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.rualesk@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 i is pivotal, then Banzhaf index for i is 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): Belgium50% Luxembourg25% 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 ( q 1 = q 2 ), and equal to 0, if positions are opposite (e.g., q 1 =0 и q 2 =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 faction Edinstvo 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 q i 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

40. Results

41. Axioms for normalized indices

42. Applications IMF Russian banks

43. Modelling preference of country i to coalesce with j Modification 1 ( Aleskerov, Kalyagin & Pogorelskiy (2008) )  Regional proximity of country j ( E r (i,j), weight W r =0.35)  Membership of the pair of countries i and j in the international political-economic blocs outside the IMF( E b (i,j), weight W b =0.65) Overall intensity p ij is defined by a generalized criterion

44. Modification 2 ( Aleskerov, Kalyagin & Pogorelskiy (2009) )  Bilateral trade with j as compared with the rest of countries in the respective constituency E.g.,  p Spain-Mexico = 0.78  p Peru-Chile = 0.84  p Belgium-Belarus =0.006 Modelling preference of country i to coalesce with j

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

46. Constituency Difference in the number of votes, % Difference in Penrose power, % Difference in κ power index, % Difference in Banzhaf power index, % Difference in normalized κ power index, % US-0.32150.2330-9.68370.1112-4.6192 Japan3.39343.8155-3.23583.68932.1900 Germany-1.3611-0.9991-8.1105-1.1201-2.9581 France-11.7741-11.5223-17.0975-11.6299-12.4482 UK-11.7741-11.5223-17.0975-11.6299-12.4482 Belgian_C-0.6072-0.2565-3.5013-0.37831.9099 Dutch_C-5.5306-5.1110-9.6803-5.2257-4.6155 Mexican_Spanish_C4.44664.87511.49824.74837.1895 Italian_C3.63644.0592-0.08893.93165.5130 China3.89274.3389-2.46794.21123.0002 Canadian_C-1.1285-0.7415-5.0647-0.86360.2578 Malaysian_C11.541211.965410.157911.828616.3349 Australian_C0.35380.7389-3.37770.61702.0414 Swedish_C-1.2835-0.8751-4.8339-0.99660.5022 Egyptian_C0.58430.8680-3.07430.74682.3603 Saudi Arabia-11.5484-11.3455-15.6741-11.4521-10.9462 South_African_C3.29523.7093-1.24533.58324.2914 Swiss_C-1.6401-1.2671-6.0891-1.3885-0.8237 Russia-11.4582-11.1646-19.8582-11.2732-15.3641 Iranian_C-6.4832-6.0975-12.2496-6.2104-7.3282 Brazilian_C15.863116.29138.304316.148114.3759 Indian_C19.247019.702512.076319.556418.3609 Argentinean_C-6.1482-5.6215-11.1899-5.7376-6.2107 Central_African_C22.468622.926520.382322.775027.1350 Changes from the status-quo: simple majority

47. Constituency Difference in Penrose power, % Difference in κ power index, % Difference in Banzhaf power index, % Difference in normalized κ power index, % US-4.5714-32.4624-1.3239-5.1472 Japan-0.0789-31.83663.3216-4.2682 Germany-4.2640-32.1715-1.0060-4.7387 France-13.9643-35.1141-11.0364-8.8713 UK-13.9643-35.1141-11.0364-8.8713 Belgian_C-3.5131-32.8369-0.2295-5.6731 Dutch_C-8.0530-36.0789-4.9239-10.2264 Mexican_Spanish_C1.2577-25.71614.70374.3276 Italian_C0.4936-31.21263.9135-3.3919 China0.7644-31.13784.1936-3.2869 Canadian_C-3.8953-32.4813-0.6248-5.1737 Malaysian_C8.1349-11.904611.814923.7251 Australian_C-2.5263-25.29090.79094.9249 Swedish_C-4.0912-25.9588-0.82723.9868 Egyptian_C-2.1488-22.85971.18138.3392 Saudi Arabia-13.8520-36.6330-10.9202-11.0045 South_African_C0.2500-20.37083.661711.8349 Swiss_C-4.3965-32.1342-1.1429-4.6862 Russia-13.8055-36.2735-10.8722-10.4996 Iranian_C-9.0766-37.7658-5.9824-12.5955 Brazilian_C12.4952-23.308016.32367.7097 Indian_C15.7199-22.103819.65809.4009 Argentinean_C-8.8410-34.7112-5.7387-8.3054 Central_African_C19.0102-10.988223.060225.0121 Changes from the status-quo: majority of 70%

48. Constituency Difference in Penrose power, % Difference in κ power index, % Difference in Banzhaf power index, % Difference in normalized κ power index, % US-13.2274-43.1242-1.3726-2.9151 Japan-11.4694-43.06130.6256-2.8078 Germany-12.9946-43.1662-1.1080-2.9868 France-18.4527-43.7226-7.3118-3.9366 UK-18.4527-43.7226-7.3118-3.9366 Belgian_C-12.6377-42.0645-0.7023-1.1062 Dutch_C-15.2373-45.1649-3.6571-6.3985 Mexican_Spanish_C-9.7743-38.37142.55235.1978 Italian_C-9.6782-42.88512.6615-2.5070 China-9.3058-42.54863.0848-1.9326 Canadian_C-12.1871-42.9707-0.1902-2.6532 Malaysian_C-4.5956-36.74698.43857.9707 Australian_C-11.6121-42.85160.4634-2.4498 Swedish_C-12.8634-43.3302-0.9589-3.2668 Egyptian_C-11.9001-42.31530.1360-1.5343 Saudi Arabia-20.4914-45.0890-9.6290-6.2690 South_African_C-8.9754-35.56743.46039.9840 Swiss_C-12.7936-42.4012-0.8796-1.6810 Russia-20.6195-45.6123-9.7745-7.1622 Iranian_C-16.9811-49.1033-5.6392-13.1212 Brazilian_C0.1716-37.634513.85706.4556 Indian_C2.6372-36.839016.65957.8134 Argentinean_C-16.7705-48.6411-5.3997-12.3324 Central_African_C6.7679-15.534421.354544.1796 Changes from the status-quo: majority of 85%

49. Constituency Difference in the number of votes Difference in Penrose power Difference in κ power index Difference in Banzhaf power index Difference in normalized κ power index US-1,1950.0015-0.01490.0232-0.4521 Japan4,5260.0067-0.00300.21170.1289 Germany-1,774-0.0017-0.0073-0.0628-0.1696 France-12,673-0.0162-0.0130-0.5379-0.6002 UK-12,673-0.0162-0.0130-0.5379-0.6002 Belgian_C-692-0.0004-0.0037-0.01850.1266 Dutch_C-5,859-0.0071-0.0093-0.2379-0.2825 Mexican_Spanish_C4,3870.00630.00130.20120.4010 Italian_C3,3080.0048-0.00010.15350.2198 China3,1590.0046-0.00140.14660.1091 Canadian_C-910-0.0008-0.0029-0.02990.0093 Malaysian_C9,0100.01220.00680.39620.6892 Australian_C2700.0007-0.00220.02020.0833 Swedish_C-979-0.0009-0.0031-0.03260.0203 Egyptian_C4140.0008-0.00180.02270.0900 Saudi Arabia-8,096-0.0104-0.0077-0.3444-0.3395 South_African_C2,2000.0032-0.00070.10250.1534 Swiss_C-1,014-0.0010-0.0026-0.0368-0.0225 Russia-6,841-0.0087-0.0083-0.2883-0.4087 Iranian_C-3,479-0.0043-0.0053-0.1428-0.2013 Brazilian_C8,5080.01140.00320.37100.3476 Indian_C10,0300.01340.00440.43650.4290 Argentinean_C-2,668-0.0032-0.0039-0.1065-0.1356 Central_African_C6,7080.00890.00480.29070.4046 Abs. changes from the status-quo: simple majority

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

51. 51 Ownership and Control Patterns: the Case of Russia Russian non-financial companies (Kapelushnikov (2005)) concentration of equity ownership and control was rather high; control of a Russian company may be held by a single shareholder, a block holder Russian banks (S&P (2007))  concentrated ownership structure: in 2007 about 60% out of 30 largest commercial banks had one major shareholder, who acquired more than 50% of the total shares

52. Ownership and Control Patterns: Sample of Top-100 Russian Banks 55 banks out of top-100 Russian banks have the single strategic owner who has absolute control. In 33 banks out of remaining 45 banks the first and the second largest stockholders are usually block holders. W1W1

53. Data Set  Russian banks: top-100 Distribution of top-100 Russian banks over ownership type  Study period: from II quarter of 2006 to II quarter of 2007

54. 54 Shareholding Concentration Ratio Our hypothesis is that banks with more concentrated ownership are less efficient (have worse performance) than those with a more dispersed ownership structure.

55. 55 Methodology (II): Definition of Pairwise Preferences to Coalesce Unified perspective on preferences of all banks’ shareholders 1. i and j are neither blockholders nor have absolute control, but jointly can form a block (25% of the total shares). Assume their preferences towards each other are equally strong (p ji =p ij =6). 2. Shareholder i is a blockholder while j is not, and jointly they either get absolute or almost absolute (47% of the total shares) control. Assume that shareholder i likes j less than in the previous case (p ij =3) If there is no alternative for j of forming a block with yet another shareholder and together with i they can get absolute control  p ji =6 If there is no alternative for j of forming a block with yet another shareholder, but i and j can get together almost absolute control  p ji =5. If there is an alternative possibility for j of creating a block with some other shareholder  p ji =3 3. Both i and j are blockholders. Assume their preferences towards each other are maximal ( p ji =p ij =9). 4. Any other possible combination. we assign p ji =p ij =1 (neutral preferences to coalesce).

56. 56 Methodology (II): Computation of Power Indices The exact number of shareholders is not usually known  some assumptions must be made. We used the approach from Leech (1988), called “most concentrated distribution” all but one non-observed holdings are assumed to coincide with the last observed share with an obvious correction for a single remaining shareholder so that the total sum of the shares is 100%. This assumption is justified, because the ownership of the Russian banks in the sample is highly concentrated

57. 57 Main Results (II): Patterns of Control Most frequently the power of the largest shareholders as a group increases with its size, ranging from as low as 20% for the largest shareholder alone to more than 60% for top-3 shareholders for all indices considered. There is a difference between the distributions obtained using the classical and preference-based indices. In particular, preference-based indices tend to assign greater power to the blocks of two and three shareholders compared to the normalized Banzhaf index. Taking into account this observation and the fact that the banks’ ownership structure usually comprises two blockholders, we conjecture that these blockholders have a similar degree of control, about 50% of total power.

58. 58 Main Results (III): Relation Between Cost Efficiency and Type of Governance  There exists a relation between the cost efficiency and the degree of control.  This relation is rather weak ( R adj 2 does not exceed 13.2%) due to moderate size of the sample (just 45 banks).  The ratio of the normalized Banzhaf index of the largest shareholder gives better results than all other cases considered.

59. 59 Main Results (IV) ownership structure of banks: Model 1: largest shareholder with absolute control, Model 2: two blockholders, having absolute control together relation between cost efficiency and type of governance: concentration and degree of control negatively influence cost efficiency of the banks Note: This relation is robust to various concentration and power indices tested. This conclusion is in line with the results received by Kapelyushnikov & Demina (2005).

60. Other works and studies in progress 1. Power in the Parliament of Russian Empire 2. Power in the Reichstag of Weimar Republic 2. The apparatus of generating functions 3. Experiments 1. Power distribution in other large organizations 2. Study of regional parliaments

61. References Aleskerov F., N.Blagoveshensky, G.Satarov, A.Sokolova, V.Yakuba “Power and Structural Stability in Russian Parliament (1905-1917 и 1993-2005)”, Moscow, Fizmatlit, 2007 (in Russian). Aleskerov F., H.Ersel, Y.Sabuncu,“Power and coalitional stability in the Turkish Parliament (1991-1999)”, Turkish Studies, v.1, no.2, 2000, 21-38 Aleskerov F. “Power indices taking into account agents’ preferences”, in “Mathematics and Democracy” (B.Simeone and F.Pukelsheim, eds.), Springer, Berlin, 2006, 1-18 F. T. Aleskerov, Power Indices Taking into Account the Agents' Preferences for Coalescence, Doklady Mathematics, 2007, v.75, №3/2 Aleskerov F., Kalyagin V., Pogorelskiy K. Multy-agent Model of Voting Power Dynamics of the IMF Members, Preprint WP7/2007/06. Moscow: State University "High School of Economics" (in Russian) Aleskerov F., Otchur O. ‘Extended Shaply-Owen Indices and Power Distribution in III State Duma’, Preprint WP7/2007/03, Moscow: State University "High School of Economics".

62. References Aleskerov, F. (2006). Power indices taking into account agents’ preferences. In: B. Simeone & F. Pukelsheim (eds), Mathematics and Democracy, Berlin: Springer, pp. 1-18 Aleskerov, F., V. Kalyagin, and K. Pogorelskiy (2008). Actual voting power of the IMF members based on their political-economic integration. Mathematical and Computer Modelling, 48:1554-1569

63. References… Шварц Д.А. О вычислении индексов влияния, учитывающих предпочтения участников. Автоматика и Телемеханика, Москва, 2009, No 3, с. 152-159. Шварц Д.А. Аксиоматика для индексов влияния, учитывающих предпочтения участников. Автоматика и Телемеханика, Москва, 2010, No 1, с. 144-158. Шварц Д.А. Аксиоматика для индексов влияния в задаче голосования с квотой. Проблемы управления, 2012, No 1, с. 33-41. Шварц Д.А. Индексы влияния как элементы проективного пространства. Доклады Академии наук, 2011, No 441 (4), с. 456-459.

64. Thank you