VALUES FOR GAMES ANALYZING VOTING WITH ROTATION

Honorata Sosnowska

Warsaw School of Economics, 02-554 Warsaw, al. Niepodleglosci 162, Poland [email protected]

Game theoreticians usually deal with some standard voting methods such as plurality voting or approval voting. In reality however, some non standard complicated voting methods are used. In this paper we shall present the voting method with rotation scheme used by the Governing Council of the European Central Bank as it enlarges to accommodate new members of the economic and monetary union. We present game theory approaches to this method and different methods of computing the Shapley and the Banzhaf values for games analyzing voting with rotation. The properties of the values are considered.

Key words: voting, EBC, Shapley value, Banzhaf value

1. Introduction A lot of different voting methods are applied in social and economic situations. The most popular are majority voting, plurality voting and ranking methods. There are also situations where very sophisticated methods are used. We analyze one of them. The method analyzed is voting with rotation, used in the European Central Bank . The construction of the method results from the predicted difficulties of voting with enlarged number of voters and guarantees voting power for the most economically important countries. Some unexpected results were observed. There are no single method of computing power indices for voting with rotation. We analyze the Shapley value (the Shapley – Shubik index in some cases) and the Banzhaf value. 1

Several methods of computing the Shapley value are used. One of them is proposed in this paper. We show that different methods lead to different results. The paper is constructed as follows. In Section 2 the voting with rotation, used by the European Central Bank, is presented. The game theory approach to this voting is presented in Section 3. In Section 4 examples are analyzed. In Section 5 properties of considered methods are presented. Conclusions are formulated in Section 6.

2. The rule of voting with rotation in the European Central Bank As of 21 March 2003 the European Council introduced voting with rotation into the statute of the European Central Bank (ECB). The presentation of the ECB voting system is based on the ECB Monthly Bulletin [3]. The voting system of the ECB is constructed in such a way that it can work in case of the enlargement (not predicted but also possible: in case of reduction) of the euro area. The declared origin of this construction was a wish to have a system that works without difficulties caused by the high number of voters. The solutions used are modeled on a system applied in the FOMC (the Federal Open Market Committee of the Federal Reserve). In the FOMC there are voting and non-voting members. The members without the permanent voting right are divided into groups and vote according to a rotation scheme [12]. The ECB Governing Council (GC) consists of the Executive Board of the ECB (EB) and the governors of the national central banks (NBC) of the countries that have adopted euro as their currency. The EB comprises of 6 members: the President, the Vice-President and 4 other members. All members are appointed by the European Council acting by a qualified majority. Each member of the GC has one vote. The GC decides by simple majority with a casting vote of the President in case of a tie. The number of members of the GC who have voting rights at any given time is limited to 21. The members of the EB have permanent voting rights . So, the number of governors with voting rights would not exceed 15. This limit was raised to 18 in 2008. Latvia has been the 18th country since January 1, 2014. The number of countries in Euro zone will exceed 18 when Lithuania joins the zone.

2

If the number of governors exceeds 18 they will be allocated to two groups on the basis of the ranking determined by the composite indicator. The first group will consist of the first five governors according to that ranking. The second group will consist of all the other governors. The first group will have 4 voting rights, the second group – 11. As of the date on which the number of governors exceeds 21, the governors will be divided into 3 groups. The first group (consisting of 5 governors) will have, again, 4 votes. The second group will consist of half of the total number of the governors, rounded up to the nearest full number. It will have 8 votes. The third group will consist of the remaining governors. It will have 3 votes. The governors will be ordered in each group in accordance with a list of their national central banks which follows the alphabetical order of the names of the Member States in the national language written in the Latin alphabet. The rotation will start at a random point in the list. The rotation period is one month with same exceptions caused by seasonality of certain votings. The number of governors gaining voting rights is constant for the first group and for other groups is defined as the difference between the number of governors and the number of voting rights minus two. All the governors, voting and non–voting, will participate in the GC meetings and will retain the right to speak. The composition of groups can alter when a new country enters the euro area or the indicator changes. The indicator is based on two parameters: the share of the country in aggregate GDP at market prices (with weight 5/6) and the country’s share in the total aggregated balance sheet of monetary financial institutions (MFIs) (with weight 1/6). The ranking of governors will be changed only when the ECB’s capital key is adjusted (every five years or when a new Member State joins the EU).

3. A game theoretical analysis of the ECB voting with rotation The standard way of modeling voting is the cooperative game theory. Usually, a game is defined and power indices or values are computed. We shall use the Shapley value [10], the Shapley –Shubik index [11], the Banzhaf 3

value and the normalized Banzhaf value.. Coalitional structures (defined in many ways) in case of suggested additional cooperation between players are considered. Let us recall definitions [8]. A set N, non-empty and finite is a set of n players. Subsets of N are called coalitions. A pair G=(N,v) is a cooperative game, where v:2N → R such that the value of v of the empty set is null. The function v is called a characteristic function of a game. A game is convex when v(S ∪ T ) ≥ v( S ) + v(T ) − v( S ∩ T ) . A game is called a monotonic game where S ⊆ T implies v(S) ≤ v(T). A convex game with nonnegative values of a characteristic function is a monotonic game. The Shapley value (Sh) is defined by the following formula: Shi (G) =

∑ (1 / n!)s!(n − s − 1)![v({S ∪ {i}) − v(S )]

.

S ⊆ N ,i∉S

The Banzhaf (B) [*] value is defined by the following formula: Bi (G) = (1/2n-1)

∑ [v({S ∪ {i}) − v(S )]

.

S ⊆ N ,i∉S

The Banzhaf value is not effective. Its normalization is called the normalized Banzhaf value (BN) . BNi(G) = BNi/(BN1+…BNn) Games such that values of their characteristic functions are only 0 and 1, are called simple games. A coalition S is called a winning coalition when v(S)=1. When the Shapley value is restricted to simple games it is called the Shapley-Shubik index (SS). Some game theory specialists, as Owen [8], deal only with convex games. Others, as Mesterton [6] deal also with games which are not convex games. As Machover [5] writes , mathematically, the Shapley value can be defined also for such games. Difficulties are connected only with some interpretations. The Shapley value measures a power of players. Some special simple games are very often used for modeling of voting. They are called weighted voting games. Each player has his weight wi. There is a threshold of voting t. A coalition S is a winning coalition where ∑ wi >t. i∈S

4

Now we shall present and compare properties of some , chosen as the most important, propositions of measuring power in the ECB while voting with rotation is applied. We concern game theory aspects. At first we present Ulrich’s model [13]. The intertemporal cooperative game (ICG) is defined by using intertemporal voting shares which are probabilities of voting, computed for each group as the quotient of the number of voting rights and the number of members. The intertemporal voting share is the weight of a player. A weighted voting game is played. The threshold is 50% of the sum of weights. Ulrich considers cases of a different number of countries in the EMU (European Monetary System). She analyzes different scenarios where the EB members vote supporting governors originated from the country which the board member represent or where the EB members care about the EMU average. Ulrich deals with cases with and without rotation . She computes the Banzhaf and the Shapley indices. The rotation scheme slightly diminishes the underrepresentation (in comparison with economic and population weight) of the countries from the first group and reduces overrepresentation of the countries from the third group. Belke and Styczynska [2] consider an enlargement to 27 countries. They apply the following assumptions. Similar to Ulrich’s model an intertemporal cooperative game (ICG) is used. The EB cooperates and is treated as one player with six votes. The other players have their intertemporal weights computed in the same way as in Ulrich’s model. The Banzhaf and the Shapley indices in case with rotation and without of rotation are computed. A shift is observed in the allocation of power during the early euro area accession phases. The power of the EB is strengthened by rotation. Kosior and others [4] analyze the generalization of Shapley-Shubik index for voting with rotation which is constructed as the weighted sum of the Shapley – Shubik indices computed for all possible auxiliary games implied by voting with rotation in the considered case . The probabilities of games are their weights. An auxiliary game is a weighted voting game with weights 1 for every player and the threshold of 50% of the number of players. We shall call this generalization the average Shapley- Shubik index (ASS; authors call it also Shapley – Shubik index) The number of considered games in case of 27 countries is 8960. It corresponds with the least common multiple of various 5

countries’ cycles of rotation among the NCB governors and the EB members. The results show that the power of the EB is strengthened by the rotation more than in the model of Belke and Styczynska. It is assumed that the difference in the methodologies of computing the Shapley – Shubik indices and their generalization is the cause of differences in power indices. They also consider a situation where some countries form additional coalitions for voting. They call these coalitions precoalitions. In spite of this name and a quotation of Owen’s paper on games with a priori unions [7] they do not use a priori unions where special construction of the Shapley value is needed1. They treat countries which form a precoalition as one player and sum their votes. Analogously to the previous case the weighted sum of Shapley – Shubik indices is computed. A priori unions cannot be used because the game with a priori unions is not defined. In this paper we used this construction for defining the average Banzhaf value (AB) and the average normalized Banzhaf value ( A(BN)). Belke and Schnuberin [1] used the preference based power index introduced by Passarelli and Barr [9]. The index measures the probability that that a player determines its outcome. The randomization scheme which is accomodated by this quasi-value is based on the multilinear extension of games. The authors confirmed the higher concentration of power in the EB. In this paper new concepts of the Shapley value and the Banzhaf value for voting with rotation will be considered2. We shall deal with a weighted value of coalition (WVC) . As in the average method we consider all possible auxiliary games. Then we construct a new game on the whole set of players where a value of a coalition is a weighted value of coalitions in considered auxiliary games. Probabilities of games are their weights. For coalitions which cannot be constructed in such a way their value is parameter (usually 0) with exception of the great coalition with value 1 or another parameter. Let us study two examples of voting with rotation to compare different methodologies. 1

Private information from A. Kosior

2

I am grateful to my colleagues from a seminar organized by the Division of Games and Decision of Institute of Computer Sciences of the Polish Academy of Sciences for the idea of this method.

6

4. Various method of defining values . Examples Example 1. There are 4 players. In each voting only 3 voters take part. In order 1,2,3,4 one of the players does not vote. Four different 3-player games are played. Each voter has one vote. For every 3-player game the Shapley Shubik index of each player equals 1/3. (i)

ICG

The method applied is based on Ulrich (2004) and Belke – Styczynska (2006) approaches. Each player has a share of 3/4. We deal with a weighted voting game (3/4, 3/4, 3/4, 3/4). The sum of weights equals 3. The winning coalitions are those in which a sum of weights of players is greater than 1.5. So, the winning coalitions are all 3-player coalitions and the great 4 player coalition. Each Shapley – Shubik index equals 1/4. Similarly, each Banzhaf index equals 3/8 and normalized Banzhaf index equals ¼. (ii)

Average method

The method applied is based on Kosior and others [4] . In each auxiliary 3-player game the Shapley - Shubik index equals 1/3. The probability of each game is 1/4 . Each player takes part in 3 games. So, a weighted sum of the Shapley – Shubik indices is equal to 3x (1/4) x(1/3) =1/4 for every player. Analogously, we compute the average sum of the Banzhaf indices (the average Banzhaf value -AB) and the normalized Banzhaf indices ( the average normalized Banzhaf vaue - A(BN)). For each i ABi=3/8, A(BN)i=1/4. The average Banzhaf is not efficient, so it can be normalized. After normalization we obtain the normalized average Banzhaf value ((AB)N) . For each i (AB)Ni=1/4. We shall see in the next section that always (AB)N=A(BN) (iii)

The weighted value of a coalition WVC)

Here we propose a new construction of the Shapley value for voting with rotation. We calculate a weighted value of a coalition. We consider all possible systems of voting rights. For each system M we construct an auxiliary game G(M). Then we construct a 4-player game where a value of a coalition is computed as a weighted sum of values of this coalition in considered

7

auxiliary 3 player games. The probability of a game is its weight. This probability equals 1/4 for each game. Let us consider 3-player games. All these games are constructed in the same way. VM (A) = 1 when #A ≥ 2 and vM(A) = 0 when #A<2. Let w be a characteristic function of the 4-player game with weighted values of coalitions (WVC). Each 3 player coalition occurs only once. So, vw(A) = 1x (1/4) x 1 = 1/4 when #A=3; Each 2-player coalition occurs twice. So, vw(A) = 2x (1/4) x 1=1/2 when #A=2. vw(A) =0 when #A<2. vw(A) = 0 when #A=1 It is impossible to define a value of the great coalition in such a way. So we define vw(A)= a for A such that #A=4. We obtained a non-simple, non-monotonic and improper game. In our game we obtained that the Shapley value equals a/4 for each player . It is the same result as in the ICG and ASS methods for a=1. The Banzhaf value equals 1/8(a-1/2) for each i, the normalized Banzhaf value equals 14 for each i . We can present the above results in the following tables. Table 1. Example 1. Games. Shapley-Shubik value computed by various methods.

I=1,2,3,4

ICG

ASS

WVC

1/4

1/4

a/4

Source: author’s work We can see that all results (with a=1 for the WVC) are similar.

Table 2. Example 1. Games. Banzhaf value and normalized Banzhaf value computed by various methods. ICG

average

WVC

Bi

3/8

3/8

1/8(a-1/2)

BNi

1/4

1/4

1/4

A(BN)i

1/4

8

Now, let us study an example where voting rights are not symmetric. Example 2. We consider a situation where there are 4 voters. Voter 1 has permanent voting right. Only 2 voters vote among voters 2,3,4. They relinquish their voting rights one by one. Four different 3-player auxiliary games are considered. We analyze the same 3 methods as in Example 1. (i)

ICG

Weights are the probabilities of voting. The weight of player one is 1, of players 2,3,4 is 2/3. So, we deal with a weighted voting game (1, 2/3, 2/3, 2/3). The sum of weights is 3. The threshold is 1,5. The characteristic function is computed as follows. v(S)=1 where #S ≥ 3 or when (#S=2 and 1 ∈ S). SS1= 1/2, SSi = 1/6 for i=2,3,4. B1=3/4, B2=B3=B4=1/4. BN1=1/2, BN2=BN3=BN4=1/6. (ii)

Average method

The following triples of voters can vote: {1,2,3}, {1,3,4}, {1,2,4} . In each auxiliary 3-player game the SSi = 1/3 for each player. The probability of each game is 1/3. The probability that player 1 takes part in 3-player game equals 1. For other players this probability is equal to 2/3. So the sum of the average Shapley - Shubik indices equals 3x(1/3)x(1/3)=1/3 (number of games x probability x SS1) for player 1. For players i=2,3,4 the average sum of the Shapley – Shubik indices equals 2x(1/3)x(1/3)=2/9. Banzhaf index in each game is equal Bi=3/4, BNi=1/3, i=2,3,4 . For average values we obtain AB1=3x(1/3)x(3/4)=3/4, ABi=2x(1/3x(3/4)=1/2, i=2,3,4, A(BN)1=1/3, A(BN)i=2/9, i=2,3,4. (AB)N1=1/3, A(BN)i=2/9, i=2,3,4. (iii)

WVC

We use the method defined in point (iii) of the Example 1. The sets of players of 3-player auxiliary games G1, G2, G3 are {1,2,3}, {1,3,4} and {1,2,4} respectively. Each game is a weighted voting game with weights 1 for every player and threshold 1,5. The winning coalitions are presented in the following table.

9

Table 3. Example 2. Winning coalitions in 3-player games in the WVC method Game →

G1

Winning

{1,2,3}, {1,3,4},

{1,2,4},

coalitions

{1,2},

{1,4},

{1,2},

{2,3},

{3,4},

{1,4},

{1,3}

{1,3}

{2,4}

G2

G3

Source:author’s work Each auxiliary game is played with the probability 1/3. So there are the following weighted values of coalitions. vw({1,2,3}) = vw({1,3,4})=vw({1,2,4}) =(1/3)x 1=1/3; vw({1,c}) = 2x(1/3)x 1=2/3, c=2,3,4; vw({c,d}) = 1x (1/3)x 1=1/3 where c,d=2,3,4, c ≠ d; vw({c})=0 for c=1,2,3,4. So, we construct game w on set {1,2,3,4} where w(S)=vw(S) if S is a coalition from one of the games G1, G2, G3 . Some subsets of {1,2,3,4} do not occur in games Gi, i=1,2,3 , so we have to define their values. We introduce parameters a and b and define vw({1,2,3,4}) = a and vw({2,3,4}) = b. The most intuitive way of describing a and b is a=1 (because we deal with the great coalition) and b=0 (because such coalition never exists). We compute the Shapley value of game G = ({1,2,3,4},vw). Sh1 (G) =(1/24)[6(a-b) +3x(4/3)]= (a-b)/4 +1/6. Shi(G) = (1/24)[6a – 2+2b - 2/3] = a/4 + b/12 – 1/18 for i=2,3,4. The following table presents the Shapley values for the three considered methods. Table 4. Example 2. The Shapley values in various methods. Method →

ICG

Average method

WVC

1

1/2

1/3

(a-b)/4 +1/6

2,3,4

1/6

2/9

a/4 +b/12 – 1/18

Player ↓

Source: author’s work Let us compute the Banzhaf and the normalized Banzhaf value.

10

B1(G) = (1/8)x[(a-b)+((1/3)-(1/3))x3+((2/3)-0)x3]=(1/8)(a-b+2) Bi(G)=(1/8)x[(a-(1/3))+((1/3)-(2/3))x2+(b-(1/3))+2/3 +(1/3)x2]=(1/8)(a-b), i=2,3,4. 4

∑

Bi =(1/4)(2a-2b+1), BN1=(a-b+2)/(2(2a-2b+1),

i =1

BNi=(a-b)/2(2a-2b+1), i=2,3,4. The following tables present The Banzhaf value and the normalized Banzhaf value for three considered methods. Table 5. Example 2. The Banzhaf values in various methods. Method →

ICG

Average method

WVC

1

3/4

3/4

(1/8)(a-b+2)

2,3,4

1/4

1/2

(1/8)(a-b)

Player ↓

Table 6. Example 2. The normalized Banzhaf values in various methods. Method →

ICG

Average method WVC

1

1/2

1/3

(a-b+2)/(2(2a-2b+1)

2,3,4

1/6

2/9

(a-b)/2(2a-2b+1)

Player ↓

This time the results of the use of three different methods are not similar. Comparing the results presented in Table 5 we see that player 1 with permanent voting right is the strongest when we use the ICG method for parameters satisfying a-b<4/3. Player 1 is the weakest when the WVC method is used (for parameters satisfying a-b ≤ 2/3). Player 1 is stronger than each of the players 2,3,4 in the ICG and ASS methods and for b ≤ 2/3 in WVC method. Let us consider a=1 and b=0. Then player 1 is stronger than each of the players 2,3,4 when we use the WCV method. Then we have Sh1=5/12 and Shi = 7/36 for i=2,3,4. In this case the results of the different methods are compared in the following table.

11

Table 7. Example 2. The Shapley values in the WVC method for a=1 and b=0 Method →

ICG

ASS

WVC

1

1/2

1/3

5/12

2,3,4

1/6

2/9

7/36

Player ↓

Source: author’s work We obtain the different results for the Banzhaf value. Player 1 is the strongest when the WVC method is used if and only if a-b>4. For a=1 and b=0 player 1 is the strongest in the ICG and average methods. If we consider the normalized Banzhaf value, he is the strongest when the WVC method is used if and only if ½ >a-b. For a=1 and b=0 player 1 is the strongest in the ICG and WVC methods. Table 8. Example 2. The Banzhaf values in various methods for a=1, b=0 Method →

ICG

Average method

WVC

1

3/4

3/4

3/8

2,3,4

1/4

1/2

1/8

Player ↓

Table 9. Example 2. The normalized Banzhaf values in various methods for a=1, b=0 Method →

ICG

Average method WVC

1

1/2

1/3

1/2

2,3,4

1/6

2/9

1/6

Player ↓

5. Various methods of defining values. Properties The results of the previous section are connected with some general properties of the WVC method. We consider the most intuitive case where we set 1 as the value of the great coalition and 0 for other which do not occur in auxiliary games. Our results yield from the symmetry of the construction of rotation scheme and anonymity of the Shapley value. First, let us deal with the situation presented in Example 1. Let us consider a voting system with rotation where there are n voters

12

and k voting rights, ni. Let us consider f:N → N, f(h)=(h+j-i)(mod n). f is a bijection and f(i)=j. Let us denote Nm={m, (m+1)mod n,…,(m+k-1)(mod n)} with n(mod n) =n, m=1,…,n. Nm is a set of voters with voting rights in the m-th voting. Let us see that (i)

f(Nm)=N(m+j-i)(modn) ,

(ii)

if i∉ S then j∉f(S) for S ⊆ N,

(iii)

#S = #f(S),

(iv)

If S ⊆ Nm then f(S) ⊆ N(m+j-i)(modn).

Let us construct game G=(N,vw) defined by WVC in the following way. (WVC1) If there exists m such that S ⊆ Nm then vw(S) = (1/n) •

∑

vm(S), where vm is a game on Nm, vm(S)=1 if #S>k/2 and vm(S)=0

m, S ⊆ N m

otherwise. (WVC2) vw(N)=1. (WVC3) vw(S)=0 for S ≠ N such that there does not exist Nm such that S ⊆ Nm.

13

Because f is a bijection and by properties (i)-(iv) we have that vw(S)=vw(f (S)). The Shapley value is anonymous ,so,

∑

s!(n-s-

i∉S , S ⊆ N

1)![vw(S ∪ {i})-vw(S)] =

∑

s!(n-s-1)![vw(fi,j(S) ∪ {j})-vw(fi,j(S))], and

j∉ f ( S ), f ( S )⊆ N

Shi(G)=Shj(G). The Banzhaf value is also anonymous [**], so (1/2n-1) •

∑

[vw(S ∪ {i})-vw(S)] = (1/2n-1) •

i∉S , S ⊆ N

∑

[vw(fi,j(S) ∪ {j})-vw(fi,j(S))]. Tne

j∉ f ( S ), f ( S )⊆ N

normalized Banzhaf values are equal because the Banzhaf values are equals. □ If the condition (WVC3) were to be replaced by vw(S) = aS for S ≠ N such that there does not exist Nm such that S ⊆ Nm then for the generalization of the above lemma the following consistency condition is needed: vw(S)=vw(f (S)) for every bijection f such that f (i)=j and every i,j∈N. Now, let us study a situation in Example 2. It is a generalization of the situation in Example 1 and Lemma 2 is a generalization of Lemma 1. Let the set of voters N be divided into r disjoint groups of voters Ni, each with ki voting rights, i=1,…,q. ni=#Ni. We assume that voters are ordered and N1={1,…,n1}, N2={n1+1,…,n1+n2},…,Ni={n1+…+ni-1+1,…,n1+…ni-1+ni}. Let us denote, m0=0, mi=n1+…+ni, i=1,…q . Then Ni={mi-1+r: r=1,…,ni}, i=1,…,q. In the following we shall use convention that m(modm)=m. In what follows we shall use for numbers x ,y a function d(x,y) where d(x,y)=1 if x ≤ y and d(x,y)=0 if x>y. Voters in part Ni have ki voting rights, ki ≤ ni. k=k1+…+kn. The rotation scheme works as follows. We assume that in the first round the first ki voters vote in Ni, i=1,…,n. They are mi-1+r, r=1,…,ki. In the second round the first voter relinquishes his vote to the voter who is the next voter after the voter mi-1+ki. If this voter is the last in Ni, then the new voter with voting right is the first voter in Ni. So his number is (mi-1+ki+1)(modmi) + d[mi-1+(mi-1 +ki+1)(modmi), mi-1]mi-1. Applying this scheme in round s the first voter in this

14

round, the voter mi-1+s(modni) relinquishes his voting rights to the voter following the ki th voter in round s. If this voter is the last in Ni then the new voter with voting rights is the first in Ni. So, in round s the set of voters with voting rights is Nis = ={[mi-1+s(modni)+t](mod mi)+mi-1d([mi-1+s(modni)+t](mod mi), mi-1): t=0,…,ki-1}, where s=1,…,K, where K is the lowest common multiple of numbers n1,…,nq. We define K in such a way because we consider all possible cases of appearing ki rights of voting in part Ni obtained by the scheme. We shall denote such system of voting rights V(n1,…,nq,k1,…,kq). For system V(n1,…,nq,k1,…,kq) let us define game Gs = (Ns, vs), s=1,…,K, q

where Ns = U Nis and vs is defined as follows. i =1

For T ⊆ Ns v(T)=1 if #T>k/2 and v(T)=0 otherwise. The family of games Gs is constructed of all games which may be played using the rotation scheme. The probability of each game is 1/K. Now we defined game G=(N,v) where v(T)=

∑ v (T ) s

=1/K

if there exists s such that T ⊆ Ns, v(T)=1 where T=N and

s:T ⊆ N s

v(T)=0 otherwise. v(T) is equal to the expected value of vs in case where it is possible to compute vs(T), 1 for the great coalition N and 0 for other coalitions where the expected value cannot be computed. The following lemma holds. Lemma 2. Let us consider a system of voting rights V(n1,…,nq,k1,…,kq) and h,j ∈ Ni. Then Shh(G)=Shj(G), Bi(G)=Bj(G), BNi(G)=BNj(G). Proof. Let j,h ∈ N, j>h. We define fi:Ni → Ni, p=mi-1+r ∈ Ni, fi(p)= [p+(j-h)(modni)](modmi) + d([p+(j-h)(modni)](modmi),mi-1)mi-1 . fi is a translation on Ni over j-h. If j,h ∈ Ni , fi(h)=j and f(h)=j. We join all fi on N into f. f:N → N, f(p)=fi(p) where p ∈ Ni, i=1,…,q.

15

Let us see that fi is a bijection on Ni. So, f is a bijection on N. Let g:{1,..K} → {1,..,K}, g(s)=(s+j-h)(mod K). g is a bijection on {1,…,K}. Let us note that fi(Nis)=Nig(s). So, T ⊆ Ns if and only if f(T) ⊆ Ng(s) and vs(T)=vg(s)(f(T)). Therefore v(T)=v(f(T)) and for h,j ∈ Ni, v(T ∪ {h})=v(f(T) ∪ {j}). By anonymity of the Shapley value we get Shh (G) =

∑ (1 / n!)t!(n − t − 1)![v({T ∪ {h}) − v(T )]

=

T ⊆ N ,h∉T

=

∑ (1 / n!)t!(n − t − 1)![v({ f (T ) ∪ { j}) − v( f (T ))] =Shj(G) . Analagously,

f (T )⊆ N , j∉ f (T )

anonymity of the Banzhaf value ([**])gives Bh (G) = (1/2n-1)

∑ [v({T ∪ {h}) − v(T )]

=

T ⊆ N ,h∉T

= (1/2n-1)

∑ [v({ f (T ) ∪ { j}) − v( f (T ))] =Bj(G) . The equality of the Banzhaf

f (T )⊆ N , j∉ f (T )

values is the cause of the equality of the normalized Banzhaf vales.□ The theses of the above lemmas are true also for other methods. Let us consider ICG. All voters from the same group have the same weights, so their Shapley values are the same. By the same reason their Banzhaf values are the same. The equality of the normalized Banzhaf values yield from the last property. So, the following lemma holds. Lemma 3. Let us consider a system of voting rights V(n1,…,nq,k1,…,kq) and the ICG method, h,j ∈ Ni. Then Shh(G)=Shj(G), Bi(G)=Bj(G), BNi(G)=BNj(G). The average method has also the above property. The following lemma holds. Lemma 4. Let us consider a system of voting rights V(n1,…,nq,k1,…,kq) and the average method. h,j∈ Ni. Then ASSh(G)=ASSj(G), ABi(G)=ABj(G), A(BN)i(G)=A(BN)j(G). Proof. Let us consider the auxiliary game G. The game is weighted majority game with k=k1+…+kq players, each with weight 1, and the threshold E(k/2)+1. So, Shh(G)=Shj(G), Bh(G)=Bj(G) and BNh(G)=BNj(G). The average value for player j is computed according to formula: probability of game • number of games with player j • value of player j in game G. Probability of game is 1/K, the number of games with player j is the same as the number of game with player h if h and j are from the same Ni. The Shapley values of all

16

players are the same in game G. So, the Banzhaf values and the normalized Banzhaf values are the same.

□

As it was mentioned in Section 4, A(BN)=(AB)N. We shall prove this fact. Lemma 5. For the average method A(BN)j=(AB)Nj for all j∈ N. Proof. Let us consider the family of games Gs =(Ns, vs), s=1,..,K, the same as in the proof of lemma 4 and use the same arguments. All games are k-person weighted voting games with weight 1 for each player and threshold E(k/2)+1. So, in every game all players are symmetrical and have the same Banzhaf value and the normalized Banzhaf value. All games Gs are the same game from the game theoretical point of view, so their Banzhaf values and normalized Banzhaf values are the same. Let b denote the Banzhaf value, b’ the normalized Banzhaf value. b’=1/k because the sum of the normalized Banzhaf values in each game Gs is equal to 1. Let j∈ Ni. #{s: j∈ Ns} is the same for all j from Ns . We denote ci=#{s: j ∈ Ns}. It is the number of games Gs in which player j plays. q

n

(AB)j=(1/K)cib,

q

∑ AB j = (1/K)b ∑∑ j∈N ci =(1/K)b ∑ nici . j =1

i =1

i

i =1

q

n

Because ci=(K/ni)ki, nici=Kki and

∑

(AB)j=(1/K)b

j =1

q

∑ Kki =b ∑ ki =bk. i =1

i =1

n

Then , N(AB)j= ABj/ ∑ AB j =(1/K) cib/(bk) =(1/K)(ci/k)=ci/kK. j =1

A(BN)j= (1/K)cib’=(1/K) ci (1/k)=ci(Kk) = N(AB)j. □ Let us consider the following example to see that the Banzhaf values and the normalized Banzhaf values can be different in ICG and WVC methods. Example 3. N={1,2,3}, N1={1}, N2={2,3}, q1=1, q2=1. In the ICG we deal with a voting weighted game (1, ½, ½). B1=3/4, B2=B3=1/4, BN1=3/5, BN2=BN3=1/5. In the WVC method 2 auxiliary games are considered with N1={1,2} and N2={1,3}, each with probability ½. So we obtain the following characteristic function v. v({1,2,3})= a, v({2,3})=b, v({1,2})=v({1.3)})=1/2 and ) for others. B1=(1/4)(a-b+1), B2=B3=(1/4)(a+b), BN1=(a-b+1)/(3a+b+1),

17

BN2=BN3=(a+b)/(3a+b+1). For a=1, b=0 we get B1=1/2, B2=B3=1/4, BN1=1/2, BN2=BN3=1/4. Conclusions The considered example of a non standard voting method is constructed with a vision of effectiveness, transparency and simplicity. It would be a very good property of voting method if well known measures of voting power were applied. Two of the most popular measures of voters’ power are the Shapley value (the Shapley –Shubik index in case of simple games) and the Banzhaf value. We show that voting with rotation used in the European Central Bank leads to controversies in defining these values for such voting. Different methods give different results. It is especially important where we compute how rotations of voters respond to the enlargement of the Euro zone. The impact of using the rotation scheme may depend on the method used for computing the values. Thus, it is difficult to measure power of voters in case of voting with rotation. We propose a new method, called weighed value of coalition (WVC), where, unlike in some considered methods, the game illustrating the rotation scheme is constructed. The method has good computational properties in spite of a complicated game form.

Literature [*] Banzhaf J. ,,”Weighted voting doesn’t work. A mathematical analysis.”, Rutgers Law Review 19(2), 317-343, 1965 [1] BELKE A., B. VON SCHUBERIN, European Monetary Policy and the EBC Rotation Model. Voting Power of the Core versus Periphery, Discussion Paper 983 ,2010, DIV Berlin, [2] BELKE A., STYCZYNSKA B.,The allocation of power in the enlarged EBC Govering Council: An Assesment of the EBC Rotation Model, Journal of Common Market Studies 2006 Vol.44, No.5,pp.865-897 [3] EBC Monthly Bulletin , European Central Bank 2009, July, Frankfurt/Main [**] Grabish M., M. Roubens, “An Axiomatic approach to the concept of interaction among players in cooperative games”, International Journal of Game Theory28, 547-565, 1999. 18

[4] KOSIOR A. , M.ROZKRUT, A TOROJ, Rotation Scheme of the EBC Governing Council: Monetary Policy Effectiveness and Voting Power Analysis, [in:] Raport na temat pełnego uczestnictwa Rzeczypospolitej Polskiej w trzecim etapie Unii Gospodarczej i Walutowej”, National Bank of Poland, 2008,pp. 53-102 [5] MACHOVER M., Notions of a priori voting power. Critique of Holler and Windgren, Homo Oeconomicus 2000, Vol.16, pp.415-425 [6] MESTERTON – GIBBONS M., An Introduction to Game Theoretic Modelling, American Mathematical Society, 2001 [7] OWEN G.,Values of games with a priori unions [in:] Hein R., Moeshlin O.,(eds) Essays in Mathematical Economics and Game Theory, Springer-Verlag, Berlin 1977, pp.76-88 [8] OWEN G., Game Theory, Third Edition, Academic Press, San Diego, 1995 [9] PASSARELLI F., BARR J., Preferences, the agenda setter, and the distribution of power in the EU, Social Choice and Welfare, 2007, Vol.28, pp.41-60 [10] SHAPLEY L.,S., A value for n-person games [in:] Contributions to the Theory of Games, Vol.II, Kuhn H.W, Tucker A.W. (eds), Annals of Mathematical Studies, 1953, Vol.28, pp.307-317 [11] SHAPLEY L.S., SHUBIK M.,A method for evaluating the distribution of power in a committee system, American Political Science Review 1954, Vol. 48, pp. 787-792 Sosnowska H., “Analysis of voting method used in the European Central Bank”, Pperations Research and Decisions, p.75-86, 2013 [12] TILMANN P., Strategic Forecasting of the FOMC, European Journal of Political Economy 2011, Vol. 27, pp.547-553 [13] ULRICH K., Decision making of the EBC. Reform and voting power, Discussion Paper 2004, No. 04-70, ZEW, Mannheim

19