1−ℓ1)≥2(√ 2−1). Here, we introduce a function f(x) =/integraldisplay0 −x(µ−⌊µ⌋)dµ x. According to numerical 14calculations (see Figure 4), inequality x≥2(√ 2−1) implies that f(x)≤ 2−√ 2. 0 1 2 3 4 5 x0.450.50.550.60.650.70.750.8f(x) Figure 4: Plot of function f(x) =/integraldisplay0 −x(µ−⌊µ⌋)dµ x. From the above, we obtain the desired result E[g(Λ)]≥(2−√ 2)n−/summationdisplay i∈N(2−√ 2) = (2−√ 2)n−(2−√ 2)n= 0. QED 5 Roughly Weighted Simple Games In this section, we discuss roughly weighted simple games. F irst, we show an upper bound of the length of a potent certificate of non-wei ghtedness. Theorem 5.1. Assume that a given simple game G= (N,W)satisfies ∅ /\e}atio\slash∈ W ∋Nand the monotonicity property (1). If a given simple game Gis not roughly weighted, then there exists a potent certificate of n on-weightedness whose length is less than or equal to 2αn+1. Proof. Let us introduce a linear inequality system: P3:/parenleftbiggA(W)1 −A(L)−1/parenrightbigg/parenleftbiggw −q/parenrightbigg ≥0, 1⊤w>0. 15First, we show that if P3 is feasible, then a given simple game is roughly weighted. Let ( q′;w′⊤) be a feasible solution of P3. We introduce a new voting weight w′′ i= max{w′ i,0}for each i∈N. We show that ( q′;w′′⊤) is a rough voting representation. As 1⊤w′>0, vector w′includes at least one positivecomponent,andthus w′′/\e}atio\slash=0. Ifacoalition Ssatisfies/summationtext i∈Sw′′ i< q′, thenq′>/summationtext i∈Sw′′ i≥/summationtext i∈Sw′ i,and thus Sis losing. Consider the case in which a coalition Ssatisfies/summationtext i∈Sw′′ i> q′. LetS′={i∈S|w′ i>0}. It is obvious that q′0, and thus Cramer’s rule states that det(B)u′= det(Bu) (Figure 5 shows an example). We multiply columns ofBucorresponding to components in ( y⊤,u) by (−1) and obtain a 0–1 matrixB′ usatisfying |det(Bu)|=|det(B′ u)|. As/tildewidec′includes at most one 0- component, Lemma 2.1 implies that |det(B′ u)| ≤2αn′−1≤2αn+1. Thus, the length of ( X;Y) satisfies /summationdisplay S∈Wx∗ S=/summationdisplay S∈W|det(B)|x′ S+|det(B)|=|det(B)|(1⊤x′+1) =|det(B)|(u′−1+1) = |det(B)|u′=|det(B)u′| =|det(Bu)|=|det(B′ u)| ≤2αn+1. QED In the rest of this section, we discuss integer voting weight s and a quota of a rough voting representation. We say that a player i∈Nis apasserif and only if every coalition S∋iis winning. 17u 0 0 1 1 0−10 0 1 0 1 0 0 0 1 0 0 1 0−10 1 1 1 0 −1−10 1 1 1 1 −1−10 1 1 1 1 0 0−1 B /tildewidec′ 0 0 1 1 0−1−1 0 1 0 1 0 0−1 1 0 0 1 0−1−1 1 1 1 0 −1−1−1 1 1 1 1 −1−10 1 1 1 1 0 0−1 Bu0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 B′ u Figure 5: Example of elementary matrix operations for D3+. Theorem 5.2. Assume that a given simple game G= (N,W)satisfies ∅ /\e}atio\slash∈ W ∋ N.If a given simple game Gis roughly weighted, then there exists an integer vector (q;w⊤)of the rough voting representation satisfying 0≤ wi≤αn−1(∀i∈N),0≤q≤αn, and1≤/summationtext i∈Nwi≤2αn. Proof. First, we show that if a given game is roughly weighted , then either P4: A(W)0 −A(L)0 −1⊤1 /parenleftbiggw u/parenrightbigg ≥ 1 −1 0 ,w≥0,u≥0, is feasible or there exists at least one passer. Suppose that a given simple game has a rough voting representation ( q;w⊤). Ifq >0, then (1 /q)w becomes a feasible solution of P4 by setting uto a sufficiently large positive number. Consider the case q≤0. Assumption ∅ /\e}atio\slash∈ Wimplies that 0 ≤q, and thus we obtain q= 0. Properties ( q,w⊤)/\e}atio\slash=0⊤andw≥0imply that ∃i◦∈N,wi◦>0, i.e., a given game Ghas a passer i◦. When a given game Ghas a passer i◦∈N, then there exists a rough 18voting representation ( q◦;w◦⊤) defined by w◦ i=/braceleftbigg1 (i=i◦), 0 (i/\e}atio\slash=i◦),q◦= 0, which produces the desired result. Lastly, we consider the case in which P4 is feasible. It is wel l-known that when P4 is feasible, there exists a basic feasible solution. Let (w′⊤,u′)⊤be a basic feasible solution of P4 and Bbe a corresponding basis matrix. It is easy to see that (1; w′⊤) is a rough voting representation of G. Assumption N∈ Wimplies the positivity of u′becauseu′≥1⊤w′≥1. Then, variable uis a basic variable, and thus Bincludes a column corresponding to u, which is called the last column. The non-singularity of Bimplies that a column corresponding to uis not the zero vector, and thus Bincludes a row corresponding to the inequality 1⊤w≤u, which is called the last row (see Figure 6). The number of rows (columns) of basis matrix B, denoted by n′, is less than or equal to n+1. Cramer’s rule states that ( q∗,w∗⊤,u∗) =|det(B)|(1,w′⊤,u′) is a non- negative integer vector. It is easy to see that ( q∗,w∗⊤,u∗) satisfies A(W)w∗=|det(B)|A(W)w′≥ |det(B)|1=q∗1, A(L)w∗=|det(B)|A(L)w′≤ |det(B)|1=q∗1,and 1⊤w∗=|det(B)|1⊤w′≤ |det(B)|u′=u∗. From the above, ( q∗;w∗⊤) is an integer vector of a rough voting represen- tation. Assumption N∈ Wimplies that 1⊤w∗≥q∗=|det(B)| ≥1. Letd′ Bbe a subvector of the right-hand-side vector of an inequalit y sys- tem in P4 corresponding to rows of B. Cramer’s rule states that det( B)u′= det(Bu),whereBuis obtained from Bbut the column corresponding to a basic variable uis replaced by d′ B(see Figure 6). We multiply rows of Bu that correspond to losing coalitions by ( −1) and multiply the last row by (−1). The resulting matrix, denoted by B′ u, is a 0–1 matrix whose last row includes exactly one 0-component (indexed by u). Lemma 2.1 (c2) implies that|det(B′ u)| ≤2αn′−1≤2αn. Thus, we obtain that 1⊤w∗≤u∗≤ |u∗|=|det(B)u′|=|det(Bu)|=|det(B′ u)| ≤2αn. By analogy with the proof of Theorem 3.1, we can prove the desi red inequal- ities:q∗=|det(B)| ≤αnandw∗ i≤αn−1(∀i∈N). QED 19w1w2w3w4w5u 1 1 1 0 1 0 0 1 0 1 1 0 0−1−1 0 0 0 −1 0 0 −1−10 0−1 0 −1 0 0 −1−1−1−1−11 Bw1w2w3w4w5u 1 1 1 0 1 1 0 1 0 1 1 1 0−1−1 0 0 −1 −1 0 0 −1−1−1 0−1 0 −1 0 −1 −1−1−1−1−10 Buw1w2w3w4w5u 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 B′ u Figure 6: Examples of elementary matrix operations for P4. 6 Conclusion In this paper, we discussed the smallest value of k∗such that every k∗-trade robust simple game would be weighted. We provided a new proof of the existence of a trading transform when a given simple game is n on-weighted. Our proof yields an improved upper bound on the required leng th of a trading transform. We showed that a given simple game Gis weighted if and only if Gisαn+1-trade robust, where αn+1denotes the maximal value of determinants of ( n+1)×(n+1) 0–1 matrices. Applying the Hadamard’s evaluation [Hadamard, 1893] of the determinant, we obtain k∗≤αn+1≤ (n+2)n+2 2(1/2)(n+1), which improves the existing bound k∗≤(n+1)nn/2 obtained by [Gvozdeva and Slinko, 2011]. Next, we discussed upper bounds for the maximum possible int eger weights and the quota needed to represent any weighted simpl e game with n players. We show that every weighted simple game (satisfyin g∅ /\e}atio\slash∈ W ∋ N) has an integer-weight representation ( q;w⊤)∈Z×ZNsuch that |wi| ≤αn (∀i∈N),|q| ≤αn+1, and 1≤/summationtext i∈Nwi≤2αn+1−1. We demonstrated the tightness of our bound on the quota when n≤5. We described a rounding method based on a linear relaxation o f an integer programming problem for finding an integer-weight r epresentation. We showed that an integer-weight representation is obtaine d by carefully rounding a solution of the linear inequality system multipl ied byλ•≤ (2−√ 2)n+(√ 2−1)<0.5858n+0.4143. Our proof of Theorem 4.1 indicates an existence of a randomized rounding algorithm for finding a n appropriate valueλ•. However, from theoretical point of view, Theorem 4.1 only s howed the existence of a real number λ•. Even if there exists an appropriate “ratio- nal” number λ•, we need to determine the size of the rational number (its numerator and denominator) to implement a naive randomized rounding algorithm. Thus, it remains open whether there exists an effic ient algo- 20rithm for finding an integer-weight representation satisfy ing the properties in Theorem 4.1. 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