A note on the growth of the dimension in complete simple games

GAMES

SASCHA KURZ

Abstract: The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games it was only recently shown in [9] that the dimension can also be exponential. Here we show that this is also the case for complete simple games with two types of voters and for the Boolean dimension of general complete simple games, which was posed as an open problem in [9].

Keywords:complete simple games, weighted games, dimension, Boolean dimension.

MSC:91B12^?, 91A12.

1. INTRODUCTION

Simple games can be viewed as binary yes/no voting systems in which a proposal is pitted against the status quo. In the subclass of weighted games each voter has a non-negative weight and a proposal is accepted if the weight sum of its supporters meets or exceeds a preset posi- tive quota. The representation complexity of weighted games is rather low, which makes them interesting candidates for real-world voting system. More precisely, for a weighted game it is sufficient to list the weights of thenvoters and the quota. Directly storing whether a proposal would be accepted or rejected for each subset of the nvoters would need2ⁿbits.¹ However, each simple game can be written as the intersection of a finite number of weighted games and the smallest possible number is called the dimension of the simple game. Unfortunately, the dimension can also be exponential in the number of voters, see e.g. [4, 11]. Complete simple games lie in between the classes of simple and weighted games. Here the voters do not admit weights but are completely ordered (which will be defined more precisely in the next section).

E.g. the voting rules of the Council of the European Union according to the Treaty of Lisbon can be modeled as a non-weighted complete simple game. In [9] it was shown that the dimension of a complete simple game can also be exponential in the number of voters. However, the stated construction requires that the number of types of different voters also increases without bound.

Here we show that the dimension of a complete simple game can be also exponential in the number of voters for just two different types of voters. If all voters are of the same type, then the game is weighted, i.e., has a dimension of 1. The concept of the dimension and the intersection of weighted games was generalized to, more general, Boolean combinations of weighted games, see e.g. [1, 4]. For simple games the corresponding Boolean dimension can also be exponential in the number of voters, see e.g. [1, 4]. Whether the Boolean dimension of a complete simple game can also be exponential in the number of voters was posed as an open problem in [9].

1Also the restriction to minimal winning coalitions, which are introduced later on, do not decrease the represen- tation complexity too much, since there can be _bn/2cⁿ

minimal winning coalitions.

1

Here we answer this question by a construction and show that the Boolean dimension is poly- nomially bounded in the number of shift minimal winning vectors and voters. We also answer another open question from [9] and analyze possible restrictions on the weights that still allow a representation of a complete simple game as the intersection of weighted games.

The paper is organized as follows. In Section 2 we introduce the necessary preliminaries. Our results are presented in Section 3.

2. PRELIMINARIES

LetN = {1, . . . , n} be a set ofnvoters. By 2^N we denote the set {S : S ⊆ N}of all subsets ofN. We also call the elementsS∈2^N coalitions.

Definition 2.1. Asimplegame is a mappingv: 2^N → {0,1}such thatv(∅) = 0, v(N) = 1, andv(S) ≤v(T)for all∅ ⊆S ⊆T ⊆N. Each coalitionS ⊆N withv(S) = 1is called a winning coalitionand each coalitionT ⊆N withv(T) = 0is called alosing coalition. IfSis a winning coalition and all proper subsets ofS are losing, thenSis called aminimal winning coalition. Similarly, we call a losing coalition amaximal losing coalitionif all proper supersets are winning. Given a simple gamev, we denote the set of minimal winning coalitions byWand the set of maximal losing coalitions byL.

A simple game is uniquely characterized by either its setWof minimal winning coalitions or its setLof maximal losing coalitions.

Example 2.2. For n = 4voters let v be the simple game with W =

{1,2},{3,4} . The corresponding set of winning coalitions is given byWand all coalitions of cardinality at least3.

We haveL=

{1,3},{1,4},{2,3},{2,4} and the other losing coalitions are those coalitions of cardinality at most1.

Definition 2.3. Given a simple gamev, we writei A j (orj @ i) for two votersi, j ∈ N if we havev

{i} ∪S\{j}

≥ v(S)for all{j} ⊆ S ⊆N\{i}and we abbreviateiA j,j A i byij. The simple gamevis called complete(simple game) if the binary relationAis a total (complete) preorder, i.e.

(1) iAifor alli∈N,

(2) eitheriAjorjAi(including “iAjandjAi”) for alli, j ∈N, and (3) iAj,jAhimpliesiAhfor alli, j, h∈N

holds.

We remark that the simple game from Example 2.2 is not complete. I.e., while we have12 and34, for eachi∈ {1,2}and eachj ∈ {3,4}we have neitheriAjnori@j.

Sinceis a equivalence relation we can partition the set of votersNinto subsetsN1, . . . , Nt

such that we haveijfor alli, j ∈N_h, where1 ≤h≤t, andijimplies the existence of an integer1 ≤ h ≤ twithi, j ∈ Nh. We call each setNh an equivalence class (of voters)andt thenumber of equivalence classes of voters. We also say thatvhasttypes of voters. Byn_i we denote the cardinality ofN_i, where1≤i≤t. Given the equivalence classes of voters, we can associate to each coalitionS ⊆N a vectorme = (m1, . . . , mt) ∈N^tviami = # (S∩Ni)for all1≤i≤t. We also callme thetypeof coalitionS. While several coalitions can be associated to the same vector, i.e. have the same types, they are either all winning or all losing, so that we speak of winning or losing vectors, respectively.

Definition 2.4. Let v be a complete simple game with equivalence classes N_h of voters for 1 ≤h ≤t. We call a vectorme = (m1, . . . , mt) ∈N^t, where0 ≤mh ≤nh for1 ≤h ≤t, a winning vectorifv(S) = 1, whereSis an arbitrary subset ofN containing exactlym_helements ofN_hfor1≤h≤t. Analogously, we callme alosing vectorifv(S) = 0.

W.l.o.g. we will always assume that for a complete simple game the equivalence classes of voters are ordered such that we havelAl⁰for alll∈N_iand alll⁰ ∈N_jwith1≤i < j≤t.

The minimal winning vectors for Example 2.2 are(2,0)and(0,2). There is a unique maximal losing vector(1,1)and the additional losing vectors are given by(0,0),(1,0), and(0,1).

Definition 2.5. Letvbe a complete simple game. We call a minimal winning coalitionS shift minimalif for every other winning coalition S⁰ with {i} = S\S⁰ and{j} = S⁰\S we have i @ j. Similarly, we call a maximal losing coalitionT shift maximalif for every other losing coalition T⁰ with{i} = T\T⁰ and {j} = T⁰\T we havei A j. Now letS be an arbitrary coalition andme = (# (S∩N1), . . . ,# (S∩Nt))be the corresponding vector. We callme shift minimal winningifS is shift minimal winning and we callme shift maximal losingifSis shift maximal losing.

In words, a coalition is a shift minimal winning coalition, if the coalition is minimal winning and the replacement of any voter by a strictly “weaker” (according to@) voter turns the coalition into a losing one.

Letvbe a complete simple game withtequivalence classes of voters. Based on the assumed ordering of the equivalence classesN₁, . . . , N_twe writeea:= (a₁, . . . , a_t) (b₁, . . . , b_t) =:eb ifP_i

h=1a_h ≥P_i

h=1b_hfor all1≤i≤t. Assumeeaeb. Ifebis a winning vector, then alsoea has to be a winning vector, while it can happen thatebis losing andeais winning. So, ifme is a winning vector inv, then there exists a shift minimal winning vectorme⁰invsuch thatme me⁰. As an abbreviation, we writeea ebifeaebandea 6=eb. Note that we can haveea eband ebeaif and only ifea=eb.

Example 2.6. Letvbe a simple game witht= 2equivalence classes of votersN₁ ={1,2}and N₂ ={3,4,5,6}such that a coalitionSis winning if# (S∩N₁)≥2or#S≥4. The minimal vectors ofvare given by(2,0),(1,3), and(0,4). Sincei A jfor all i∈ N1 and allj ∈ N2

the simple gamevis complete and the shift minimal winning vectors are(2,0)and(0,4). Note that(1,3)(0,4), while we have neither(2,0)(1,3)nor(1,3)(2,0). The unique shift maximal losing vector is given by(1,2).

Definition 2.7. A simple game v is weighted if there exists a quota q ∈ R>0 and weights wi ∈R≥0, where1≤i≤n, such thatv(S) = 1iffw(S) :=P

i∈Swi ≥qfor every coalition S ⊆N. We also writev= [q;w₁, . . . , w_n].

Sincew_i≥w_j impliesiAj, every weighted game is complete, so that the simple game from Example 2.2 is not weighted.

Definition 2.8. A sequence of coalitions

T = (X₁, . . . , X_j;Y₁, . . . , Y_j) of a simple gamevis called atrading transformof lengthjif

#{i :h∈X_i}= #{i : h∈Y_i}

for all h ∈ N. A trading transform T is called a certificate of non-weightedness for v if X1, . . . , Xj are winning andY1, . . . , Yj are losing coalitions.

The absence of a certificate of non-weightedness of any length is a necessary and sufficient condition for the weightedness of a simple gamev, see e.g. [11].

A certificate of non-weightedness for Example 2.6 is given by ({1,2},{3,4};{1,3},{2,4}) and by

({1,2},{3,4,5,6};{1,3,4},{2,5,6}) for Example 2.6.

Definition 2.9. Letv1, . . . , vdbedsimple games with the same set of votersN. Theirintersec- tionv=v₁∧ · · · ∧v_dis defined viav(S) = min{v_i(S) : 1≤i≤d}for allS⊆N. Similarly, theirunionv =v1∨ · · · ∨vdis defined viav(S) = max{v_i(S) : 1≤i≤d}for allS ⊆N.

It can be easily checked that the intersection and the union of a list of simple games is a simple game itself. It is well known that each simple game can be written as the intersection as well as the union of a finite list of weighted games, see e.g. [11].

Definition 2.10. Letvbe a simple game. The smallest integerdsuch thatvis the intersection ofdweighted games is called thedimensionofv. Similarly, the smallest numberdof weighted games such thatvis the union ofdweighted games is called thecodimensionofv.

The simple game of Example 2.2 can be written as

[2; 1,1,2,0]∧[2; 1,1,0,2] or [2; 1,1,0,0]∨[2; 0,0,1,1].

Since we already know that the game is not weighted, both the dimension and the codimension are equal to2. For the simple game from Example 2.6 we have the representations

[8; 5,3,2,2,2,2]∧[8; 3,5,2,2,2,2] and [2; 1,1,0,0,0,0]∨[4; 1,1,1,1,1,1], so that, again, both the dimension and the codimension are equal to2.

A useful criterion for a lower bound for the dimension of a simple game is:

Lemma 2.11. ([6, Observation 1],[9, Theorem 1])

Letvbe a simple game and letT1, . . . , T_dbe losing coalitions such that for all1 ≤i < j ≤j there is no weighted gamev^i,j for which every winning coalition ofvis winning inv^i,j butT_i andTj are both losing inv^i,j. Then, the dimension ofvis at leastd.

Definition 2.12. Letv₁, . . . , v_dbe simple games. ABoolean combinationofv₁, . . . , v_dis given by v1∧v⁰ orv1 ∨v⁰, where v⁰ is a Boolean combination ofv2, . . . , v_d. For the special case d= 1we say that a simple game is a Boolean combination of itself. TheBoolean dimensionof a simple gamevis the smallest integerdsuchvis a Boolean combination ofdweighted games v₁, . . . , v_d.

In words, the Boolean dimension of a simple gamev is the smallest number of weighted games that are needed to expressv by a logical formula connecting the weighted games using

∧and∨. As an example we mention that the voting rules of the Council of the European Union according to the Treaty of Lisbon can be written as(v₁∧v₂)∨v₃, wherev₁,v₂, andv₃are suitable weighted games. In [6] it was shown that the dimension is at least7and the codimension is at least2000, so that the Boolean dimension is indeed equal to3.

3. RESULTS

It is well known that the dimension of a simple game is upper bounded by the number#Lof maximal losing coalitions, see e.g. [11]. If there exists an equivalence class with many voters this upper bound can be lowered:

Lemma 3.1. Letv be a simple game withtequivalence classesN1, . . . , Ntof voters and 1≤ i ≤ tbe fix but arbitrary. For each maximal losing coalitionS ∈ Llet a(S) := # (S∩N_i), S⁰ =S\N_i, and the weighted gamev^S=

q^S;w^S

be defined by

• q^S=a+ 1;

• wj = 1for allj∈Ni;

• w_j =a+ 1for allj ∈N\(S⁰∪N_i); and

• wj = 0for allj∈S⁰.

With this, the intersectionv⁰of the weighted gamesv^SforS ∈ Lequalsv.

PROOF. LetSbe an arbitrary maximal losing coalition inv. Sincew^S(S) =aandq^S =a+ 1 we havev^S(S) = 0, so thatv⁰(S) = 0 = v(S). Sincev⁰is a simple game any losing coalition of v is also losing in v⁰. Now let T be an arbitrary winning coalition and S be an arbitrary maximal losing coalition inv. SinceT 6⊆S there either exists a voterj ∈T\N_iwithj /∈S or

# (T∩N_i) ≥a(S) + 1. In both cases we havew^S(T) ≥a(S) + 1 =q^S, so thatv^S(T) = 1.

Thus, we havev⁰(T) = 1 =v(T), which then impliesv⁰ =v.

While we have constructed a weighted gamev^Sfor each maximal losing coalitionS, we have v^S=v^T ifS\N_i=T\N_iand# (S∩Ni) = # (T ∩Ni).

So, it is indeed possible to represent each complete simple game as the intersection of weighted games where the voters of one arbitrary equivalence class of voters always have equal weights.

However, in general it is not possible to restrict the intersection to weighted games respecting the strict ordering of the voters:

Proposition 3.2. There exists a complete simple gamevsuch that for every representation

v= q¹;w¹

∧ · · · ∧h q^d;w^d

i

as the intersection of weighted games there exists an index1 ≤h ≤jand two votersi, jfrom different equivalence classes of voters withiAjandw_i^h< w_j^h.

PROOF. Letvbe the complete simple game witht= 4equivalence classes of voters,n₁ =n₂= n3 = n4 = 20, and a unique shift maximal losing vector(4,4,4,4). Choose a losing coalition T ⊆ N with# (T ∩N_p) = 4for all1 ≤ p ≤ 4and an index1 ≤ h ≤ dsuch that coalition T is also losing in

q^h;w^h

. By eventually scaling the quotaq^h and the weightsw^h we assume w^h(T) ≤ q^h−1. We set a_p = w^h(T∩N_p)/# (T ∩N_p)andx_p = w^h(N_p\T)/# (N_p\T) for all 1 ≤ p ≤ 4, i.e., the average weight of members ofT or non-members of T in each equivalence class of voters. Note that (0,9,0,0)is a winning vector and choose a coalition T ∩N2 ⊆ S¹ ⊆ N2 with cardinality9 and minimum weight. With this,S¹ is winning and q^h ≤w^h S¹

≤4a₂+ 5x₂. Sincew^h(T) = 4a₁+ 4a₂+ 4a₃+ 4a₄ ≤q^h−1, we have 4a₂+ 5x₂≥4a₁+ 4a₂+ 4a₃+ 4a₄+ 1,

which is equivalent to

5x₂−4a₁−4a₃−4a₄≥1. (1)

Note that (0,0,0,17)is a winning vector and choose T ∩N₄ ⊆ S² ⊆ N₄ with cardinality 17and minimum weights. With this, S² is winning andq^h ≤ w^h S²

≤ 4a4+ 13x4. Since w^h(T) = 4a₁+ 4a₂+ 4a₃+ 4a₄ ≤q^h−1, we have

4a4+ 13x2 ≥4a1+ 4a2+ 4a3+ 4a4+ 1, which is equivalent to

13

4 x₄−a₁−a₂−a₃ ≥ 1

4. (2)

Assuming thatw^h_i ≥ w^h_j for all votersi, j from different equivalence classes withi A j, we especially havea1 ≥x2anda3 ≥x4, which is equivalent to

5a1−5x2 ≥0 (3)

and

5a₃−5x₄ ≥0. (4)

Adding the left and the right hand sides of inequalities (1)-(4) yields

−7

4x₄−a₂−4a₄≥1.25,

which is a contradiction, sincea2, a4, x4 ≥ 0. Thus, there exist votersiandj from different equivalence classes of voters withw_i^h< w_j^handiAj.

Proposition 3.2 gives a negative answer to the the second question from the conclusion of [9], where it is additionally assumed that in an arbitrary equivalence class of voters all weights are equal. The first question from the conclusion of [9] concerns the worst case behavior of the Boolean dimension of a complete simple game.

A lower bound for worst-case Boolean dimension of a simple or a complete simple game with nvoters can be concluded from a simple counting argument. First note that there are at least

2(bn/2cⁿ ) >2^{√2πn1 2n} (5)

simple games withnvoters, see e.g. [4] for tighter estimates, and at least 2

q2 3π·2ⁿ

/(^n√n)

(6) complete simple games withnvoters, see [10].

However, there are not too many possibilities for Boolean combinations:

Proposition 3.3. ([1, Proposition 1])

The total number of Boolean combinations ofsweighted games withnvoters is at most2^O(^sn2log(sn)). So, as observed in [1, Corollary 2] and [4], the Boolean dimension of a simple game withn voters can be exponential inn. Actually, almost all simple games have an exponential dimension.

Using the same reasoning we can also conclude that the Boolean dimension of a complete simple game can be exponential in the number of voters. This answers an open question from [9], where it was shown that the dimension of a complete simple game can be exponential in the number of voters.

Lemma 3.4. Letvbe a complete simple game withtequivalence classesN₁, . . . , N_tof voters.

Ifvhas exactlyr shift minimal winning vectorsme¹, . . . ,me^r ∈N^t, then the Boolean dimension ofvis at mostrt.

PROOF. For each index1≤i≤rwe will define a complete simple gamevⁱ as the intersection oftweighted games whose unique minimal winning vector coincides with meⁱ. The union of thosevⁱgive a representation ofvas a Boolean combination ofrtweighted games.

So, letmeⁱ = mⁱ₁, . . . , mⁱ_t

. With this, we define the weighted gamesv^i,j =

q^i,j, w^i,j by

• q^i,j =

j

P

h=1

mⁱ_h;

• w^i,j_l = 1ifl∈ ∪^j_h=1N_handw_l= 0otherwise.

Now let vⁱ = v^i,1 ∧ · · · ∧v^i,t. First we check l A l⁰ for all l ∈ N_h and all l⁰ ∈ N_h⁰ with 1≤h < h⁰ ≤t, i.e., the simple gamevⁱ is complete. Ifme = (m1, . . . , mt) ∈N^tis a winning vector invⁱ, then we haveme meⁱ. So, if me = (m1, . . . , mt)is a winning vector inv⁰, then there exists an index1 ≤ i ≤ rsuch thatme meⁱ. Since that is exactly the condition for me

being a winning vector inv, we havev⁰ =v.

Corollary 3.5. The Boolean dimension of a complete simple gamevwithnvoters andtequiva- lence classes of voters is at mosttn^t. Ift= 2, then the Boolean dimension is at most₂

3(n+ 3) . PROOF. The number of possible winning vectors(m₁, . . . , m_t)is at mostn^tsince1≤m₁ ≤n ift= 1and0 ≤mi ≤n−1for all1 ≤ i≤tift ≥2. Thus, the numberr of shift minimal winning vectors also is at mostn^t, so that we can apply Lemma 3.4 to conclude that the Boolean dimension ofvis at mostrt≤tn^t. For the special caset= 2we can concluden≥3r−3from [7, Lemma 1]. Thus, the Boolean dimension ofvis at most₂

3(n+ 3)

, again using Lemma 3.4.

We remark that it is well known that complete simple games with a unique equivalence class of voters, i.e.,t= 1, are weighted. The maximum numberrof shift minimal winning vectors of a complete simple game withnvoters can indeed be exponential inn, see [5] for an exact formula for the maximum value ofr(depending onn).

Next we want to consider the dimension of complete simple games. We remark that the exact dimension is only known for very few simple games. In [8] a large family of simple games was constructed, where the dimension could be determined exactly. This yields an explicit description of a sequence of simple games with dimension2^n−o(n).

Proposition 3.6. Let d ≥ 2 be an integer and v^d be the complete simple game witht = 2 equivalence classes of voters, wheren₁ = dandn₂ ≥ 2d, and r = 2shift minimal winning vectorsme¹= (2,0),me²= (0,4). Then, the dimension ofvis exactlyd.

PROOF. W.l.o.g. we number the voters so that N1 = {1, . . . , d}and N2 = {d+ 1, . . . , n}, wheren = n₁ +n₂ ≥ 3d. For each 1 ≤ i ≤ iwe define a weighted gamevⁱ =

qⁱ;wⁱ by qⁱ = 8,w_iⁱ = 3,w_jⁱ = 5for allj ∈N₁\{i}, andw_jⁱ = 2for allj ∈N₂. LetS be an arbitrary winning coalition ofv. If# (S∩N₁) ≥2, thenwⁱ(S) ≥ 3 + 5 = 8 =qⁱ for all1 ≤ i≤ d.

If# (S∩N1) ≤1, then#S ≥ 4, so thatwⁱ(S) ≥#S·2≥8 = qⁱ for all1 ≤i≤d. Thus, every winning coalition ofv is also winning invⁱ for all 1 ≤ i ≤ d. Now letT be a losing coalition ofv. IfT ∩N1 = ∅, then#T ≤ 3, so that wⁱ(T) = 2·#T ≤ 6 < 8 = qⁱ for all 1≤i≤d. IfT∩N₁ 6=∅, thenT∩N₁ ={i}for a voter1≤i≤dand# (T\{i})≤2, so that wⁱ(T) = 3 + 2(#T−1)≤7<8 =qⁱ. Thus, we havev=v¹∧ · · · ∧v^d, so that the dimension ofvis at mostd.

For the other direction we setTi ={i, d+2i−1, d+2i}for all1≤i≤d. SinceTi∩N₁ ={i}

andT_i∩N₂ ={d+ 2i−1, d+ 2i}the coalitionT_i is losing inv, where1 ≤i ≤ d. For all

1≤i < i⁰ ≤i⁰

T^i,i0 = {i, i⁰},{2i−1,2i,2i⁰−1,2i⁰};Ti, T_i⁰

is a certificate of non-weightedness. Thus, we can apply Lemma 2.11 to conclude that the

dimension ofvis at leastd.

The complete simple games in the prof of Lemma 3.6 generalize the complete simple game from Example 2.6. The argument for the lower bound for the dimension ofvis the same as in [9, Proposition 2].

Our next aim is to prove that there exist complete simple games with two equivalence classes of voters whose dimension is exponential in the number of voters. To this end, we have to introduce some notation from coding theory. A (binary) code is a subsetCofFⁿ2whose elements c ∈C are calledcodewords. TheHamming weightwt(c)of a codewordc ∈ Cis the number

#{1≤i≤n : ci6= 0}of non-zero coordinates. TheHamming distanced(c, c⁰)between two codewordsc, c⁰ ∈Cis the number#{1≤i≤n : c_i 6=c⁰_i}of coordinates wherecandc⁰differ.

Theminimum Hamming distanced(C)of a codeC is the minimum ofd(c, c⁰)for all pairs of different codewordsc, c⁰ ∈ C. ByA(n,2δ;w) we denote the maximum cardinality of a code C in Fⁿ₂ with minimum Hamming distanced(C) ≥ 2δ such that all codewords c ∈ C have Hamming weightwt(c) = w. Those codes are calledconstant weight codes. It is well known thatA(n,2;w) = _wⁿ

and ⁿ_w

·_n¹ ≤A(n,4;w)≤ _w−1ⁿ

·_w¹, see e.g. [3].

Theorem 3.7. Ifn≥4is divisible4, then there exists a complete simple gamevwithnvoters, t= 2equivalence classes of voters, and dimension at least

A(n/2,4;n/4)≥ n/2

n/4

· 2

n ≥ 4·2^n/2 n² .

PROOF. Let v be a complete simple game with n = 4kvoters, t = 2equivalence classes of voters,n1 =n2 = 2k,N1 ={1, . . . ,2k},N2 = {2k+ 1, . . . ,4k}, andr = 2shift-minimal winning vectorsme¹ = (k,0),me²= (0,2k).

LetC₂ be a code in F^2k2 withd := A(2k,4;k) codewords of constant weight k and min- imum Hamming distanced(C₂) ≥ 4. Since #C₂ ≤ _k−1^2k

· ¹_k we can choose a code C₁ in F^2k₂ with A(2k,4;k) ≤ _k−1^2k

= A(2k,2;k−1)codewords of constant weight k−1 and minimum Hamming distance d(C₁) ≥ 2. To each codeword c ∈ C₁ we associate the set {i : c_i= 1,1≤i≤2k}. This givesdsetsT_i¹⊆N₁. Similarly, we associate to each codeword c ∈ C₂ the set{i+ 2k : c_i = 1,1≤i≤2k}. This givesdsetsT_i² ⊆ N₂. With this, we set T_i :=T_i¹∪T_i², so that# Tⁱ∩N₁

=k−1,# Tⁱ∩N₂

=k, andTⁱ is a losing coalition of vfor all1≤i≤d.

For each pair(i, j)with1≤i < j ≤dwe consider the two losing coalitionsT_i =T_i¹∪T_i² andTj =T_j¹∪T_j². Since the codewords ofC1have Hamming distance at least2, there exists a votera^i,j witha^i,j ∈T_i¹ anda^i,j ∈/ T_j¹. Since the codewords ofC₂have Hamming distance at least4, there exist two different votersb^i,j₁ ,b^i,j₂ withb^i,j₁ , b^i,j₂ ∈T_j²andb^i,j₁ , b^i,j₂ ∈/ T_j¹. With this,

T^i,j = T_i\

a^i,j ∪n

b^i,j₁ , b^i,j₂ o , T_j\n

b^i,j₁ , b^i,j₂ o

∪

a^i,j ;T_i, T_j

is a certificate of non-weightedness. Thus, we can apply Lemma 2.11 to conclude that the dimension ofvis at leastd, whered=A(2k,4;k) =A(n/2,4;n/4).

Due to Lemma 3.4 the complete simple game constructed in the proof of Theorem 3.7 has a Boolean dimension of at most4. A null voter in a simple gamevis a voterisuch thatv(S) = v(S\{i}) for allS ⊆ N. By adding up to three null voters, the construction of Theorem 3.7 gives a complete simple game withnvoters at dimension at least4·2^(n−3)/2/n²for eachn≥4.

We remark that the number of complete simple games withnvoters andt= 2types of voters is F ib(n+ 6)−(n²−4n+ 8), see e.g. [7, Theorem 4], whereF ib(n)denotes thenth Fibonacci number, and at mostⁿ₁₅⁵+4n⁴of them are weighted, see [2, Theorem 5.2]. Nevertheless, there are much more complete simple games with two types of voters than weighted games with two types of voters we cannot directly use this to lower bound the worst-case behavior of the dimension.

Given a representationv=v¹∧ · · · ∧v^dof a (complete) simple gamevwith dimensiondas the intersection ofdweighted games, the partition of the voters into equivalence classes typically differ widely across the weighted gamesvⁱ.

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Fakult¨at f¨ur Mathematik, Physik und Informatik, Universit¨at Bayreuth, Germany, email: sascha.kurz@uni-bayreuth.de, tel: +49 921 557353.