Trading Transforms of Non-weighted Simple Games and Integer Weights of Weighted Simple Games
aa r X i v : . [ c s . G T ] J a n Trading Transforms of Non-weighted Simple Gamesand Integer Weights of Weighted Simple Games ∗ Akihiro Kawana Tomomi MatsuiJanuary 20, 2021
Abstract
This paper is concerned with simple games. One of the fundamentalquestions regarding simple games is that of what makes a simple gamea weighted majority game. Taylor and Zwicker (1992) showed thata simple game is non-weighted if and only if there exists a tradingtransform of finite size. They also provided an upper bound on the sizeof such a trading transform, if it exists. Gvozdeva and Slinko (2009)improved on that upper bound. Their proof employs a property oflinear inequalities demonstrated by Muroga (1971). We provide a newproof of the existence of a trading transform when a given simple gameis non-weighted. Our proof employs Farkas’ lemma (1894), and yieldsan improved upper bound on the size of a trading transform.We also discuss an integer weights representation of a weightedsimple game, and improve on the bounds obtained by Muroga (1971).We show that our bounds are tight when the number of players is lessthan or equal to five, based on the computational results obtained byKurz (2012).Lastly, we deal with the problem of finding an integer weights rep-resentation under the assumption that we have minimal winning coali-tions and maximal losing coalitions. We discuss a performance of arounding method. ∗ Graduate School of Engineering, Tokyo Institute of Technologypreliminary version of this paper was presented at Seventh International Workshop onComputational Social Choice (COMSOC-2018), Rensselaer Polytechnic Institute, Troy,NY, USA, 25-27 June, 2018.This work was supported by JSPS KAKENHI Grant Numbers JP26285045, JP26242027,JP20K04973. Introduction
A simple game consists of a pair G = ( N, W ) where N is a finite set ofplayers and W ⊆ N is an arbitrary collection of subsets of N . Throughoutthis paper, we denote | N | by n . Usually, the property(monotonicity): if S ′ ⊇ S ∈ W , then S ′ ∈ W , (1)is assumed. Subsets in W are called winning coalitions . We denote 2 N \ W by L , and subsets in L are called losing coalitions . A simple game ( N, W )is said to be weighted if there exists a weight vector w ∈ R N and q ∈ R satisfying(weightedness): for any S ⊆ N, S ∈ W if and only if X i ∈ S w i ≥ q. (2)The necessary and sufficient conditions that guarantee the weightednessof a simple game are known. [Elgot, 1961] and [Chow, 1961] investigatedthe theory of threshold logic, and showed a condition in terms of asumma-bility . [Muroga, 1971] described a proof of the sufficiency of asummabilitybased on the theory of linear inequality systems and discussed some varia-tions of their results in cases of a few variables. [Taylor and Zwicker, 1992,Taylor and Zwicker, 1999] obtained necessary and sufficient conditions inde-pendently in terms of a trading transform . A trading transform of size j is acoalition sequence ( X , X , . . . , X j ; Y , Y , . . . , Y j ), which may contain repe-titions of coalitions, satisfying that ∀ p ∈ N , |{ i | p ∈ X i }| = |{ i | p ∈ Y i }| .A simple game will be called k -trade robust if there is no trading trans-form of size j satisfying (1) 1 ≤ j ≤ k , (2) X , X , . . . , X j ∈ W , and (3) Y , Y , . . . , Y j ∈ L . A simple game will be called trade robust if it is k -traderobust for all positive integers k .Taylor and Zwicker showed that a given simple game G with n players isweighted if and only if G is 2 n -trade robust. In 2009, [Gvozdeva and Slinko, 2011]showed that a given simple game G is weighted if and only if G is ( n +1) n n/ -trade robust. The relations between results in the field of threshold logicand the field of simple games are clarified in [Freixas et al., 2017].In Section 2, we show that a given simple game G is weighted if andonly if G is α n +1 -trade robust where α n +1 denotes the maximal value ofdeterminants of ( n + 1) × ( n + 1) 0-1 matrices. It is well-known that α n +1 ≤ ( n + 2) n +22 (1 / ( n +1) .Our definition of a weighted simple game allows for an arbitrary realnumber of weights. However, it is easy to see that any weighted sim-2le game is represented by integer weights. An integer weights represen-tation of a weighted simple game consists of an integer vector w ∈ Z N and some q ∈ Z satisfying property (2). [Isbell, 1956] found an exampleof weighted simple game of 12 players without a unique minimum sum in-teger weights representation. Examples for 9, 10, or 11 players are givenin [Freixas and Molinero, 2009, Freixas and Molinero, 2010]. In the field ofthreshold logic, examples of threshold functions requiring large weights arediscussed in [Myhill and Kautz, 1961, Muroga, 1971, H˚astad, 1994]. Thereexist some previous studies that enumerate (minimal) integer weights repre-sentations of simple games with small numbers of players (e.g., [Muroga et al., 1962,Winder, 1965, Muroga et al., 1970, Krohn and Sudh¨olter, 1995]). In the caseof n = 9 players, we refer the reader to results in [Kurz, 2012]. In gen-eral, [Muroga, 1971] showed that every weighted simple game has an integerweight representation satisfying (1) w i ≤ ( n + 1) n +12 (1 / n ( ∀ i ∈ N ) and(2) q ≤ n ( n + 1) n +12 (1 / n simultaneously.In Section 3, we slightly improve Muroga’s result and show that everyweighted simple game has an integer weight representation ( w , q ) ∈ Z N × Z satisfying (1) w i ≤ α n ( ∀ i ∈ N ) and (2) q ≤ α n +1 simultaneously. Based onthe computational results of [Kurz, 2012], we also demonstrate the tightnessof our bounds (1) and (2) when n ≤
5. Here we note that α n denotes themaximal value of determinants of n × n α n ≤ ( n + 1) n +12 (1 / n .When we have a family of minimal winning coalitions, [Peled and Simeone, 1985]proposed a polynomial time algorithm for checking the weightedness of agiven simple game. They also showed that for weighted simple games repre-sented by minimal winning coalitions, all maximal losing coalitions can becomputed in polynomial time. When we have minimal winning coalitionsand maximal losing coalitions, there exists a linear inequality system whosesolution gives a weight vector w ∈ R N and q ∈ R satisfying property (2).However, it is less straightforward to find an integer weights representationwhere all the weights are integers as the problem transforms from linearprogramming to integer programming.In Section 4, we deal with the problem of finding an integer weights rep-resentation under the assumption that we have minimal winning coalitionsand maximal losing coalitions. We show that an integer weights representa-tion is obtained by carefully rounding a solution of linear inequality systemmultiplied by at most (2 − √ n + ( √ − Trading Transforms of Non-weighted Simple Games
In this section, we discuss the size of a trading transform that guaranteesthe non-weightedness of a given simple game. Throughout this section, wewill not need to assume the monotonicity property (1). First, we introduce alinear inequality system for determining the weightedness of a given simplegame. For any nonempty family of player subsets ∅ 6 = N ⊆ N , we introducea 0-1 matrix A ( N ) = ( a ( N ) Si ) whose rows are indexed by subsets in N andcolumns are indexed by players in N defined by a ( N ) Si = (cid:26) i ∈ S ∈ N ) , . It is obvious that a given simple game G = ( N, W ) is weighted if and onlyif the linear inequality systemP1: (cid:18) A ( W ) − A ( L ) − − (cid:19) w − qε ≥ ,ε > , is feasible, where ( ) denotes a zero vector (all-one vector) of the appro-priate dimension.Farkas’ Lemma [Farkas, 1902] says that P1 is infeasible if and only if thefollowing systemD1: A ( W ) ⊤ − A ( L ) ⊤ ⊤ − ⊤ ⊤ − ⊤ (cid:18) xy (cid:19) = − , x ≥ , y ≥ , is feasible. For simplicity, we denote the linear inequality system D1 by A z = c , z ≥ where A = A ( W ) ⊤ − A ( L ) ⊤ ⊤ − ⊤ ⊤ − ⊤ , z = (cid:18) xy (cid:19) , and c = − . In the following, we consider the case that the linear inequality system A z = c , z ≥ is feasible. Let f A z = e c be a linear equality system obtainedfrom A z = c by repeatedly removing redundant equalities. It is well-knownthat if the linear inequality system f A z = e c , z ≥ is feasible, then it has a4asic feasible solution, i.e., there exists an index subset J of the index setof the vector z satisfying that (1) a column submatrix B of f A consistingof column vectors of f A indexed by J is a square invertible matrix and (2) B − e c ≥ . We introduce a subvector z J ( z J ) consisting of components of z indexed by indices in J (not in J ). Then, we have a basic feasible solution z ∗ = (cid:18) z ∗ J z ∗ J (cid:19) = (cid:18) B − e c (cid:19) when A z = c , z ≥ is feasible. By Cramer’s rule, z ∗ j = det( B j ) / det( B )for each j ∈ J where B j is obtained from B with the column that is indexedby j replaced by e c . Because B j is an integer matrix, det( B ) z ∗ j = det( B j ) isan integer for any j ∈ J . Thus, | det( B ) | z ∗ is an integer feasible solution of A z = | det( B ) | c , z ≥ . Let (cid:18) x ∗ y ∗ (cid:19) be an integer vector corresponding to | det( B ) | z ∗ . Let us recall that x ∗ is indexed by W and y ∗ is indexed by L .Then, the pair of integer vectors x ∗ and y ∗ satisfies that A ( W ) ⊤ x ∗ = A ( L ) ⊤ y ∗ , X S ∈W x ∗ S = X S ∈L y ∗ S = | det( B ) | , x ∗ ≥ , y ∗ ≥ . In the following, we construct a trading transform corresponding to the pair x ∗ and y ∗ . Let X = ( X , X , . . . , X | det ( B ) | ) be a sequence of winning coali-tions satisfying that each winning coalition S ∈ W appears in X x ∗ S -times.Similarly, we introduce a sequence Y = ( Y , Y , . . . , Y | det ( B ) | ) satisfying thateach losing coalition S ∈ L appears in Y y ∗ S -times. Then, the equality A ( W ) ⊤ x ∗ = A ( L ) ⊤ y ∗ implies that ( X ; Y ) is a trading transform of size | det( B ) | . From the above discussion, we have shown that if D1 is feasible,then a given simple game G = ( N, W ) is not | det( B ) | -trade robust.Finally, we provide an upper bound on | det( B ) | . Let α n be the maximumof the determinant of an n × n B is indexed by acomponent of x (i.e., indexed by a winning coalition), then each componentof the column is either 0 or 1. Otherwise, a column (of B ) is indexed by acomponent of y (i.e., indexed by a losing coalition) whose components areeither 0 or −
1. Now, we apply elementary matrix operations to B . Foreach column of B indexed by a component of y , we multiply the column by( − B ′ , is a 0-1 matrix satisfying | det( B ) | = | det( B ′ ) | .Let us recall that B is a submatrix of A , and the number of rows of A isequal to n + 2. Thus, the number of rows (columns) of the basis matrix B isless than or equal to n +2. In case that the number of rows (columns) of B is5ess than n +2, we obtain the desired result that | det( B ) | = | det( B ′ ) | ≤ α n +1 .Consider the case that the basis matrix B has n + 2 rows. Then, B has a rowvector corresponding the equality ⊤ x − ⊤ y = 0, which satisfies that eachcomponent is either 1 or −
1, and thus B ′ has an all-one row vector. (Because B ′ is invertible, B ′ includes exactly one all-one row vector.) Now, we applythe following elementary row operations to B ′ . For each row vector of B ′ except a unique all-one row vector, if the first component is equal to one,then we multiply the row by ( −
1) and add the all-one row vector. Then, theobtained matrix, denoted by B ′′ , is an ( n + 2) × ( n + 2) 0-1 matrix satisfyingthat | det( B ) | = | det( B ′ ) | = | det( B ′′ ) | and the first column is a unit vector.Thus, it is obvious that | det( B ′′ ) | ≤ α n +1 .From the above discussion, we obtain the following theorem (withoutthe assumption of monotonicity property (1)). Theorem 2.1.
A given simple game G = ( N, W ) with n players is weightedif and only if G is α n +1 -trade robust, where α n is the maximum of determi-nants of n × n Proof. If a given simple game is not α n +1 -trade robust, then it is nottrade robust and thus not weighted, as shown by [Taylor and Zwicker, 1992,Taylor and Zwicker, 1999]. We have discussed the inverse implication thata given simple game G = ( N, W ) is not weighted. Then, the linear in-equality system P1 is infeasible. Farkas’ lemma [Farkas, 1902] implies thatD1 is feasible. From the above discussion, we have a trading transform( X , . . . , X j ; Y , . . . Y j ) satisfying that (1) j ≤ α n +1 , (2) X , . . . , X j ∈ W ,and (3) Y = 1 , . . . , Y j ∈ L . Q.E.D.The application of Hadamard’s evaluation [Hadamard, 1893] of the de-terminant leads to the following. Theorem 2.2.
For any positive integer n , α n ≤ ( n + 1) n +12 (1 / n . The exact values of α n for small positive integers n appear in “The On-Line Encyclopedia of Integer Sequences (A003432)” [Sloane et al., 2018] andTable 1.In the above discussion, we have shown the following property. Corollary 2.3. If B is an n × n n ≥ and atlease one row vector of B is the all-one vector, then | det( B ) | ≤ α n − . In this section, we discuss the integer weight representations of weightedsimple games. Throughout this section, we will not need to assume the6onotonicity property (1), except in Table 1.
Theorem 3.1.
For any weighted simple game G = ( N, W ) , there exists aninteger vector w ∈ Z N and some q ∈ Z satisfying (W0) ∀ S ⊆ N, S ∈ W if and only if P x ∈ S w ( x ) ≥ q , (W1) | w i | ≤ α n ( ∀ i ∈ N ) , and (W2) | q | ≤ α n +1 ,simultaneously. Proof. It is easy to show that a given simple game G = ( N, W ) is weightedif and only if the linear inequality systemP2: A ( W ) w ≥ q ,A ( L ) w ≤ q − , is feasible. We define A = (cid:18) A ( W ) − A ( L ) − (cid:19) , v = (cid:18) w − q (cid:19) , d = (cid:18) (cid:19) , and simply denote the inequality system P2 by A v ≥ d . In the following,we discuss the case that A v ≥ d is feasible.Given a pair ( I, J ) of subsets of row indices and column indices of thematrix A , A ( I, J ) denotes a submatrix of A indexed by I × J , A ( ∗ , J )denotes a column submatirx of A consisting of column vectors of A indexedby J , and d I denotes a subvector of d indexed by I . When the inequalitysystem A v ≥ d is feasible, it is well-known that there exists a basic feasiblesolution, i.e., there exists a pair ( I, J ) of subsets of row and column indicesof matrix A satisfying that (1) A ( I, J ) is a square invertible matrix, and(2) a vector A ( I, J ) − d I satisfies A ( ∗ , J ) A ( I, J ) − d I ≥ d . We introducea subvector v J ( v J ) of v consisting of components indexed by indices in J (not in J ). In the remainder of this proof, we denote A ( I, J ) by B forsimplicity. Then, we have a basic feasible solution v ∗ = (cid:18) v ∗ J v ∗ J (cid:19) = (cid:18) B − d I (cid:19) of the inequality system A v ≥ d . By Cramer’s rule, v ∗ j = det( B j ) / det( B )for each j ∈ J where B j is obtained from B with the column indexed by j replaced by d I . Because B j is an integer matrix, det( B ) v ∗ j = det( B j ) is7n integer for any j ∈ J . Thus, | det( B ) | v ∗ is an integer vector satisfying A | det( B ) | v ∗ = | det( B ) | A v ∗ ≥ | det( B ) | d ≥ d .For each j ∈ J , we apply the following elementary matrix operations to B j . First, we multiply the j -th column of B j (which is equal to d I ) by ( − − B ′ j , is0-1 valued and satisfies | det( B j ) | = | det( B ′ j ) | . If we denote the number ofrows (columns) of B by n ′ , it is obvious that n ′ ≤ n + 1 and thus || det( B ) | v ∗ j | = | det( B ) v ∗ j | = | det( B j ) | = | det( B ′ j ) | ≤ α n ′ ≤ α n +1 ( ∀ j ∈ J ) . Because v ∗ J = , we obtain that − α n +1 ≤ − α n ′ ≤ | det( B ) | v ∗ ≤ α n ′ ≤ α n +1 . (3)Let (cid:18) w ∗ − q ∗ (cid:19) be an integer vector corresponding to | det( B ) | v ∗ . Because | det( B ) | v ∗ is a feasible solution of A v ≥ d , the pair w ∗ and q ∗ is feasiblefor P2, and satisfies property (W0). The inequalities in (3) directly implythat | q ∗ | ≤ α n +1 , and thus the property (W2) holds. When n ′ , the numberof rows (columns) of B , is less than or equal to n , the inequalities in (3)imply that | w ∗ i | ≤ α n ( ∀ i ∈ N ). Finally, we demonstrate the property (W1)in the case that n ′ = n + 1. In this case, B is a row submatrix of A , andthus contains a column corresponding to original variable − q , whose entriesconsist of 1 or −
1. We only need to consider the case that the column index j ∈ J corresponds to a player i ∈ N , which implies that the index j doesnot correspond to the original variable − q , and thus the matrix B j includesa column whose entries consisting of 1 or −
1. Then, the matrix B ′ j containsa column vector that is equal to the all-one vector. Corollary 2.3 impliesthat | w ∗ i | = || det( B ) | v ∗ j | = | det( B j ) | = | det( B ′ j ) | = | det(( B ′ j ) ⊤ ) | ≤ α n ′ − = α n and thus we obtain the desired result. QED[Kurz, 2012] exhaustively generated all weighted voting games satisfyingthe monotonicity property (1) for up to nine voters. Table 1 shows max-ima of the exact values of minimal integer weights representations obtainedin [Kurz, 2012] and our upper bounds. The table shows that our bounds aretight when n ≤
5. 8able 1: Exact values of integer weights representations. n α n † ( N, W ) min [ q ; w ] max i w i ‡ Our bound ( α n ) 1 1 2 3 5 9 32 56 144 320 1458max ( N, W ) min [ q ; w ] q ‡ Our bound ( α n +1 ) 1 2 3 5 9 32 56 144 320 1458 † [Sloane et al., 2018], ‡ [Kurz, 2012]. This section deals with the problem of finding integer weights representa-tions. Throughout this section, we assume the monotonicity property (1).In this section, a weighted simple game is given by a triplet ( N, W m , L M)where W m and L M denote the set of minimal winning coalitions and the setof maximal losing coalitions, respectively. We also assume that the emptyset is a losing coalition, N is a winning coalition, and every player in N is not a null player. Thus, there exists an integer weights representationsatisfying that q ≥ w i ≥ ∀ i ∈ N ).In this section, we discuss a problem for finding an integer weights repre-sentation, which is formulated by the following integer programming prob-lem; Q: find a vecter ( q ; w )satisfying X i ∈ S w i ≥ q ( ∀ S ∈ W m) , (4) X i ∈ S w i ≤ q − ∀ S ∈ L M) , (5) q ≥ , w i ≥ ∀ i ∈ N ) , (6) q ∈ Z , w i ∈ Z ( ∀ i ∈ N ) . (7)A linear relaxation problem Q is obtained from Q by dropping the integerconstraints (7).Let ( q ∗ , w ∗⊤ ) be a basic feasible solution of linear inequality system Q.Our proof in the previous section showed that | det( B ∗ ) | ( q ∗ , w ∗⊤ ) gives asolution of Q (i.e., an integer weight representation) where B ∗ denotes thecorresponding base matrix of Q. When | det( B ∗ ) | > n , there exists a sim-ple method for generating a smaller integer weights representation. For9ny weight vector w = ( w , w , . . . , w n ) ⊤ , we denote the integer vector( ⌊ w ⌋ , ⌊ w ⌋ , . . . , ⌊ w n ⌋ ) ⊤ by ⌊ w ⌋ . Given a solution ( q ∗ , w ∗⊤ ) of Q. we in-troduce an integer vector w ′ = ⌊ n w ∗ ⌋ and an integer q ′ = ⌊ n ( q ∗ − ⌋ + 1.For any minimal winning coalition S ∈ W m, we have that X i ∈ S w ′ i > X i ∈ S ( nw ∗ i − ≥ n X i ∈ S w ∗ i − n ≥ nq ∗ − n = n ( q ∗ − ≥ ⌊ n ( q ∗ − ⌋ , X i ∈ S w ′ i ≥ ⌊ n ( q ∗ − ⌋ + 1 = q ′ . Each maximal losing coalition S ∈ L M satisfies that X i ∈ S w ′ i ≤ X i ∈ S nw ∗ i ≤ n ( q ∗ − , X i ∈ S w ′ i ≤ ⌊ n ( q ∗ − ⌋ = q ′ − . Thus, the pair w ′ and q ′ gives an integer weights representation satisfying( q ′ , w ′⊤ ) ≤ n ( q ∗ , w ∗⊤ ). In the remainder of this section, we show that thereexists an integer weights representation (vector) which is less than or equalto ((2 − √ n + ( √ − q ∗ , w ∗⊤ ) < (0 . n + 0 . q ∗ , w ∗⊤ ) for anysolution ( q ∗ , w ∗⊤ ) of Q. Theorem 4.1.
Let ( q ∗ , w ∗⊤ ) be a solution of Q . We define ℓ = (2 −√ n − ( √ − and u = (2 − √ n + ( √ − . Then, there exists a real number λ • ∈ [ ℓ , u ] satisfying that the pair Q = ⌊ λ • ( q ∗ − ⌋ + 1 and W = ⌊ λ • w ∗ ⌋ gives a solution of Q (i.e., an integer weights representation). Proof. For any positive real λ , it is easy to see that each maximal losingcoalition S ∈ L M satisfies that X i ∈ S ⌊ λw ∗ i ⌋ ≤ X i ∈ S λw ∗ i ≤ λ ( q ∗ − , X i ∈ S ⌊ λw ∗ i ⌋ ≤ ⌊ λ ( q ∗ − ⌋ . (8)In the following, we discuss the weights of minimal winning coalitions.We introduce a function g ( λ ) = λ − P i ∈ N ( λw ∗ i − ⌊ λw ∗ i ⌋ ). In the latterpart of this proof, we will show that if we choose Λ ∈ [ ℓ , u ] uniformlyat random, then E[ g (Λ)] ≥
0. This implies that ∃ λ • ∈ [ ℓ , u ] satisfying g ( λ • ) >
0, because g ( λ ) is right-continuous, piecewise linear, and not a10onstant function. Then, for any minimal winning coalition S ∈ W m, wehave that λ • > X i ∈ N ( λ • w ∗ i − ⌊ λ • w ∗ i ⌋ ) ≥ X i ∈ S ( λ • w ∗ i − ⌊ λ • w ∗ i ⌋ ) , X i ∈ S ⌊ λ • w ∗ i ⌋ > X i ∈ S λ • w ∗ i − λ • = λ • X i ∈ S w ∗ i − ! ≥ λ • ( q ∗ − ≥ ⌊ λ • ( q ∗ − ⌋ , X i ∈ S ⌊ λ • w ∗ i ⌋ ≥ ⌊ λ • ( q ∗ − ⌋ + 1 . Finally, we show that E[ g (Λ)] ≥
0. It is obvious thatE[ g (Λ)] = E[Λ] − X i ∈ N E[(Λ w ∗ i − ⌊ Λ w ∗ i ⌋ )] = ℓ + u − X i ∈ N Z u ℓ ( λw ∗ i − ⌊ λw ∗ i ⌋ )d λu − ℓ = (2 − √ n − X i ∈ N Z u ℓ ( λw ∗ i − ⌊ λw ∗ i ⌋ )d λu − ℓ . Now we discuss the last term appearing above. By substituting µ for λw ∗ i ,we obtain that Z u ℓ ( λw ∗ i − ⌊ λw ∗ i ⌋ )d λu − ℓ = Z u w ∗ i ℓ w ∗ i ( µ − ⌊ µ ⌋ )d µw ∗ i ( u − ℓ ) ≤ Z − w ∗ i ( u − ℓ ) ( µ − ⌊ µ ⌋ )d µw ∗ i ( u − ℓ ) = Z − x ( µ − ⌊ µ ⌋ )d µx where the last equality is obtained by setting x = w ∗ i ( u − ℓ ). Since u − ℓ =2( √ −
1) and w ∗ i ≥
1, it is clear that x = w ∗ i ( u − ℓ ) ≥ √ − f ( x ) = Z − x ( µ − ⌊ µ ⌋ )d µx . By numerical calculations(see Figure 1), the inequality x ≥ √ −
1) implies that f ( x ) ≤ − √ g (Λ)] ≥ (2 − √ n − X i ∈ N (2 − √
2) = (2 − √ n − (2 − √ n = 0 . x f(x) Figure 1: Plot of function f ( x ) = Z − x ( µ − ⌊ µ ⌋ )d µx . QEDThe above proof indicates an existence of a randomized rounding algo-rithm for finding an appropriate value λ • . However, from theoretical pointof view, our computer does not actually sample uniformly from an intervalof real numbers. Thus, it remains open whether there exists an efficientdeterministic algorithm for finding an appropriate value λ • . In this paper, we discussed the smallest value k ∗ , such that every k ∗ -traderobust simple game to be weighted. We provided a new proof of the existenceof a trading transform when a given simple game is non-weighted. Our proofyields an improved upper bound on the required length of a trading trans-form. More precisely, we showed that k ∗ ≤ ( n + 2) n +22 (1 / ( n +1) , which im-proves the existing bound k ∗ ≤ ( n +1) n n/ obtained by [Gvozdeva and Slinko, 2011].Next, we discussed upper bounds for the maximum possible integerweights and quota needed to represent any weighted simple game on n play-ers. We obtained upper bounds based on the maximal value of determinantsof n × n n ≤ λ • ≤ (2 −√ n + ( √ − < . n + 0 . References [Chow, 1961] Chow, C.-K. (1961). On the characterization of thresholdfunctions. In , pages 34–38. IEEE.[Elgot, 1961] Elgot, C. C. (1961). Decision problems of finite automata de-sign and related arithmetics.
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