txt/2101.07621.txt

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′</summationtext
i∈Sw′′
i=/summationtext
i∈S′w′′
i=/summationtext
i∈S′w′
iand thus S′is winning.
The monotonicity property (1) and S′⊆Simply that Sis winning.
From the above discussion, it is obvious that if a given simpl e game is
not roughly weighted, then P3 is infeasible. Farkas’ Lemma [ Farkas, 1902]
states that
D3:/parenleftbiggA(W)⊤−A(L)⊤
1⊤−1⊤/parenrightbigg/parenleftbiggx
y/parenrightbigg
=/parenleftbigg−1
0/parenrightbigg
,
x≥0,y≥0,
is feasible if and only if P3 is infeasible. By introducing an artificial non-
negative variable u≥0 and equality 1⊤x=u−1, we obtain a linear
inequality system:
D3+:
A(W)⊤−A(L)⊤0
1⊤−1⊤0
1⊤0⊤−1

x
y
u
=
−1
0
−1
,
x≥0,y≥0, u≥0.
It is obvious that D3 is feasible if and only if D3+is feasible.
Next, we construct a trading transform from a basic feasible solution
of D3+. Let/tildewiderA3z=/tildewidec′be a linear equality system obtained from D3+by
repeatedly removing redundant equalities. As D3+is feasible, there exists a
basic feasible solution, denoted by z′, and a corresponding basis matrix B
of/tildewiderA3. From Cramer’s rule, z′
S/\e}atio\slash= 0 implies that z′
S= det(BS)/det(B) for
eachS⊆N,whereBSis obtained from Bwith a column corresponding to
variable zSreplaced by/tildewidec′. Obviously, |det(B)|z′is a non-negative integer
vector. We denote by ( x′⊤,y′⊤,u′)⊤the basic feasible solution z′. We recall
that∅ /\e}atio\slash∈ W ∋ Nandintroduceapairofnon-negative integer vectors ( x∗,y∗)
defined as follows:
x∗
S=/braceleftbigg|det(B)|x′
S (ifS∈ W \{N}),
|det(B)|(x′
N+1) (if S=N),
y∗
S=/braceleftbigg|det(B)|y′
S (ifS∈ N \{∅} ),
|det(B)|(y′
∅+1) (if S=∅).
16Subsequently, χ(S) denotes the characteristic vector of a coalition S. It is
easy to see that pair ( x∗,y∗) satisfies
A(W)⊤x∗−A(L)⊤y∗=/summationdisplay
S∈Wχ(S)x∗
S−/summationdisplay
S∈Lχ(S)y∗
S
=/summationdisplay
S∈Wχ(S)|det(B)|x′
S+χ(N)|det(B)|−/summationdisplay
S∈Lχ(S)|det(B)|y′
S−χ(∅)|det(B)|
=|det(B)|/parenleftBigg/parenleftBigg/summationdisplay
S∈Wχ(S)x′
S−/summationdisplay
S∈Lχ(S)y′
S/parenrightBigg
+χ(N)−χ(∅)/parenrightBigg
=|det(B)|/parenleftBig/parenleftBig
A(W)⊤x′−A(L)⊤y′/parenrightBig
+1−0/parenrightBig
=|det(B)|(−1+1−0) =0
and
/summationdisplay
S∈Wx∗
S−/summationdisplay
S∈Ly∗
S=/summationdisplay
S∈W|detB|x′
S+|det(B)|−/summationdisplay
S∈L|det(B)|y′
S−|det(B)|
=|det(B)|/parenleftBigg/parenleftBigg/summationdisplay
S∈Wx′
S−/summationdisplay
S∈Ly′
S/parenrightBigg
+1−1/parenrightBigg
=|det(B)|(0+1−1) = 0.
Next, we can construct a trading transform ( X;Y) corresponding to the
pair ofx∗andy∗by analogy with the proof of Theorem 2.2. Both x∗
Nand
y∗
∅are positive and ∅ /\e}atio\slash∈ W ∋ N; therefore ( X;Y) is a potent certificate of
non-weightedness.
Lastly, we discuss the length of ( X;Y). The number of rows (columns)
ofB, denoted by n′, is less than or equal to n+2. The basic feasible solution
z′satisfies that u′= 1+1⊤x′≥1>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.
Lastly, we showed that a roughly weighted simple game (satis fying∅ /\e}atio\slash∈
W ∋N) has an integer vector ( q;w⊤) of the rough voting representation
satisfying 0 ≤wi≤αn−1(∀i∈N), 0≤q≤αn, and 1≤/summationtext
i∈Nwi≤2αn.
When a given simple game is not roughly weighted, we showed th at (under
the the monotonicity property (1) and ∅ /\e}atio\slash∈ W ∋ N) there existed a potent
certificate of non-weightedness whose length is less than or equal to 2 αn+1.
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