Finding extremal voting games via
integer linear programming
Sascha Kurz
(Joint work with Josep Freixas)
Business mathematics
University of Bayreuth
sascha.kurz@uni-bayreuth.de
Dagstuhl 2012 – Computation and Incentives in Social Choice
Universitat Politècnica de Catalunya
Finding extremal voting games via integer linear programming
Introduction
Voting games and their properties
1 / 18
Finding extremal voting games via integer linear programming
Introduction
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0, 1}, where 2N denotes the set of subsets
of N := {1, 2, . . . , n}.
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Finding extremal voting games via integer linear programming
Introduction
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0, 1}, where 2N denotes the set of subsets
of N := {1, 2, . . . , n}.
Definition – Simple game
Binary voting game χ with χ(∅) = 0, χ(N) = 1, and χ(S) ≤ χ(T )
for all S ⊆ T .
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Finding extremal voting games via integer linear programming
Introduction
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0, 1}, where 2N denotes the set of subsets
of N := {1, 2, . . . , n}.
Definition – Simple game
Binary voting game χ with χ(∅) = 0, χ(N) = 1, and χ(S) ≤ χ(T )
for all S ⊆ T .
Isbell’s desirability relation
i A j for two voters i, j ∈ N if and only if χ {i} ∪ S\{j} ≥ χ S
for all {j} ⊆ S ⊆ N\{i}.
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Finding extremal voting games via integer linear programming
Introduction
Binary voting games
Definition – Complete simple game
Simple game χ where 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 (including “i A j and 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.
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Finding extremal voting games via integer linear programming
Introduction
Binary voting games
Definition – Complete simple game
Simple game χ where 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 (including “i A j and 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.
Definition – Weighted voting game
Simple game (or complete simple game) χ such that there are
weights wi ∈ R≥0 for all i ∈ N and a quota q ∈ R>0 satisfying
P
i∈S wi ≥ q exactly if χ(S) = 1, where ∅ ⊆ S ⊆ N.
Notation: [q; w1 , w2 , . . . , wn ].
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Finding extremal voting games via integer linear programming
Introduction
Example of a weighted voting game
Winning (χ(S) = 1) and losing (χ(S) = 0) coalitions of the
weighted voting game [4; 3, 2, 1, 1]:
{1, 2}
{1, 2, 3, 4}
tP
PP
@
P
{1, 2, 4} @ {1,P
{1, 2, 3} t
@X
tX3, 4} PPt{2, 3, 4}
tX
X
X
X
X
X
X
XX X
XHH
XX
XXX
XXXH
X
X
XH
XXt
XXt{2, 4}
X
{1, 3} tX
{1, 4} X
t
t
Xt
H
{2, 3} X
{3, 4}
X
X
HX
X
X
X
XXX X
HHXXX X
X
X
X
X X
XX
HH
XX
X
X
Xt
t X{2}
t
t
{1} PP
{3}
PP @
{4}
@
PP
@t
P
{}
4 / 18
Finding extremal voting games via integer linear programming
Introduction
5 / 18
Enumeration results
n123 4
5
6
7
8
9
108
1018
1036
#S 1 3 8 28 208 16351 > 4.7 ·
> 1.3 ·
> 2.7 ·
#C 1 3 8 25 117 1171
44313 16175188 284432730174
#W 1 3 8 25 117 1111
29373 2730164
989913344
Tabelle: Number of distinct simple games, complete simple games, and
weighted voting games up to symmetry, i.e. orbits under the symmetric
group on n elements.
Finding extremal voting games via integer linear programming
ILP formulation
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An integer linear programming
formulation of a simple game
x∅ = 0
(1)
xN = 1
(2)
xS ≤ xT
∀S ⊆ T ⊆ N
(3)
xS ∈ {0, 1}
∀S ⊆ N
(4)
Finding extremal voting games via integer linear programming
Inverse Power Index Problem
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How to measure power
Shapley-Shubik power index (there are more power indices)
The power of a player is measured by the fraction of the possible
voting sequences in which that player casts the deciding vote,
that is, the vote that first guarantees passage or failure. The
power index is normalized between 0 and 1.
Example: A → 3, B → 2, C → 1, D → 1, quota= 4
ABCD
BACD
CABD
DABC
Pow(A) =
ABDC
BADC
CADB
DACB
12
24 ,
ACBD
BCAD
CBAD
DBAC
Pow(B) =
4
24 ,
ACDB
BCDA
CBDA
DBCA
ADBC
BDAC
CDAB
DCAB
Pow(C) =
4
24 ,
ADCB
BDCA
CDBA
DCBA
Pow(D) =
4
24
Finding extremal voting games via integer linear programming
Inverse Power Index Problem
Inverse Power Index Problem
Problem 1
Find a voting game whose Shapley-Shubik vector has minimal
L1 -distance to a given ideal power distribution σ, e.g.
2
2
2
1
σn =
,
,...,
,
.
2n − 1 2n − 1
2n − 1 2n − 1
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Finding extremal voting games via integer linear programming
Inverse Power Index Problem
Inverse Power Index Problem
Problem 1
Find a voting game whose Shapley-Shubik vector has minimal
L1 -distance to a given ideal power distribution σ, e.g.
2
2
2
1
σn =
,
,...,
,
.
2n − 1 2n − 1
2n − 1 2n − 1
ILP approach
I
wS = (|S|! · (n − |S| − 1)!)/n! (constants for the
Shapley-Shubik index)
I
xS ∈ {0, 1}: is coalition S ⊆ N winning?
I
yi,S ∈ {0, 1}: is coalition S a swing for voter i?
I
pi : Shapley-Shubik power for voter i
I
di : |pi − σi |
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Finding extremal voting games via integer linear programming
Inverse Power Index Problem
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An ILP formulation for the inverse
Shapley-Shubik problem
min
n
X
di
(5)
i=1
s.t.
σi − di ≤ pi ≤ σi + di
X
pi =
wS · yi,S
∀i ∈ N,
(6)
∀i ∈ N,
(7)
yi,S = xS∪{i} − xS
∀i ∈ N, S ⊆ N\{i},
(8)
xS ≥ xS\{j}
∀∅ =
6 S ⊆ N, j ∈ S,
S∈N\{i}
x∅ = 0
xN = 1
xS ∈ {0, 1}
yi,S ∈ {0, 1}
di , pi ≥ 0
(9)
(10)
(11)
∀S ⊆ N,
(12)
∀i ∈ N, S ⊆ N\{i},
(13)
∀i ∈ N.
(14)
Finding extremal voting games via integer linear programming
Inverse Power Index Problem
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Results – Minimal deviations
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
k · k1
0.000000
0.333333
0.266667
0.214286
0.144444
0.096970
0.084249
0.761905
0.658263
0.054386
0.050216
0.046377
0.042937
0.037759
Tabelle: Optimal deviation for σn in the set of complete simple games.
Finding extremal voting games via integer linear programming
α-roughly weighted games
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α-roughly weighted games
Definition (Gvozdeva, L. Hemaspaandra, Slinko; 2012)
A simple game χ is α-roughly weighted if there are weights
wi ∈ R≥0 such that
P
(1)
wi ≥ 1 if χ(S) = 1;
Pi∈S
(2)
i∈S wi ≤ α if χ(S) = 0
for all ∅ ⊆ S ⊆ N.
The critical threshold value µ(χ) of χ is the minimum value α ≥ 1
such that χ is α-roughly weighted.
Finding extremal voting games via integer linear programming
α-roughly weighted games
11 / 18
α-roughly weighted games
Definition (Gvozdeva, L. Hemaspaandra, Slinko; 2012)
A simple game χ is α-roughly weighted if there are weights
wi ∈ R≥0 such that
P
(1)
wi ≥ 1 if χ(S) = 1;
Pi∈S
(2)
i∈S wi ≤ α if χ(S) = 0
for all ∅ ⊆ S ⊆ N.
The critical threshold value µ(χ) of χ is the minimum value α ≥ 1
such that χ is α-roughly weighted.
Problem 2
What is the maximal critical threshold value of a complete simple
game on n voters?
Finding extremal voting games via integer linear programming
α-roughly weighted games
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LP formulation for the critical threshold
value
µ(χ) = min α
w(S) ≥ 1 ∀S ⊆ N : χ(S) = 1
w(S) ≤ α ∀S ⊆ N : χ(S) = 0
α≥1
w1 , . . . , wn ∈ R≥0
Remark
The conditions can be restricted to minimal winning and maximal
losing coalitions.
Finding extremal voting games via integer linear programming
α-roughly weighted games
13 / 18
Using duality
General linear program
max c T x
Ax ≤ b
x ≥0
... as a feasibility problem
c T x = bT y
Ax ≤ b
AT y ≥ c
x ≥0
y ≥0
Finding extremal voting games via integer linear programming
α-roughly weighted games
14 / 18
An ILP approach for the determination of
the maximum critical threshold value
max
X
uS
(15)
S⊆N
x∅ = 1 − xN
X
{i}⊆S⊆N
= 0
xS − xS\{i} ≥ 0 ∀∅ =
6 S ⊆ N, i ∈ S
X
uS −
vT ≤ 0 ∀i ∈ N
(16)
(17)
(18)
{i}⊆T ⊆N
X
vT
T ⊆N
uS − xS
≤ 1
(19)
≤ 0 ∀S ⊆ N
(20)
vT + xT
≤ 1 ∀T ⊆ N
(21)
∈ {0, 1} ∀S ⊆ N
(22)
xS
uS , vS ≥ 0 ∀S ⊆ N
(23)
Finding extremal voting games via integer linear programming
α-roughly weighted games
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The maximum critical threshold value
s(n) of a complete simple game
n
1-6
s(n)
1
7
8
9
10
11
12
13
14
15
16
8
7
26
21
4
3
38
27
22
15
14
9
33
20
111
64
123
68
15
8
Finding extremal voting games via integer linear programming
Local monotonicity of the PGI
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Local monotonicity of the Public Good
Index
Definition – PGI
Given a simple game χ. A coalition S called minimal winning
coalition if χ(S) = 1 but all proper subsets of S are losing. By
MWi we denote the number of minimal winning coalitions
containing player i. The Public Good Index (PGI) for player i is
given by
MW
PGI(i) = P i .
MWj
j∈N
Finding extremal voting games via integer linear programming
Local monotonicity of the PGI
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Local monotonicity of the Public Good
Index
Definition – PGI
Given a simple game χ. A coalition S called minimal winning
coalition if χ(S) = 1 but all proper subsets of S are losing. By
MWi we denote the number of minimal winning coalitions
containing player i. The Public Good Index (PGI) for player i is
given by
MW
PGI(i) = P i .
MWj
j∈N
Problem 3
Let i A j in a complete simple game χ.
How large can MWi − MWj be?
Finding extremal voting games via integer linear programming
Local monotonicity of the PGI
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An ILP formulation
Variables
yS ∈ {0, 1} with yS = 1 is coalition ∅ ⊆ S ⊆ N is a minimal
winning coalition.
Inequalities
yS ≤ xS ∀S ⊆ N
yS ≥ 1 − XS\{min i:i∈S} ∀∅ =
6 S⊆N
yS ≤ 1 − XS\{i} ∀∅ =
6 S ⊆ N, i ∈ S
yS ∈ {0,P
1} ∀∅ =
6 S⊆N
MWi = {i}⊆S⊆N} yS ∀i ∈ N
Finding extremal voting games via integer linear programming
Local monotonicity of the PGI
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Thank you very much for your attention!
Conclusion
Whenever a certain class of voting games should be analyzed
concerning a certain parameter, the extremal values may be
obtained using integer linear programming techniques –
enlarging the scope of exhaustive search methods.
Proposals are highly welcome.