Finding extremal voting games via integer linear programming

Transcription

1 Finding extremal voting games via integer linear programming Sascha Kurz Business mathematics University of Bayreuth Euro 2012 Vilnius

2 Finding extremal voting games via integer linear programming Introduction 1 / 18 Voting games and their properties

3 Finding extremal voting games via integer linear programming Introduction 2 / 18 Definition Binary voting games Binary voting games A mapping χ : 2 N {0, 1}, where 2 N denotes the set of subsets of N := {1, 2,..., n}.

4 Finding extremal voting games via integer linear programming Introduction 2 / 18 Definition Binary voting games Binary voting games A mapping χ : 2 N {0, 1}, where 2 N 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.

5 Finding extremal voting games via integer linear programming Introduction 2 / 18 Definition Binary voting games Binary voting games A mapping χ : 2 N {0, 1}, where 2 N 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 j for two voters i, j N if and only if χ ( {i} S\{j} ) χ ( S ) for all {j} S N\{i}.

6 Finding extremal voting games via integer linear programming Introduction 3 / 18 Definition Complete simple game Binary voting games Simple game χ where the binary relation is a total preorder, i.e. (1) i i for all i N, (2) i j or j i (including i j and j i ) for all i, j N, and (3) i j, j h implies i h for all i, j, h N.

7 Finding extremal voting games via integer linear programming Introduction 3 / 18 Definition Complete simple game Binary voting games Simple game χ where the binary relation is a total preorder, i.e. (1) i i for all i N, (2) i j or j i (including i j and j i ) for all i, j N, and (3) i j, j h implies i h for all i, j, h N. Definition Weighted voting game Simple game (or complete simple game) χ such that there are weights w i R 0 for all i N and a quota q R >0 satisfying i S w i q exactly if χ(s) = 1, where S N. Notation: [q; w 1, w 2,..., w n ].

8 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 the weighted voting game [4; 3, 2, 1, 1]: {1, 2, 3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {}

9 Finding extremal voting games via integer linear programming Introduction 5 / 18 Enumeration results n #S > > > #C #W 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.

10 Finding extremal voting games via integer linear programming ILP formulation 6 / 18 An integer linear programming formulation of a simple game x = 0 (1) x N = 1 (2) x S x T S T N (3) x S {0, 1} S N (4)

11 Finding extremal voting games via integer linear programming Inverse Power Index Problem 7 / 18 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 ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA Pow(A) = 12 24, Pow(B) = 4 24, Pow(C) = 4 24, Pow(D) = 4 24

12 Finding extremal voting games via integer linear programming Inverse Power Index Problem 8 / 18 Problem 1 Inverse Power Index Problem Find a voting game whose Shapley-Shubik vector has minimal L 1 -distance to a given ideal power distribution σ, e.g. ( ) 2 σ n = 2n 1, 2 2n 1,..., 2 2n 1, 1. 2n 1

13 Finding extremal voting games via integer linear programming Inverse Power Index Problem 8 / 18 Problem 1 Inverse Power Index Problem Find a voting game whose Shapley-Shubik vector has minimal L 1 -distance to a given ideal power distribution σ, e.g. ( ) 2 σ n = 2n 1, 2 2n 1,..., 2 2n 1, 1. 2n 1 ILP approach w S = ( S! (n S 1)!)/n! (constants for the Shapley-Shubik index) x S {0, 1}: is coalition S N winning? y i,s {0, 1}: is coalition S a swing for voter i? p i : Shapley-Shubik power for voter i d i : p i σ i

14 Finding extremal voting games via integer linear programming Inverse Power Index Problem 9 / 18 An ILP formulation for the inverse Shapley-Shubik problem n min d i (5) i=1 s.t. σ i d i p i σ i + d i i N, (6) p i = w S y i,s i N, (7) S N\{i} y i,s = x S {i} x S i N, S N\{i}, (8) x S x S\{j} S N, j S, (9) x = 0 (10) x N = 1 (11) x S {0, 1} S N, (12) y i,s {0, 1} i N, S N\{i}, (13) d i, p i 0 i N. (14)

15 Finding extremal voting games via integer linear programming Inverse Power Index Problem 10 / 18 Results Minimal deviations n Tabelle: Optimal deviation for σ n in the set of complete simple games.

16 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 w i R 0 such that (1) i S w i 1 if χ(s) = 1; (2) i S w i 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.

17 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 w i R 0 such that (1) i S w i 1 if χ(s) = 1; (2) i S w i 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?

18 Finding extremal voting games via integer linear programming α-roughly weighted games 12 / 18 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 w 1,..., w n R 0 Remark The conditions can be restricted to minimal winning and maximal losing coalitions.

19 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 = b T y Ax b A T y c x 0 y 0

20 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 {i} S N max u S (15) S N x = 1 x N = 0 (16) x S x S\{i} 0 S N, i S (17) u S v T 0 i N (18) {i} T N v T 1 (19) T N u S x S 0 S N (20) v T + x T 1 T N (21) x S {0, 1} S N (22) u S, v S 0 S N (23)

21 Finding extremal voting games via integer linear programming α-roughly weighted games 15 / 18 The maximum critical threshold value s(n) of a complete simple game n s(n)

22 Finding extremal voting games via integer linear programming Local monotonicity of the PGI 16 / 18 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 MW i we denote the number of minimal winning coalitions containing player i. The Public Good Index (PGI) for player i is given by PGI(i) = MW i. MW j j N

23 Finding extremal voting games via integer linear programming Local monotonicity of the PGI 16 / 18 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 MW i we denote the number of minimal winning coalitions containing player i. The Public Good Index (PGI) for player i is given by PGI(i) = MW i. MW j j N Problem 3 Let i j in a complete simple game χ. How large can MW i MW j be?

24 Finding extremal voting games via integer linear programming Local monotonicity of the PGI 17 / 18 An ILP formulation Variables y S {0, 1} with y S = 1 is coalition S N is a minimal winning coalition. Inequalities y S x S S N y S 1 X S\{min i:i S} S N y S 1 X S\{i} S N, i S y S {0, 1} S N MW i = {i} S N} y S i N

25 Finding extremal voting games via integer linear programming Local monotonicity of the PGI 18 / 18 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.