Power distribution: theory and applications

Fuad Aleskerov

State University “Higher School of Economics”and Institute of Control Sciences

Moscow, Russiaalesk@hse.ru, alesk@ipu.ru

EXAMPLE

Parliament with 99 seats3 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

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.

Banzhaf index

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

ib

jj

i

bb

i)(

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.

Example (continued)

Then Banzhaf index for А which is pivotal in each three coalitions is evaluated as follows

5

3

113

3)(

A

Similarly, for В and С, each of which is pivotal in only one coalition, one can obtain

5

1

113

1)()(

CB

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

Power distribution of some parties in Russian parliament (1994-2003)

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

jan.

94

apr.9

4

jul.9

4

oct.9

4

jan.

95

apr.9

5

jul.9

5

oct.9

5

jan.

96

apr.9

6

jul.9

6

oct.9

6

jan.

97

apr.9

7

jul.9

7

oct.9

7

jan.

98

apr.9

8

jul.9

8

oct.9

8

jan.

99

apr.9

9

jul.9

9

oct.9

9

jan.

-feb

.00

may

.00

sep.

00

dec.

00

may

.01

sep.

01

dec.

01

mar

.02

jun.

02

nov.

02

feb.

03

may

.03

oct.0

3

Ban

zhaf

ind

ex

Communists Liberal-Democrats Russia Regions Yabloko Agrarians

Power distribution in the 3d Duma

0

0,05

0,1

0,15

0,2

0,25

Ban

zh

af

ind

ex

Communists Edinstvo OVRSPS Liberal-Democrats YablokoAgrariants Narodnyi Deputat Regions of Russia

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

3

2

12

2)(

A

0)( B

3

1

12

1)(

C

;

;

.

Consistency index

2211

2121 1,,1,max

1,qqqq

qqqqc

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

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

0,000

0,100

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

the

con

sist

ency

ind

ex

Communists_Edinstvo Edinstvo_OVR SPS_Yabloko Communists_Agrariants

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

0

0,05

0,1

0,15

0,2

0,25

0,3

Ban

zh

af

ind

ex

Communists Edinstvo Narodnyi Deputat

Communists_share Edinstvo_share Narodnyi Deputat_share

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

Ban

zh

af

ind

ex

SPS Liberal-Democrats Yabloko

SPS_share Liberal-Democrats_share Yabloko_share

Power distribution of small factions

(SPS, Liberal-Democrats, Yabloko), scenario 0,4

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

0,000

0,100

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

the c

on

sis

ten

cy in

dex

SPS_Communists SPS_Commusists_authSPS_Edinstvo SPS_Edinstvo_auth

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

0,000

0,100

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

the c

on

sis

ten

cy in

dex

Edinstvo_Communists Edinstvo_Communists_authOVR_Edinstvo OVR_Edinstvo_auth

Power distribution on the authority issue for faction Edinstvo and Communists, scenario 0.4

0,000

0,050

0,100

0,150

0,200

0,250

0,300

Ban

zh

af

ind

ex

Communists Communists_auth Edinstvo Edinstvo_auth

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

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

pro-reform - anti-reform

lib

era

l -

sta

te o

rie

nte

d

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

pro-reform - anti-reform

lib

era

l -

sta

te o

rien

ted

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

05.03 06.03

09.03

10.03

11.03

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

pro-reform - anti-reform

lib

era

l -

sta

te o

rien

ted

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

05.03

06.03 09.03

10.03

11.03

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

pro-reform - anti-reform

lib

era

l -

sta

te o

rien

ted

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

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

pro-reform - anti-reform

lib

eral -

sta

te o

rie

nte

dTrajectory of largest group of MPs of Communists and Yabloko belonging to one cluster

Consistency Index on Political Map

11

21

2

1

ijij d

d

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.

,!i

i

qSOV

n

Shapley-Owen index

ExtensionThe average value of i’s weight

The power index of player i

where is the share of votes, and is thenumber of votes of party i.

1( )

t

imm

i

wv t

t

n

jjj

ii

tv

tviPI

1

1

)(

)()(

i i jj

n n in

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

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Jan.

, 200

0

Mar

, 200

0

May

, 200

0

Jul,

2000

Oct

., 20

00

Dec

., 20

00

Feb

., 20

01

Apr

., 20

01

Jun,

200

1

Sep

., 20

01

Nov

., 20

01

Jan.

, 200

2

Mar

, 200

2

May

, 200

2

Sep

., 20

02

Nov

., 20

02

Jan.

, 200

3

Mar

,200

3

May

, 200

3

Sep

., 20

03

Nov

., 20

03

PI1

ind

ex

CPRF Edinstvo

Ordinal and cardinal indices

We define the intensity of connection ),i(f of the agent with other members of is defined.

Then for such agent i the value i is evaluated as

,,ifi

i.e. the sum of intensities of connections of i over those coalitions in which i is pivotal. Naturally, other functions instead of summation can be considered.

Then the power indices are constructed as

jj

ii .

The very idea of i is the same as for Banzhaf index, with the difference that in Banzhaf index we evaluate the number of coalitions in which i is pivotal.

Ordinal indices How to construct the intensity functions ),i(f ? For each coalition and each agent i construct now an intensity ),i(f of

connections in this coalition. In other words, f is a function which maps

N (= N2 \{Ø}) into 1R , 1RN:f . a) Intensity of i’s preferences. In this form only preferences of i’s agent over

other agents are evaluated, i.e.,

j

ijpif ),(

b) Intensity of preferences for i. In this case consider the sum of ranks of i given by other members of coalition

j

jipif ),(

Cardinal intensity functions

Assume now that the desire of party i to coalesce with party j is given as real number ijp , njip

jij ,,1, ,1 . In general, it is not assumed that jiij pp .

One can call the value ijp as an intensity of connection of i with j. It may

be interpreted as, for instance, a probability for i to form a coalition with j. We define now several intensity functions g) minimal intensity of i's connections

ijj

pif min),(min ;

h) maximal intensity of i's connections

ijj

pif max),(max ;

i) maximal fluctuation of i's connections

)maxmin(2

1),( ij

jij

jmf ppif

Using these intensity functions one can define now the corresponding power

indices )i( . Let i be a pivotal agent in a winning coalition . Denote

as i the number equal to the value of the intensity function for a given coalition

and agent i. Then the power index is defined as follows

Njj

j

i

i

i

in pivotal is ,

in pivotal is ,

)(

As we already mentioned this index is similar to the Banzhaf index. The

difference is that i in the Banzhaf index is equal to 1, in the case under study

i represents some intensity value.

Example 3. Consider the case when 3 parties A, B and C have 50, 49 and 1 seats, respectively. Assume that decision making rule is simple majority, i.e. 51 votes. Then the winning coalitions are A+B, A+C and A+B+C. Note that A is pivotal in all three coalitions, B and C are pivotal in one coalition each. Then

5/3)( A , 5/1)()( CB . Consider now the case with the preferences of agents given

below: BC:PA ; BC:PA and BA:PC . Then the values of 1 and 2 (constructed by ),( if and ),( if are as

follows 8/1)( ,5/1)( ,8/5)( 111 CBA ,

and the values of 2 are equal to 1 . Consider another preference profile: BC:PA , BC:PA ; BA:PC , i.e., only agent C changes her preferences. Then one can easily evaluate that

7/1)( )( ,7/5)( 111 CBA ; 19/10)(2 A , 19/3)(2 B , 19/6)(2 C .

Axiomatic construction of a cardinal

intensity function First, we define an intensity function depending on intensities ijp of

connections of i with other members of coalition , i.e., if },,1{ m , nm , ),,,,,,,,,,,(),( 1221111 mmimimmi pppppppfif

However, we will restrict this function in a way which is similar to independence of irrelevant alternatives [3]: ),( if will depend on connections of agent i with other members of coalition only, i.e.,

).,,(),( 1 imii ppfif

For the sake of simplicity we put 0ijp for all j,i and

jijpi =1.

I would like to emphasize that in this formulation the sum of

ijp is equal to

1 in each , i.e., now connections are defined by 12 N matrices

ijp for each

coalition .

Axioms which reasonable function should satisfy to.

Axiom 1. For any m – tuple of values ),,( 1 imi pp there exist a function ),( if such that 1),(0 if , f is continious differentiable function of each of its arguments.

Axiom 2. If 0ijp for any j, then ),( if =0.

Axiom 3. (Monotonicity). A value of ),( if increases if any value ijp increases,

and a value of ),( if decreases if ijp decreases. Moreover, equal changes in

intensities ijp lead to equal changes of ),( if . This means that

iij

i

p

f

for any j

and 0

lj

i

p

f for any il .

Theorem. An intensity function ),i(f satisfies Axioms 1–3 iff it is represented

in the form

j

ijp

),i(f .

Axioms for power indices defined on the games with preferencesVoting with quota: 𝑣 = (𝑞;𝜔1,…,𝜔𝑛). Ωi(𝑣) – the set of coalitions in which the player 𝑖 is pivotal.

Oligarchic game ሺ𝑢𝑆ሻ – with the only winning coalition 𝑆.

A game satisfies the condition of the uniqueness of the voting outcome if for any winning coalition 𝑆, 𝑁\𝑆 is losing coalition

Erasing coalition: Denote as 𝑣−𝑆 the game in which the winning coalition 𝑆 is announced the losing one.

Players’ preferences - 𝑛× 𝑛-matrix 𝑃. 0 ≤ 𝑝𝑖𝑗 ≤ 1,𝑝𝑖𝑖 = 0.

Axioms for power indices defined on the games with preferences

The weight of coalition 𝑆 for the player 𝑖 depends on the player, coalition and the preference matrix and is denoted as 𝑓ሺ𝑖,𝑆ሻ. The set of simple games with preferences for 𝑛 players is denoted as 𝑆𝐺𝑃𝑛, the set of symmetric simple games with preferences (𝑓(𝑖,𝑆) does not depend on 𝑖) is denoted as 𝑆𝑆𝐺𝑃𝑛.

Power index Φ:𝑆𝐺𝑃𝑛 →ℝ𝑛 (𝑆𝑆𝐺𝑃𝑛 →ℝ𝑛), put into correspondence to each game with preferences the vector Φ(𝑣), 𝛼-power index is defined as

𝛼𝑖(𝑣) = 𝑓(𝑖,𝑆)𝑆∈𝛺𝑖(𝑣)

Normalized power index is defined as

𝑁𝛼𝑖(𝑣) = 𝛼𝑖(𝑣)σ 𝛼𝑖(𝑣)𝑗∈𝑁 .

Axioms for power indices defined on the games with preferences

Null Player (NP). The gain of the null player does not depend on the preferences and always equal to 0.

Transfer (T). For any games 𝑣,𝑤∈𝑆𝐺𝑃𝑛 and for any coalition 𝑆∈𝑀(𝑣) ∩𝑀(𝑤) and any 𝑖 Φi(𝑣) − Φi(𝑣−𝑆) = Φ𝑖(𝑤) − Φ𝑖(𝑤−𝑆).

Strong Transfer (ST). For any game 𝑣 ∈𝑆𝐺𝑃𝑛 and for any coalition 𝑆∈𝑀(𝑣) and any 𝑖 ∈𝑆

Φi(𝑣) − Φi(𝑣−𝑆) = 𝑓(𝑖,𝑆).

Axioms for power indices defined on the games with preferences

Symmetric Gain-Loss (SymGL). For any game 𝑣 ∈𝑆𝐺𝑃𝑛, any coalition 𝑆∈𝑀(𝑣) and any 𝑖,𝑗∈𝑆:

Φi(𝑣) − Φi(𝑣−𝑆) = Φ𝑗(𝑣) − Φ𝑗(𝑣−𝑆). Total Power (TP).

Φi(𝑣)𝑛𝑖=1 = 𝑓(𝑖,𝑆)𝑆∈𝑊𝑖(𝑣)

𝑛𝑖=1

Results

Theorem. The power index Φ(𝑣) satisfies axioms NP и ST iff Φ(𝑣) = 𝛼(𝑣).

Theorem. The power index Φ(𝑣), defined on 𝑆𝑆𝐺𝑃𝑛 satisfies axioms NP, TP, T and SymGL iff Φ(𝑣) = 𝛼(𝑣).

Axioms for normalized indices

Normalization axiom (N). For any game ∈𝑆𝐺𝑃𝑛 σ Φ(𝑣)𝑛𝑖=1 = 1.

Axiom NP is the same as for 𝛼−index.

Weak Anonimity (WAn). For any oligarchic game 𝑢𝑆 ∈𝑆𝐺𝑃𝑛 and any players 𝑖,𝑗∈𝑆 Φ𝑖(𝑢𝑆) = Φ𝑗(𝑢𝑆). Transfer axiom ሺ𝐓𝒏ሻ. For any game 𝑣 ∈𝑆𝐺𝑃𝑛 there is a positive number 𝑐(𝑣) such that for any coalition 𝑆∈𝑀(𝑣) 𝑐(𝑣)Φ(𝑣) − 𝑐(𝑣−𝑆)Φ(𝑣−𝑆) = 𝑤𝑆 Theorem. Let Φ be an index defined on the set of games with symmetric preferences saticfying the condition of the uniqueness of voting. Then Φ satisfies axioms NP, WAn, T𝑛 and N iff Φ(𝑣) = 𝛼(𝑣).

Applications IMF Russian banks

Modelling preference of country i to 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 i and j in the

international political-economic blocs outside the IMF(Eb(i,j), weight Wb=0.65)

Overall intensity pij is defined by a generalized criterion jiEjiEp brrij ,W,W b

Modification 2 (Aleskerov, Kalyagin & Pogorelskiy (2009)) Bilateral trade with j as compared with the rest of

countries in the respective constituency

ikVk

ik

ij

ikVk

iik

iijij X

X

TXX

TXXp

/

/

E.g., pSpain-Mexico = 0.78 pPeru-Chile = 0.84 pBelgium-Belarus =0.006

Modelling preference of country i to coalesce with j

Preference-based voting power indices

VCSNC

VSNS SC

CCV\

for swing a is

yes'' votesPr1yes'' votesPr

ViVi ,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

1

j

ij

i

p

f

1

j

ji

i

p

f

i

ii ififf

}{\}{\21

l

winningisV

f

V2

yes'' votesPr

otherwise1,

if ,2 1

for swing a is qv(i)

f

il

iV

i

E.g.,f+

Argentina(Argentina+Chile)= 0.66

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 -0.3215 0.2330 -9.6837 0.1112 -4.6192Japan 3.3934 3.8155 -3.2358 3.6893 2.1900Germany -1.3611 -0.9991 -8.1105 -1.1201 -2.9581France -11.7741 -11.5223 -17.0975 -11.6299 -12.4482UK -11.7741 -11.5223 -17.0975 -11.6299 -12.4482Belgian_C -0.6072 -0.2565 -3.5013 -0.3783 1.9099Dutch_C -5.5306 -5.1110 -9.6803 -5.2257 -4.6155Mexican_Spanish_C 4.4466 4.8751 1.4982 4.7483 7.1895Italian_C 3.6364 4.0592 -0.0889 3.9316 5.5130China 3.8927 4.3389 -2.4679 4.2112 3.0002Canadian_C -1.1285 -0.7415 -5.0647 -0.8636 0.2578Malaysian_C 11.5412 11.9654 10.1579 11.8286 16.3349Australian_C 0.3538 0.7389 -3.3777 0.6170 2.0414Swedish_C -1.2835 -0.8751 -4.8339 -0.9966 0.5022Egyptian_C 0.5843 0.8680 -3.0743 0.7468 2.3603Saudi Arabia -11.5484 -11.3455 -15.6741 -11.4521 -10.9462South_African_C 3.2952 3.7093 -1.2453 3.5832 4.2914Swiss_C -1.6401 -1.2671 -6.0891 -1.3885 -0.8237Russia -11.4582 -11.1646 -19.8582 -11.2732 -15.3641Iranian_C -6.4832 -6.0975 -12.2496 -6.2104 -7.3282Brazilian_C 15.8631 16.2913 8.3043 16.1481 14.3759Indian_C 19.2470 19.7025 12.0763 19.5564 18.3609Argentinean_C -6.1482 -5.6215 -11.1899 -5.7376 -6.2107Central_African_C 22.4686 22.9265 20.3823 22.7750 27.1350

Changes from the status-quo: simple majority

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 -5.1472Japan -0.0789 -31.8366 3.3216 -4.2682Germany -4.2640 -32.1715 -1.0060 -4.7387France -13.9643 -35.1141 -11.0364 -8.8713UK -13.9643 -35.1141 -11.0364 -8.8713Belgian_C -3.5131 -32.8369 -0.2295 -5.6731Dutch_C -8.0530 -36.0789 -4.9239 -10.2264Mexican_Spanish_C 1.2577 -25.7161 4.7037 4.3276Italian_C 0.4936 -31.2126 3.9135 -3.3919China 0.7644 -31.1378 4.1936 -3.2869Canadian_C -3.8953 -32.4813 -0.6248 -5.1737Malaysian_C 8.1349 -11.9046 11.8149 23.7251Australian_C -2.5263 -25.2909 0.7909 4.9249Swedish_C -4.0912 -25.9588 -0.8272 3.9868Egyptian_C -2.1488 -22.8597 1.1813 8.3392Saudi Arabia -13.8520 -36.6330 -10.9202 -11.0045South_African_C 0.2500 -20.3708 3.6617 11.8349Swiss_C -4.3965 -32.1342 -1.1429 -4.6862Russia -13.8055 -36.2735 -10.8722 -10.4996Iranian_C -9.0766 -37.7658 -5.9824 -12.5955Brazilian_C 12.4952 -23.3080 16.3236 7.7097Indian_C 15.7199 -22.1038 19.6580 9.4009Argentinean_C -8.8410 -34.7112 -5.7387 -8.3054Central_African_C 19.0102 -10.9882 23.0602 25.0121

Changes from the status-quo: majority of 70%

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 -2.9151Japan -11.4694 -43.0613 0.6256 -2.8078Germany -12.9946 -43.1662 -1.1080 -2.9868France -18.4527 -43.7226 -7.3118 -3.9366UK -18.4527 -43.7226 -7.3118 -3.9366Belgian_C -12.6377 -42.0645 -0.7023 -1.1062Dutch_C -15.2373 -45.1649 -3.6571 -6.3985Mexican_Spanish_C -9.7743 -38.3714 2.5523 5.1978Italian_C -9.6782 -42.8851 2.6615 -2.5070China -9.3058 -42.5486 3.0848 -1.9326Canadian_C -12.1871 -42.9707 -0.1902 -2.6532Malaysian_C -4.5956 -36.7469 8.4385 7.9707Australian_C -11.6121 -42.8516 0.4634 -2.4498Swedish_C -12.8634 -43.3302 -0.9589 -3.2668Egyptian_C -11.9001 -42.3153 0.1360 -1.5343Saudi Arabia -20.4914 -45.0890 -9.6290 -6.2690South_African_C -8.9754 -35.5674 3.4603 9.9840Swiss_C -12.7936 -42.4012 -0.8796 -1.6810Russia -20.6195 -45.6123 -9.7745 -7.1622Iranian_C -16.9811 -49.1033 -5.6392 -13.1212Brazilian_C 0.1716 -37.6345 13.8570 6.4556Indian_C 2.6372 -36.8390 16.6595 7.8134Argentinean_C -16.7705 -48.6411 -5.3997 -12.3324Central_African_C 6.7679 -15.5344 21.3545 44.1796

Changes from the status-quo: majority of 85%

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 -1,195 0.0015 -0.0149 0.0232 -0.4521Japan 4,526 0.0067 -0.0030 0.2117 0.1289Germany -1,774 -0.0017 -0.0073 -0.0628 -0.1696France -12,673 -0.0162 -0.0130 -0.5379 -0.6002UK -12,673 -0.0162 -0.0130 -0.5379 -0.6002Belgian_C -692 -0.0004 -0.0037 -0.0185 0.1266Dutch_C -5,859 -0.0071 -0.0093 -0.2379 -0.2825Mexican_Spanish_C 4,387 0.0063 0.0013 0.2012 0.4010Italian_C 3,308 0.0048 -0.0001 0.1535 0.2198China 3,159 0.0046 -0.0014 0.1466 0.1091Canadian_C -910 -0.0008 -0.0029 -0.0299 0.0093Malaysian_C 9,010 0.0122 0.0068 0.3962 0.6892Australian_C 270 0.0007 -0.0022 0.0202 0.0833Swedish_C -979 -0.0009 -0.0031 -0.0326 0.0203Egyptian_C 414 0.0008 -0.0018 0.0227 0.0900Saudi Arabia -8,096 -0.0104 -0.0077 -0.3444 -0.3395South_African_C 2,200 0.0032 -0.0007 0.1025 0.1534Swiss_C -1,014 -0.0010 -0.0026 -0.0368 -0.0225Russia -6,841 -0.0087 -0.0083 -0.2883 -0.4087Iranian_C -3,479 -0.0043 -0.0053 -0.1428 -0.2013Brazilian_C 8,508 0.0114 0.0032 0.3710 0.3476Indian_C 10,030 0.0134 0.0044 0.4365 0.4290Argentinean_C -2,668 -0.0032 -0.0039 -0.1065 -0.1356Central_African_C 6,708 0.0089 0.0048 0.2907 0.4046

Abs. changes from the status-quo: simple majority

Cost Efficiency and Shareholders’ Voting

Power in Russian Banking

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

Ownership and Control Patterns:

Sample of Top-100 Russian Banks

?5% 5-10% 10-20% 20-30%

30-40%

40-50%

Total

?5% 1 1 24 9 8 2 455-10% 1 1 0 0 0 0 210-20%

0 1 30 4

20-30% 24 5 4

0 33

?30% 3 1 2 6

W2

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.

W1

Data Set

Russian banks: top-100Distribution of top-100 Russian banks over ownership type

Study period: from II quarter of 2006 to II quarter of 2007

28%

34%

17%

21%SBERBANK

OTHER STATE-OWNEDBANKS

FOREIGN-OWNEDBANKS

DOMESTIC PRIVATE-OWNED BANKS

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

Methodology (II): Definition of Pairwise Preferences

to Coalesce Unified perspective on preferences of all banks’ shareholders1. 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 (pji =pij =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 (pij =3) If there is no alternative for j of forming a block with yet another shareholder and

together with i they can get absolute controlpji =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 pji =5. If there is an alternative possibility for j of creating a block with some other

shareholder pji =33. Both i and j are blockholders.

Assume their preferences towards each other are maximal (pji =pij =9).4. Any other possible combination.

we assign pji =pij =1 (neutral preferences to coalesce).

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

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

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 ( Radj2 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

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).

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

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".

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

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.

Thank you