Classes of Complete Simple Games that are All Weighted
Sascha Kurz
1
and Nikolas Tautenhahn
1,2
1
Department of Mathematics, Physics, and Computer Science, University of Bayreuth, 95440 Bayreuth, Germany
2
LivingLogic AG, Markgrafenallee 44, 95448 Bayreuth, Germany
Keywords:
Complete Simple Games, Weighted Games, Voting, Group Decision Making.
Abstract:
Decisions are likely made by groups of agents. Transparent group aggregating rules are given by weighted
voting. Here a proposal is accepted if the sum of the weights of the supporting agents meets or exceeds a given
quota. We study a more general class of binary voting systems – complete simple games – and propose an
algorithm to determine which sub classes, parameterized by the agent’s type composition, are weighted.
1 INTRODUCTION
Weighted voting is a method for group decision mak-
ing. For simplicity, we assume that for each proposal
on a certain issue the group members, called agents
for brevity, options are either to vote “yes” or “no”.
The aggregated group decision then is also either
“yes” or “no”. Those procedures are called binary
voting systems or games in the literature. The special
case of a weighted game consists of a quota q > 0 and
weights w
i
≥ 0 for every participating agent. With
this, the aggregated group decision is “yes” if and
only if the summed weights of the supporters of a
given proposal meets or exceeds the quota. For n
agents, such a game is denoted by [q;w
1
, . . . , w
n
].
Weighted voting systems are commonly applied
whenever not all agents are considered to be equal.
Reasons may lie in heterogeneous competencies for
different issues, see e.g. (Grofman et al., 1983). In
stock corporations, weights can arise as the number
of shares that each shareholder owns, see e.g. (Leech,
2013).
In Section 2, we introduce the class of complete
simple games, where some are weighted and others
are not. Restricting to symmetric (complete) simple
games, where all agents have equal capabilities, sim-
plifies things dramatically, as first found out in (May,
1952): All such games
1
are weighted.
In this paper we aim to generalize May’s Theo-
rem by providing a strategy to classify all classes of
complete simple games, according to the agent’s type
1
More precisely, May’s Theorem applies to (complete)
simple games with one type of agents (see Section 2).
composition (see Definition 5), with the property that
every class member is weighted. It will turn out that
this can happen only if at most five different types of
agents are present and from all but one type there have
to be very few agents, see lemmas 4 and 5.
Exact formulas for the number of sub classes of
weighted games are rather rare. From May’s Theo-
rem, one can conclude that the number of weighted
games with n agents all of the same type
2
is given by
n. In (Kurz and Tautenhahn, 2013), the authors have
presented an algorithm that can compute an exact enu-
meration formula for complete simple games with t
types of agents and r, so-called, shift-minimal win-
ning vectors depending on the number of agents n. No
such algorithm is known for weighted games. Having
our classification result at hand, we can enumerate the
corresponding sub classes of weighted games since
it will turn out, that in each case the number of oc-
curring shift-minimal winning vectors is bounded by
a small integer and the restrictions from the agent’s
type composition can be easily incorporated into the
enumeration algorithm.
2 COMPLETE SIMPLE GAMES
A binary voting procedure can be modeled as a func-
tion v : 2
N
→ {0, 1} mapping the coalition S of sup-
porting agents to the aggregated group decision v(S),
where N = {1, . . . , n} and 2
N
denotes the set of sub-
sets of N. Quite naturally, several assumptions of
a binary voting procedure are taken for granted: If
2
They all have the form [q; 1, . . . ,1] with q ∈ {1, . . . , n}.
288
Kurz S. and Tautenhahn N..
Classes of Complete Simple Games that are All Weighted.
DOI: 10.5220/0004921902880293
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 288-293
ISBN: 978-989-758-017-8
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

no agent’s supports the proposal, reject it. If all
agent’s supports the proposal, accept it. If the sup-
porting clique of agents is enlarged by some addi-
tional agents, the group decision should not change
from acceptance to rejection.
Definition 1. A pair (v, N) is called simple game if
N is a finite set, v : 2
N
→ {0, 1} satisfies v(
/
0) = 0,
v(N) = 1, and v(S) ≤ v(T ) for all S ⊆ T ⊆ N.
A more demanding assumption is to require that
the agents are linearly ordered according to their ca-
pabilities to influence the final group decision. This
can be formalized with the desirability relation intro-
duced in (Isbell, 1956).
Definition 2. Let (v, N) be a simple game. We write
i A j (or j @ i) for two agents i, j ∈ N if we have
v

{i} ∪ S\{ j}

≥ v(S) for all { j} ⊆ S ⊆ N\{i} and
we abbreviate i A j, j A i by i j.
The relation  partitions the set of agents N into
equivalence classes N
1
, . . . , N
t
.
Example 1. For the weighted game [4; 5, 4, 2, 2, 0] we
have N
1
= {1, 2}, N
2
= {3, 4}, and N
3
= {5}.
Agents having the same weight are contained in
the same equivalence class, while the converse is not
necessarily true. But there always exists a different
weighted representation of the same game such that
the agents of each equivalence class have the same
weight. For Example 1, such a representation is e.g.
given by [2; 2, 2, 1, 1, 0].
Definition 3. A simple game (v, N) is called complete
if the binary relation A is a total preorder, i.e.,
(1) i A i for all i ∈ N,
(2) i A j or j A i for all i, j ∈ N, and
(3) i A j, j A h implies i A h for all i, j, h ∈ N.
All weighted games are obviously simple and
complete.
Definition 4. For a simple game (v, N) a coalition S ⊆
N is called winning if v(S) = 1 and losing otherwise.
If v(S) = 1, v(T ) = 0 for all T ( S, then S is called
minimal winning. A coalition with v(S) = 0, v(T ) = 1
for all S ( T ⊆ N is called maximal losing.
In Example 1, coalition {2, 3} is winning, {2} is
minimal winning, {3} is losing, and {3, 5} is maximal
losing.
Definition 5. For a complete simple game (v, N),
the vector (n
1
, . . . , n
t
) ∈ N
t
>0
, where n
i
=
|
N
i
|
, is
called type composition. The number t of equivalence
classes is called number of types (of agents).
The type composition of Example 1 is given by

2, 2, 1

consisting of three types. Agents within the
same equivalence class are interchangeable, i.e., since
coalition {2, 3} is winning also the coalitions {1, 3},
{1, 4}, and {2, 4} have to be winning. The combina-
torial explosion of the set of corresponding winning
coalitions can be partially captured by:
Definition 6. Let (v, N) be a complete simple game
with type composition (n
1
, . . . , n
t
). Each vector s =
(s
1
, . . . , s
t
) ∈ N
t
with 0 ≤ s
i
≤ n
i
for all 1 ≤ i ≤ t is
called coalition vector of (v, N). Coalition vector s is
winning if we have v(S) = 1 for coalitions S ⊆ N with
|
S ∩ N
i
|
= s
i
for all 1 ≤ i ≤ t, and losing otherwise.
The just mentioned four winning coalitions can be
condensed to the winning vector (1, 1, 0).
Definition 7. For two vectors a = (a
1
, . . . , a
t
) ∈ N
t
and b = (b
1
, . . . , b
t
) ∈ N
t
we write a ≤ b if a
i
≤ b
i
for
all 1 ≤ i ≤ t.
If a is a winning vector of a complete simple game
and a ≤ b, then b is winning too. Next, we define
a tightening of the concept of minimal winning and
maximal losing coalitions for coalition vectors. To
this end we have to assume 1 A 2 A ··· A n in the
following.
Definition 8. For two vectors a = (a
1
, . . . , a
t
) ∈ N
t
and b = (b
1
, . . . , b
t
) ∈ N
t
we write a  b if
∑
i
j=1
a
j
≤
∑
i
j=1
b
j
for all 1 ≤ i ≤ t. Vector a is called shift-
minimal winning (SMW), if a is winning and all b  a,
with b 6= a, are losing. Similarly, vector a is called
shift-maximal losing (SML), if a is losing and all
b  a, with b 6= a, are winning.
An example is given by (0, 1, 0)  (1, 0, 0). The
SMW vectors of Example 1 are given by (1, 0, 0)
and (0, 2, 0). The unique SML vector is given by
(0, 1, 1). We remark that each complete simple game
is uniquely characterized by its type composition and
its full list of SMW vectors. Of course, not every col-
lection of coalition vectors for a given type composi-
tion is a feasible set of SMW vectors.
Definition 9. Let a = (a
1
, . . . , a
t
) ∈ N
t
and b =
(b
1
, . . . , b
t
) ∈ N
t
be two vectors. We write a  b if
neither a  b nor a  b, i.e., when they are incom-
parable. Mimicking the lexicographic order, we write
a m b if there exists an index k ∈ {0, . . . , n − 1} such
that a
j
= b
j
for all 1 ≤ j ≤ k and a
j+1
> b
j+1
.
A parameterization theorem for complete simple
games with t types of agents has been given in (Car-
reras and Freixas, 1996):
Theorem 1.
(a) Let vector
b
n = (n
1
, . . . , n
t
) ∈ N
t
>0
and a matrix
S =




s
1,1
s
1,2
. . . s
1,t
s
2,1
s
2,2
. . . s
2,t
.
.
.
.
.
.
.
.
.
.
.
.
s
r,1
s
r,2
. . . s
r,t




=




b
s
1
b
s
2
.
.
.
b
s
r




ClassesofCompleteSimpleGamesthatareAllWeighted
289

satisfy the following properties:
(i) 0 ≤ s
i, j
≤ n
j
, s
i, j
∈ N for 1 ≤ i ≤ r, 1 ≤ j ≤ t,
(ii)
b
s
i

b
s
j
for all 1 ≤ i < j ≤ r,
(iii) for each 1 ≤ j < t there is at least one row-
index i such that s
i, j
> 0, s
i, j+1
< n
j+1
if t > 1
and s
1,1
> 0 if t = 1, and
(iv)
b
s
i
m
b
s
i+1
for 1 ≤ i < r.
Then, there exists a complete simple game (v, N)
associated to (
b
n, S ).
(b) Two complete simple games (
b
n
1
, S
1
) and (
b
n
2
, S
2
)
are isomorphic if and only if
b
n
1
=
b
n
2
and S
1
= S
2
.
Besides being rather technical, there is some easy
interpretation for the stated conditions. Condition (i)
simply states that the
b
s
i
are feasible with respect to the
type composition
b
n. If we would not have
b
s
i

b
s
j
, then
either
b
s
i

b
s
j
or
b
s
i

b
s
j
, so that one of both vectors can
not be shift-minimal. Condition (iii) is necessary to
enforce equivalence classes according to
b
n and con-
dition (iv) prevents from row permutations. We call
two complete simple games isomorphic if there exists
a bijection for the respective agent’s names preserving
winning and losing coalitions.
Definition 10. A simple game (v, N) is called
weighted if and only if there exist weights w
i
∈ R
≥0
,
for all i ∈ N, and a quota q ∈ R
>0
such that v(S) = 1
is equivalent to
∑
i∈S
w
i
≥ q for all S ⊆ N.
3 (NON-) WEIGHTEDNESS
We have mentioned in the introduction that some
complete simple games are weighted while others are
not. In this section, we want to provide a method to
decide which case occurs
3
.
Example 2. Let
b
n = (2, 4) and S =

2 0
0 4

, then the
complete simple game (
b
n, S ) is not weighted.
Similar to the matrix S of the shift-minimal vec-
tors, one can write down a matrix L of the shift-
maximal losing vectors. If running time is not an is-
sue, this can be easily done algorithmically:
For each coalition vector a=(a_1,...a_t)
determine whether a is winning or losing
End
For each losing vector a=(a_1,...a_t)
ok=True
For i from 1 to t
If a_i<n_i and a+e_i is losing
Then ok=False
3
Several algorithms to decide whether a given simple
game is weighted or not are known in the literature, see e.g.
(Taylor and Zwicker, 1999) for an overview.
End
If ok==True Then output a
End
Here e
i
denotes the ith unit vector and we have
L =

1 2

in Example 2.
Lemma 1. For a complete simple game (v, N) let
˜
S be
a matrix of (some) winning vectors and
˜
L be a matrix
of (some) losing vectors. If there exist (row) vectors
x, y with non-negative real entries, kxk
1
= kyk
1
> 0
and x
˜
S ≤ y
˜
L, then (v, N) can not be weighted.
Proof. Combining the fact that the weight of each
winning coalition is larger than the weight of each
losing coalitions with the weighting of coalitions in-
duced by x and y gives a contradiction.
A well known fact from the literature is the in-
verse statement, i.e., for each non-weighted (com-
plete) simple game there exists a set of winning coali-
tions (or winning vectors) and a set of losing coali-
tions (or losing vectors) with multipliers x, y certify-
ing non-weightedness. The underlying concepts are
trading transforms, see (Taylor and Zwicker, 1999), or
dual multipliers in the theory of linear programming.
An example of a non-weighted complete simple game
can be extended to other type compositions:
Lemma 2. Let G
1
= (v, N) be a complete simple
game with type composition
b
n = (n
1
, . . . , n
t
) and
˜
S a
matrix of winning vectors,
˜
L a matrix of losing vec-
tors, and x, y be vectors according to Lemma 1, which
certify non-weightedness of (v, N). For each vector
b
m ≥
b
n there exists a non-weighted complete simple
game G
2
= (v
0
, N
0
) with type composition
b
m.
Proof. We choose N
0
such that N ⊆ N
0
and set v
0
(S) =
v(S) for all S ⊆ N, i.e., winning vectors of G
1
are also
winning in G
2
and losing vectors of G
1
are also losing
in G
2
. All coalition vectors of G
2
that are comparable
to the already assigned vectors, i.e., to the coalition
vectors of G
1
, are accordingly set to be either win-
ning or losing. For the remaining vectors we have
some freedom, but for simplicity determine them to
be losing vectors. We can easily check that G
2
is com-
pletely characterized and is indeed a complete simple
game. Since the rows of
˜
S are also winning vectors
in G
2
and the rows of L are also losing vectors in G
2
,
we can apply Lemma 1 with the original vectors x,y
to deduce that G
2
is non-weighted.
As an example, let (v, N) be uniquely character-
ized by
b
n = (3, 4) and S =

2 2

. The matrix of
shift-maximal losing vectors is given by L =

3 0
1 4

and we have x = (2), y = (1, 1) as a certificate for
non-weightedness. For
b
m = (6, 6) the construction of
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
290

Lemma 2 gives the game with type composition
b
m
and S =

2 2

. Now the matrix of shift-maximal
losing vectors is given by L =

3 1 0
0 4 6

T
. From
the reused vectors x, y we can conclude the non-
weightedness of the larger complete simple game.
The degree of freedom in the proof of Lemma 2 al-
lows us to also conclude that the complete game given
by type composition
b
m and S =

2 2
0 6

is also non-
weighted. Here we have L =

3 0
1 4

. The common
parts
˜
S and
˜
L have to be chosen accordingly.
Definition 11. We call a type composition
b
n =
(n
1
, . . . , n
t
) weighted if all complete simple games,
given by a matrix S of its SMW vectors and
b
n, are
weighted. Otherwise we call
b
n non-weighted.
Since (3, 4) ≥ (2, 4) we do not learn anything new,
i.e., Lemma 2 alone is sufficient to prove:
Lemma 3. Each type composition
b
n = (n
1
, n
2
) with
n
1
≥ 2 and n
2
≥ 4 is non-weighted
Lemma 4. Each type composition
b
n = (n
1
, . . . , n
t
) ∈
N
t
>0
with t ≥ 6 is non-weighted.
Proof. The game (
b
m, S ) with
S =





1 1 0 0 0 0
1 0 1 0 0 1
1 0 0 1 1 0
0 1 1 1 0 0
0 0 1 1 1 1





is complete, non-weighted, and has
b
m =
(1, 1, 1, 1, 1, 1) as its type composition with 6
types. For t > 6 we consider the complete simple
non-weighted game with type composition with
b
m = (1, . . . , 1) ∈ N
t
>0
uniquely characterized by its
matrix
S =





1 1 0 0 0 0 | 1 0 1 0 . . .
1 0 1 0 0 1 | 0 1 0 1 . . .
1 0 0 1 1 0 | 1 0 1 0 . . .
0 1 1 1 0 0 | 0 1 0 1 . . .
0 0 1 1 1 1 | 1 0 1 0 . . .





of SMW vectors. Lemma 2 transfers the result to ar-
bitrary type compositions
b
n with t ≥ 6 types.
With the help of lemmas 2 and 4, we can pro-
pose the following strategy to classify all weighted
type compositions
b
n. For small n, determine all com-
plete simple games with at most 5 types of agents and
determine which ones are weighted or non-weighted.
Taking only the smallest examples, we obtain a gener-
alized version of Lemma 3 and may hope that all other
cases correspond to weighted type compositions. Do-
ing that we obtain:
Lemma 5. For each type composition
b
n = (2, 4),
(2, 2, 2), (1, 1, 5), (1, 2, 3), (1, 3, 2), (2, 1, 4), (2, 4, 1),
(1, 1, 1, 3), (1, 1, 3, 1), (2, 1, 2, 1), (2, 3, 1, 1), (1, 2, 2, 1),
(1, 2, 1, 2), (1, 1, 2, 2), (1, 2, 1, 1, 1), (1, 1, 2, 1, 1),
(1, 1, 1, 2, 1), (1, 1, 1, 1, 2) there exists a non-weighted
complete simple game attaining
b
n.
Conjecture 1. Each type composition
b
n is either
weighted or there exists a type composition
b
m, con-
tained in the list of Lemma 5, with
b
n ≥
b
m.
In the next section, we propose an algorithmic ap-
proach to prove Conjecture 1. As an example, we
prove some special cases in Subsection 4.5.
4 WEIGHTED TYPE
COMPOSITIONS
Due to Lemma 3, for t = 2 types of agents, the only
possible candidates for weighted type compositions
are of the form (1, ), (, 1), (, 2), and (, 3), where
 stands for an arbitrary positive integer.
Definition 12. A set ω = (n
1
, . . . , n
i−1
, , n
i+1
, . . . , n
t
)
of type compositions is called i family with t types.
We call ω weighted if all of its elements are weighted.
Corollary 1. (of Conjecture 1) (), (1, ), (, 1),
(, 2), (, 3), (, 1, 1), (, 1, 2), (, 1, 3), (, 2, 1),
(, 3, 1), (1, , 1), (1, 1, 4), (1, 2, 2), (, 1, 1, 1),
(, 1, 1, 2), (, 2, 1, 1), (1, , 1, 1), (1, 1, 2, 1), and
(, 1, 1, 1, 1) are weighted.
In this section, we propose an algorithm capable to
prove that the type compositions of a given i family
ω with t types are weighted (if true).
4.1 Step 1: SMW Vectors
Consider a complete simple game with type compo-
sition
b
n ∈ ω and matrix S of its SMW vectors. We
aim to write down a finite set of parameterized can-
didates for the rows of S only depending on ω. With
α =

a = (a
1
, . . . , a
i−1
) | 0 ≤ a
j
≤ n
j
, 1 ≤ j ≤ i − 1

we introduce the injective function τ : α → N by
τ(a) =
∑
i−1
j=1
a
i
∏
i−1
k= j+1
(n
k
+ 1), i.e., we are just num-
bering the elements of α in a convenient way. Addi-
tionally we set τ(ω) := τ

(n
1
, . . . , n
i−1
)

+ 1 = |α|.
Lemma 6. Given an i family ω with t types let C(ω)
=

a, m
τ(a)
− c, b

| a ∈ α, b ∈ β, c ∈ N, c ≤ Λ

,
where β =

(b
i+1
, . . . , b
n
) ∈ N
n−i
| b
h
≤ n
h

, Λ =
∑
n
h=i+1
n
h
, and the m
j
are free variables. For each
simple game with type composition
b
n ∈ ω and ma-
trix S = ( ˜s
1
, . . . , ˜s
r
)
T
of its SMW vectors, there exists
an allocation of the m
j
, such that ˜s
h
∈ C(ω) for all
1 ≤ h ≤ r.
ClassesofCompleteSimpleGamesthatareAllWeighted
291

Proof. Given a complete simple game with type com-
position
b
n ∈ ω and matrix S = ( ˜s
1
, . . . , ˜s
r
)
T
of its
SMW vectors. For each a ∈ α let
˜
t
a
= (a, m, b), where
b = (b
i+1
, . . . , b
n
) ∈ β, the row-vector of S whose first
i − 1 coordinates coincide with a and whose ith coor-
dinate m ∈ N is maximal. If
˜
t
a
exists, we set m
τ(a)
= m
and m
τ(a)
= −1 otherwise. Now let ˜s
j
= (a, m
0
, b
0
) be
an arbitrary SMW vector whose first i −1 components
coincide with a. Clearly m
0
∈ N, m
0
≤ m, and b
0
=
(b
0
i+1
, . . . , b
0
n
) ∈ β. If m
0
< m then there exists an index
k ∈ {i +1, . . . , n} with m
0
+
∑
k
h=i+1
b
0
h
> m+
∑
k
h=i+1
b
h
since
∑
i−1
h=1
a
h
+ m
0
<
∑
i−1
h=1
a
h
+ m and
˜
t
a
 ˜s
j
. With
∑
k
h=i+1
b
h
≥ 0 and
∑
k
h=i+1
b
0
h
≤
∑
n
h=i+1
n
h
we have
m
0
> n − Λ.
Thus we can parameterize the potential SMW vec-
tors of a complete simple game attaining ω using at
most τ(ω) parameters as elements in C(ω), where
|C(ω)| ≤

∑
n
j=1, j6=i
n
j

·
∏
n
j=1, j6=i
(n
j
+ 1), i.e., the
number rows r is bounded by ω.
4.2 Step 2: Matrices of All
Shift-minimal Winning Vectors
Given an i family ω with t types, the sets of SMW
vectors of a complete simple game with type com-
position
b
n ∈ ω are subsets of C(ω). Thus, we can
loop over all elements of 2
C(ω)
and need to check
whether the selected subsets satisfy the conditions of
Theorem 1(a). The entries can linearly depend on the
parameters m
j
. In (Kurz and Tautenhahn, 2013) the
similar situation, where all entries s
i, j
are parameters,
has been treated. There it is shown that all feasible
cases, meeting the conditions of Theorem 1(a), can
be formulated as a union of systems of linear inequal-
ity systems in terms of the parameters. Exemplarily,
condition (a)(ii) is satisfied for a pair of indices 1 ≤
i < j ≤ r, if two further indices 1 ≤ h, k ≤ t exist with
∑
h
u=1
s
i,u
+1 ≤
∑
h
u=1
s
j,u
and
∑
k
u=1
s
i,u
≥ 1 +
∑
k
u=1
s
j,u
.
Performing these steps yields a finite list S
1
, S
2
, . . .
of matrices, whose entries are linear functions of the
parameters m
j
, such that the rows of each matrix S
h
are the (parametric) SMW vectors of complete sim-
ple games with type composition in ω whenever the
parameters m
i
satisfy the linear constraints of the cor-
responding polytope P
h
. Moreover, all complete sim-
ple games with type composition in ω are captured by
one of the pairs S
h
, P
h
.
4.3 Step 3: Losing Vectors
For simplicity, we assume that we are given a single
pair (S
h
, P
h
) according to Subsection 4.2. In (Kurz
and Tautenhahn, 2013) the parametric Barvinok al-
gorithm was applied to count the respective number
of complete simple games. Here we want to study
weightedness so that we also need a description (not
necessarily the most compact description) of the set
of losing vectors. To this end, we mention that the
SML vectors of a complete simple game are either in-
comparable to all SMW vectors, and so contained in
C(ω), or arise as so-called shifts of one of the SMW
vectors, i.e., special vectors that have a fairly small
k · k
1
-distance to one of the SMW vectors. Due to
space limitations we just mention, that it is possible
to exactly describe a set of all candidates for SML
vectors, similar as C(ω) for the set of SMW vectors.
Then we can again consider subsets of the set of can-
didates and have to check that the implications, with
respect to be a winning or a losing vector, are non-
contradicting and that the state of each vector can be
deduced in any case. If properly implemented with all
technical details, things boil down to a splitting of a
sub case (S
h
, P
h
) into a finite list of sub sub cases
(S
h
, L
h,1
, P
h,1
), (S
h
, L
h,2
, P
h,2
), . . . ,
where the rows of the L
h, j
correspond to (not neces-
sarily shift-maximal) losing vectors and the P
h, j
are
sub polytopes of P
h
.
4.4 Step 4: Weighted Representation
For simplicity, we assume that we are given a single
triple Γ =

S
h
, L
h
, P
h

according to Subsection 4.3.
For each integral choice of the parameters m
j
in P
h
,
we have a unique complete simple game at hand and
can check whether it is weighted with the help of a
linear program, see e.g. (Taylor and Zwicker, 1999).
If at least one of such games is non-weighted, then
we can use the methods of Section 3 to deduce that
a certain class of type compositions is non-weighted.
So let us assume that all games corresponding to Γ are
indeed weighted.
Thus, for each (of the possibly infinitely many)
complete simple games corresponding to Γ, there ex-
ist feasible weights obtained at a basis solution of the
corresponding linear program, which is uniquely de-
termined by Γ but depends on the parameters m
i
. Nev-
ertheless, the number of possible basis solutions is fi-
nite. Thus, we can loop over all possible paramet-
ric basis solutions δ
j
and determine the correspond-
ing list of polytopes P
h,k
, such that δ
j
yields a feasible
weighting of the complete simple games correspond-
ing to S
h
, L
h
whenever the parameters m
l
are in P
h,k
.
If ∪
k
P
h,k
= P
h
, then ω is weighted for sub case Γ.
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292

4.5 Examples
In the previous four subsections we have sketched an
algorithm that is capable to prove that a given i fam-
ily ω with t types is weighted (if the statement is in-
deed true). Due to space limitations, we have not
given all technical, sometimes non-trivially, details.
Instead, we want to give examples for special cases.
Lemma 7. ω = () is weighted.
Proof. According to Step 1 we have τ(ω) = 1 param-
eter m
0
, Λ = 0, and C(ω) = {(m
0
)} with |C(ω)| = 1.
Since each complete simple game consists of a least
one shift-minimal winning coalition we obtain the
one-element list

S
1
= (m
0
), P
1
= {(m
0
) ∈ R
1
| 1 ≤
m
0
≤ n
1
}

. Since all elements of C(ω) have already
been assigned to be shift-minimal winning, there re-
mains the unique shifted vector (m
0
− 1) to be losing
in Step 3. In Step 4, we can obtain the basis solution
q = m
0
, w
1
= 1, which is feasible for the entire poly-
tope P
1
. (We only state the weights for each type of
agents, numbered from 1 to t.)
Lemma 8. ω = (1, ) is weighted.
Proof. According to Step 1 we have τ(ω) = 2 param-
eters m
0
, m
1
, Λ = 0, and C(ω) = {(0, m
0
), (1, m
1
)}
with |C(ω)| = 2. In Step 2 we have to consider
the three non-empty subsets of C(ω). For the case
S =

0 m
0

we observe that condition (a)(iii) of
Theorem 1 can not be met for any allocation of the
parameters m
0
, m
1
. Thus, there remain only two cases
with non-empty polytopes for the parameters:
S
1
=

1 m
1

, P
1
=
{
(m
1
) ∈ R | 0 ≤ m
1
≤ n
2
− 1
}
and
S
2
=

1 m
1
0 m
0

, P
2
=




m
0
m
1

∈R
2
|
m
1
≥ 0
m
1
+ 2 ≤ m
0
m
0
≤ n
2



.
In Step 3, each of the two sub cases is split
into two sub sub cases Γ
1
= (S
1
, L
1,1
, P
1,1
),
Γ
2
= (S
1
, L
1,2
, P
1,2
), Γ
3
= (S
2
, L
2,1
, P
2,1
),
Γ
3
= (S
2
, L
2,2
, P
2,2
), with L
1,1
=

1 m
1
− 1
0 n
2

,
L
1,2
=

0 n
2

, L
2,1
=

1 m
1
− 1
0 m
0
− 1

,
L
2,2
=

0 m
0
− 1

,
P
1,1
=
{
(m
1
)∈R | m
1
≥ 1
}
∩ P
1
,
P
1,2
=
{
(m
1
)∈R | m
1
= 0
}
∩ P
1
,
P
2,1
=

(m
0
, m
1
)∈R
2
|m
1
≥ 1

∩ P
2
,
P
2,2
=

(m
0
, m
1
)∈R
2
|m
1
= 0

∩ P
2
.
In Step 4, fortunately, no further splitting is neces-
sary, and we can even condense sub cases. For Γ
1
, Γ
2
we have the weighted representation [n
2
+ 1; n
2
+ 1 −
m
1
, 1] and for Γ
3
, Γ
4
we have the weighted represen-
tation [m
0
;m
0
− m
1
, 1].
Using the parametric Barvinok algorithm or ele-
mentary summation formulas with case differentia-
tion, we conclude the well known fact that the number
of n-agent weighted games with type composition ()
is n. Similarly, we conclude, that the number of n-
agent weighted games with type composition (1, ) is
given by
n
3
−n
6
=
n(n−1)(n+1)
6
=

n+1
3

for all n ∈ N.
5 FUTURE WORK
Conjecture 1 has to be proven in the first run. Next,
one can ask for a similar classification for roughly-
weighted games, i.e., for games where the weight of a
winning coalition can equal the weight of another los-
ing coalition, or for complete simple games of small
dimension, i.e., games which can be represented as
the intersection of a small number of weighted games.
A possibly more challenging open problem is to pro-
vide explicit enumeration results for other sub classes
of weighted games.
ACKNOWLEDGEMENTS
The authors thank the two anonymous referees for
their helpful comments and suggestions on the paper.
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293