Power distribution: theory and applications

1. Power distribution: theory and applications Fuad Aleskerov State University “Higher School of Economics” and Institute of Control Sciences Moscow, Russia alesk@hse.ru, alesk@ipu.ru

2. EXAMPLE Parliament with 99 seats 3 parties: A – 33 seats, B – 33 seats, C – 33 seats. Decision rule – simple majority, i.e. 50 votes. Winning coalitions are А+В, А+С, В+С, А+В+С

3. Another example Distribution of seats has changed: A and B have 48 votes each, C has 3 votes. However, winning coalitions are the same, i.e. each party can equally influence an outcome.

4. Banzhaf index If is the number of coalitions in which party iis pivotal, then Banzhaf index for iis evaluated as follows:

5. EXAMPLE Parliament with 100 seats, and 3 parties A, B, С with 50, 49 и 1, resp. Decision rule is a simple majority one. Then winning coalitions are A+В, A+С, A+B+С.

6. Example (continued) Then Banzhaf index for А which is pivotal in each three coalitions is evaluated as follows Similarly, for В and С, each of which is pivotal in only one coalition, one can obtain

7. Voting power: another example • European Economic Community (1958-1972) • Belgium (2 votes), France (4), Italy (4), Luxembourg (1), Netherlands(2), West Germany (4). • «Power» (%, wrt West Germany): • Belgium 50% • Luxembourg 25% • Population (%, wrt West Germany): • Belgium ~16.7% • Luxembourg ~0.6% • Decision-making threshold: 12 votes • Actual (formal) power of Luxembourg is 0 • Luxembourg could only be decisive if the combined total of the votes cast by the other five members was 11 • Impossible since they were all even numbers

9. Power distribution in the 3d Duma

10. What if not coalitions are posible? Three parties А, В and С, distribution of seats: A - 50, В – 49 and С - 1. Parties А and В do not coalesce. Then, if grand coalition is admissible ; ; .

11. Consistency index C is equal to 1, if positions of groups coincide (q1= q2), and equal to 0, if positions are opposite (e.g., q1=0 иq2=1).

12. Consistency of key pairs of factions in the third Duma(Communists_Edinstvo, Edinstvo_OVR, SPS_Yabloko, Communists_Agrariants)

13. Power distribution of large factions (Communists, Edinstvo, Narodnyi Deputat),scenario 0.4

14. Power distribution of small factions (SPS, Liberal-Democrats, Yabloko), scenario 0,4

15. Consistency of factions in the votings related to the authority issue for SPS with Communists and Edinstvo

16. Consistency of factions in the votings related to the authority issue for Edinstvo with Communists and OVR

17. Power distribution on the authority issue for factionEdinstvo and Communists, scenario 0.4

18. Trajectory of largest group of MPs of Communists belonging to one cluster

19. Trajectory of largest group of MPs of Edinstvo belonging to one cluster

20. Trajectory of largest group of MPs of Yabloko belonging to one cluster

21. Trajectory of largest group of MPs of SPS belonging to one cluster

22. Trajectory of largest group of MPs of Communists and Yabloko belonging to one cluster

23. Consistency Index on Political Map

24. Shapley-Owen index The power index for player i where qi is the number of orderings, for which player i is pivotal, n! is the total number of all possible orderings.

25. Shapley-Owen index

26. Extension The average value of i’s weight The power index of player i where is the share of votes, and is the number of votes of party i.

27. Extended power index values for third Duma (Edinstvo, CPRF)

28. Ordinal and cardinal indices

29. Ordinal indices

30. Cardinal intensity functions

33. Axiomatic construction of a cardinal intensity function

34. Axioms which reasonable function should satisfy to.

36. Axioms for power indices defined on the games with preferences

37. Axioms for power indices defined on the games with preferences

38. Axioms for power indices defined on the games with preferences

39. Axioms for power indices defined on the games with preferences

40. Results

41. Axioms for normalized indices

42. Applications • IMF • Russian banks

43. Modelling preference of country ito coalesce with j • Modification 1 (Aleskerov, Kalyagin &Pogorelskiy (2008)) • Regional proximity of country j (Er(i,j), weight Wr=0.35) • Membership of the pair of countries iand jin the international political-economic blocs outside the IMF(Eb(i,j), weight Wb=0.65) • Overall intensity pij is defined by a generalized criterion

44. Modelling preference of country ito coalesce with j • Modification 2 (Aleskerov, Kalyagin &Pogorelskiy (2009)) • Bilateral trade with jas compared with the rest of countries in the respective constituency • E.g., • pSpain-Mexico = 0.78 • pPeru-Chile = 0.84 • pBelgium-Belarus =0.006

45. Preference-based voting power indices (1) (3) E.g., f+Argentina(Argentina+Chile)= 0.66 (4) (2) (5) (6) (7)

46. Changes from the status-quo: simple majority

47. Changes from the status-quo: majority of 70%

48. Changes from the status-quo: majority of 85%

49. Abs. changes from the status-quo: simple majority

50. Cost Efficiency and Shareholders’ Voting Power in Russian Banking