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Finding extremal voting games viainteger linear programming
Sascha KurzBusiness mathematicsUniversity of [email protected]
Euro 2012 – Vilnius
Finding extremal voting games via integer linear programming Introduction 1 / 18
Voting games and their properties
Finding extremal voting games via integer linear programming Introduction 2 / 18
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0,1}, where 2N denotes the set of subsetsof N := {1,2, . . . ,n}.
Definition – Simple gameBinary voting game χ with χ(∅) = 0, χ(N) = 1, and χ(S) ≤ χ(T )for all S ⊆ T .
Isbell’s desirability relationi A j for two voters i , j ∈ N if and only if χ
({i} ∪ S\{j}
)≥ χ
(S)
for all {j} ⊆ S ⊆ N\{i}.
Finding extremal voting games via integer linear programming Introduction 2 / 18
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0,1}, where 2N denotes the set of subsetsof N := {1,2, . . . ,n}.
Definition – Simple gameBinary voting game χ with χ(∅) = 0, χ(N) = 1, and χ(S) ≤ χ(T )for all S ⊆ T .
Isbell’s desirability relationi A j for two voters i , j ∈ N if and only if χ
({i} ∪ S\{j}
)≥ χ
(S)
for all {j} ⊆ S ⊆ N\{i}.
Finding extremal voting games via integer linear programming Introduction 2 / 18
Binary voting games
Definition – Binary voting games
A mapping χ : 2N → {0,1}, where 2N denotes the set of subsetsof N := {1,2, . . . ,n}.
Definition – Simple gameBinary voting game χ with χ(∅) = 0, χ(N) = 1, and χ(S) ≤ χ(T )for all S ⊆ T .
Isbell’s desirability relationi A j for two voters i , j ∈ N if and only if χ
({i} ∪ S\{j}
)≥ χ
(S)
for all {j} ⊆ S ⊆ N\{i}.
Finding extremal voting games via integer linear programming Introduction 3 / 18
Binary voting gamesDefinition – Complete simple gameSimple 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 gameSimple game (or complete simple game) χ such that there areweights wi ∈ R≥0 for all i ∈ N and a quota q ∈ R>0 satisfying∑
i∈S wi ≥ q exactly if χ(S) = 1, where ∅ ⊆ S ⊆ N.Notation: [q;w1,w2, . . . ,wn].
Finding extremal voting games via integer linear programming Introduction 3 / 18
Binary voting gamesDefinition – Complete simple gameSimple 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 gameSimple game (or complete simple game) χ such that there areweights wi ∈ R≥0 for all i ∈ N and a quota q ∈ R>0 satisfying∑
i∈S wi ≥ q exactly if χ(S) = 1, where ∅ ⊆ S ⊆ N.Notation: [q;w1,w2, . . . ,wn].
Finding extremal voting games via integer linear programming Introduction 4 / 18
Example of a weighted voting game
Winning (χ(S) = 1) and losing (χ(S) = 0) coalitions of theweighted voting game [4;3,2,1,1]:
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{1, 2, 3, 4}
{1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4}
{1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4}
{1} {2} {3} {4}
{}
Finding extremal voting games via integer linear programming Introduction 5 / 18
Enumeration results
n 1 2 3 4 5 6 7 8 9#S 1 3 8 28 208 16351 >4.7 · 108 >1.3 · 1018 >2.7 · 1036
#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, andweighted voting games up to symmetry, i.e. orbits under the symmetricgroup on n elements.
Finding extremal voting games via integer linear programming ILP formulation 6 / 18
An integer linear programmingformulation 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 7 / 18
How to measure powerShapley-Shubik power index (there are more power indices)
The power of a player is measured by the fraction of the possiblevoting sequences in which that player casts the deciding vote,that is, the vote that first guarantees passage or failure. Thepower index is normalized between 0 and 1.
Example: A→ 3, B → 2, C → 1, D → 1, quota= 4ABCD ABDC ACBD ACDB ADBC ADCBBACD BADC BCAD BCDA BDAC BDCACABD CADB CBAD CBDA CDAB CDBADABC DACB DBAC DBCA DCAB DCBA
Pow(A) = 1224 , Pow(B) = 4
24 , Pow(C) = 424 , Pow(D) = 4
24
Finding extremal voting games via integer linear programming Inverse Power Index Problem 8 / 18
Inverse Power Index ProblemProblem 1Find a voting game whose Shapley-Shubik vector has minimalL1-distance to a given ideal power distribution σ, e.g.
σn =
(2
2n − 1,
22n − 1
, . . . ,2
2n − 1,
12n − 1
).
ILP approachI 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 iI di : |pi − σi |
Finding extremal voting games via integer linear programming Inverse Power Index Problem 8 / 18
Inverse Power Index ProblemProblem 1Find a voting game whose Shapley-Shubik vector has minimalL1-distance to a given ideal power distribution σ, e.g.
σn =
(2
2n − 1,
22n − 1
, . . . ,2
2n − 1,
12n − 1
).
ILP approachI 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 iI di : |pi − σi |
Finding extremal voting games via integer linear programming Inverse Power Index Problem 9 / 18
An ILP formulation for the inverseShapley-Shubik problem
minn∑
i=1
di (5)
s.t. σi − di ≤ pi ≤ σi + di ∀i ∈ N, (6)
pi =∑
S∈N\{i}wS · yi,S ∀i ∈ N, (7)
yi,S = xS∪{i} − xS ∀i ∈ N, S ⊆ N\{i}, (8)
xS ≥ xS\{j} ∀∅ 6= S ⊆ N, j ∈ S, (9)
x∅ = 0 (10)
xN = 1 (11)
xS ∈ {0, 1} ∀S ⊆ N, (12)
yi,S ∈ {0, 1} ∀i ∈ N, S ⊆ N\{i}, (13)
di , pi ≥ 0 ∀i ∈ N. (14)
Finding extremal voting games via integer linear programming Inverse Power Index Problem 10 / 18
Results – Minimal deviations
n ‖ · ‖11 0.0000002 0.3333333 0.2666674 0.2142865 0.1444446 0.0969707 0.0842498 0.7619059 0.658263
10 0.05438611 0.05021612 0.04637713 0.04293714 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 11 / 18
α-roughly weighted gamesDefinition (Gvozdeva, L. Hemaspaandra, Slinko; 2012)A simple game χ is α-roughly weighted if there are weightswi ∈ R≥0 such that(1)
∑i∈S wi ≥ 1 if χ(S) = 1;
(2)∑
i∈S wi ≤ α if χ(S) = 0for all ∅ ⊆ S ⊆ N.The critical threshold value µ(χ) of χ is the minimum value α ≥ 1such that χ is α-roughly weighted.
Problem 2What is the maximal critical threshold value of a complete simplegame on n voters?
Finding extremal voting games via integer linear programming α-roughly weighted games 11 / 18
α-roughly weighted gamesDefinition (Gvozdeva, L. Hemaspaandra, Slinko; 2012)A simple game χ is α-roughly weighted if there are weightswi ∈ R≥0 such that(1)
∑i∈S wi ≥ 1 if χ(S) = 1;
(2)∑
i∈S wi ≤ α if χ(S) = 0for all ∅ ⊆ S ⊆ N.The critical threshold value µ(χ) of χ is the minimum value α ≥ 1such that χ is α-roughly weighted.
Problem 2What is the maximal critical threshold value of a complete simplegame on n voters?
Finding extremal voting games via integer linear programming α-roughly weighted games 12 / 18
LP formulation for the critical thresholdvalue
µ(χ) = min αw(S) ≥ 1 ∀S ⊆ N : χ(S) = 1w(S) ≤ α ∀S ⊆ N : χ(S) = 0α ≥ 1w1, . . . ,wn ∈ R≥0
RemarkThe conditions can be restricted to minimal winning and maximallosing coalitions.
Finding extremal voting games via integer linear programming α-roughly weighted games 13 / 18
Using duality
General linear program
max cT xAx ≤ bx ≥ 0
... as a feasibility problem
cT x = bT yAx ≤ bAT y ≥ cx ≥ 0y ≥ 0
Finding extremal voting games via integer linear programming α-roughly weighted games 14 / 18
An ILP approach for the determination ofthe maximum critical threshold value
max∑S⊆N
uS (15)
x∅ = 1− xN = 0 (16)xS − xS\{i} ≥ 0 ∀∅ 6= S ⊆ N, i ∈ S (17)∑
{i}⊆S⊆N
uS −∑
{i}⊆T⊆N
vT ≤ 0 ∀i ∈ N (18)
∑T⊆N
vT ≤ 1 (19)
uS − xS ≤ 0 ∀S ⊆ N (20)vT + xT ≤ 1 ∀T ⊆ N (21)
xS ∈ {0,1} ∀S ⊆ N (22)uS, vS ≥ 0 ∀S ⊆ N (23)
Finding extremal voting games via integer linear programming α-roughly weighted games 15 / 18
The maximum critical threshold values(n) of a complete simple game
n s(n)1-6 1
7 87
8 2621
9 43
10 3827
11 2215
12 149
13 3320
14 11164
15 12368
16 158
Finding extremal voting games via integer linear programming Local monotonicity of the PGI 16 / 18
Local monotonicity of the Public GoodIndex
Definition – PGIGiven a simple game χ. A coalition S called minimal winningcoalition if χ(S) = 1 but all proper subsets of S are losing. ByMWi we denote the number of minimal winning coalitionscontaining player i . The Public Good Index (PGI) for player i isgiven by
PGI(i) =MWi∑
j∈NMWj
.
Problem 3Let 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 16 / 18
Local monotonicity of the Public GoodIndex
Definition – PGIGiven a simple game χ. A coalition S called minimal winningcoalition if χ(S) = 1 but all proper subsets of S are losing. ByMWi we denote the number of minimal winning coalitionscontaining player i . The Public Good Index (PGI) for player i isgiven by
PGI(i) =MWi∑
j∈NMWj
.
Problem 3Let 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 17 / 18
An ILP formulation
VariablesyS ∈ {0,1} with yS = 1 is coalition ∅ ⊆ S ⊆ N is a minimalwinning coalition.
Inequalities
yS ≤ xS ∀S ⊆ NyS ≥ 1− XS\{min i:i∈S} ∀∅ 6= S ⊆ NyS ≤ 1− XS\{i} ∀∅ 6= S ⊆ N, i ∈ SyS ∈ {0,1} ∀∅ 6= S ⊆ NMWi =
∑{i}⊆S⊆N} yS ∀i ∈ N
Finding extremal voting games via integer linear programming Local monotonicity of the PGI 18 / 18
Thank you very much for your attention!
ConclusionWhenever a certain class of voting games should be analyzedconcerning a certain parameter, the extremal values may beobtained using integer linear programming techniques –enlarging the scope of exhaustive search methods.
Proposals are highly welcome.