Finding extremal voting games via integer linear programming fileFinding extremal voting games via integer linear programming Sascha Kurz Business mathematics University of Bayreuth

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


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}.


Binary voting games





({i} ∪ S\{j}

)≥ χ

(S)



Binary voting games





({i} ∪ S\{j}

)≥ χ

(S)



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].


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].


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}

{}


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


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 |


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 |


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)


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?


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?


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.


Using duality

General linear program

max cT xAx ≤ bx ≥ 0

... as a feasibility problem

cT x = bT yAx ≤ bAT y ≥ cx ≥ 0y ≥ 0


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)


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?


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?


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


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.

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Finding extremal voting games via integer linear programming fileFinding extremal voting games via integer linear programming Sascha Kurz Business mathematics University of Bayreuth