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Voting power in an ideological spectrum
Mathematical Social Sciences 40 (2000) 215–226 www.elsevier.nl / locate / econbase
Voting power in an ideological spectrum ´ The Markov-Polya index Thommy Perlinger* Lund University, Department of Statistics, POB 7008, 220 07 Lund, Sweden Received December 1997; received in revised form December 1998; accepted August 1999
Abstract Power distributions of voting games in which only certain coalitions are allowed, due to the fact that the players are spread across an ideological spectrum, are considered. The paper extends Edelman’s model [Edelman, P., 1997. A note on voting. Math. Soc. Sci. 34, 37–50] concerning ´ allowable coalitions. The Markov-Polya index is introduced as a parametrized family of power indices which has Edelman’s extension of the Shapley-Shubik index as a special case. Unlike the extended Shapley-Shubik index, it puts different weights on the allowable coalitions. For particular parameter values, simple explicit power formulas are derived. 2000 Elsevier Science B.V. All rights reserved. ´ distribution Keywords: Voting power index; Connected coalitions; Initiator; Markov-Polya
1. Introduction Game theoretic solutions to voting situations, in the form of common power indices, do not take into account that the voters are often spread across an ideological spectrum. Voting power indices in common use are, for example, the Banzhaf-Coleman index, (Banzhaf, 1965) which treats all coalitions as equally probable and the Shapley- Shubik index, (Shapley and Shubik, 1954) where all permutations (i.e. vote sequences) are treated as equally probable. In these two power indices, voters are assumed anonymous and treated symmetrically. When in a voting situation, a coalition comes into being, this procedure can be regarded as the formation of a series of subcoalitions, where one voter at a time joins the coalition. When ideological positions matter, the problem arises of how to model this *Tel.: 146-46-222-3653; fax.: 146-46-222-4220. E-mail address: [email protected] (T. Perlinger) 0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0165-4896( 99 )00044-X
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process in a reasonably realistic way. Should, for instance, a coalition that consists of two voters from opposite extremes be as likely as a coalition that consists of two voters who are ideological neighbors? Is it at all reasonable to believe that the former coalition will arise? These questions suggest that in ideological voting situations, all coalitions should not be treated as equally probable to occur. The problem then is: which are the least likely coalitions and how do we restrict them? An interesting approach to this kind of problem is suggested by Edelman (1997). His model of coalition formation is based on the concept of convex sets, and only a certain type of coalitions are allowed to occur. A specific application of the model is when the players are spread across an ideological spectrum, such as the usual left–right ideology scale. The idea is that a coalition is permissible if and only if there are no ideological gaps between members of the coalition; in other words, there can be no non-member who is ideologically in an intermediate position between any two coalition members. Edelman shows that the Shapley-Shubik index very naturally extends to accommodate these restricted games, which henceforth will be referred to as spectrum games. In voting situations modeled by spectrum games, one would expect the player in the median position to be the strongest in terms of voting power, since he is connected to both flanks and thus have more possibilities to form and join coalitions. This is not the case, however, when voting power in spectrum games is measured by the extended Shapley-Shubik index. Somewhat surprisingly, it is instead the players in quartile positions who are the strongest. Only the outer players in the most extreme positions on the ideology scale are weaker than the median player. The purpose of the present paper is to study voting power in spectrum games by combining Edelman’s ideas with those of Berg (1997). To do this, we first introduce the concept of an initiator as the first member of a coalition. Edelman’s model determines implicitly an initiator distribution, which favors (or rather dis-favors) the player in the median position. Taking the cue from Berg, we then introduce a model for coalition formation, in which each grand coalition is seen as the realization of a sequence of ´ Markov-Polya binary random variables. In this manner, we derive a parametrized family of Shapley-Shubik type indices, and in the process we also obtain a simple and flexible family of initiator distributions. Edelman’s model of voting is a special case, and other special cases have simple and appealing probabilistic interpretations. The rest of this paper is organized as follows. To establish notation, Section 2 gives a brief overview of n-person weighted voting games and voting power indices. The definition of spectrum games, i.e. Edelman’s model regarding allowable coalitions applied to voting games on an ideological spectrum, is given in Section 3 along with some basic results. In this section we also define Edelman’s extension of the ShapleyShubik index. In Section 4, which is the main part of the paper, we introduce the ´ Markov-Polya index. We discuss properties of the index, and we provide illustrations and interpretations. Section 5, finally, concludes the paper.
2. n-person weighted voting games and power indices A normalized n-person weighted voting game is a cooperative n-person game in which the players possess a varying number of votes ( p1 , . . . , pn ), and where a coalition
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S is either winning or losing. A coalition is winning if and only if the aggregated votes of the coalition members exceed a certain quota q. Such a game is represented by the notation [q; p1 , . . . , pn ] and the characteristic function is 1
v(S) 5
5
if
O p $q O p , q. i
;i [S
0
if
(1)
i
;i [S
When the influence of the individual players is measured in these games, it is common to use some power index. A power index determines the players’ power mainly on the basis of so called swing sets. A swing set for player i is a winning coalition where i is an influential member, i.e. by defecting he can turn the coalition over into a losing one. For such a coalition we thus have v(S) 2 v(S\hij) 5 1,
(2)
and we say that the player i is pivotal in the coalition S. The Shapley-Shubik index is of particular interest here. This index focuses on permutations on the set of players (vote sequences). A permutation is an arranged row consisting of all the n players, which could be interpreted as the way in which the grand coalition (i.e. the coalition consisting of all the players) is coming into being. This interpretation means that player i is expected to join the coalition formed by the players in front of him in the row. The Shapley-Shubik index then registers if player i is pivotal in this coalition. There are n! such arrangements, but many of them will give the same value because the value of a coalition does not depend on the particular order of the coalition forming process. Taking account of this, the Shapley-Shubik index for the game v is given by
O
(s 2 1)!(n 2 s)! ]]]]] [v(S) 2 v(S\hij)], i [ N. n! S #N,i [S
wi [v] 5
(3)
where uSu 5 s and N 5 h1,2, . . . , nj. This means that in terms of the Shapley-Shubik index, it is better for a player to hold the balance of power in small coalitions (which indicates that the player is strong), rather than being pivotal in coalitions with approximately half of the players (which the weaker players are capable of also). In each permutation there is exactly one player who is pivotal, which guarantees that the sum of the players’ Shapley-Shubik indices Eq. (3) equals 1, i.e.
O w [v] 5 1. n
i
(4)
k51
The Shapley-Shubik index Eq. (3) can be given a probabilistic interpretation. If all the n! permutations are regarded as equally probable, Eq. (3) represents the expected number of times player i will be in a pivotal position. Henceforth, this interpretation will be used. For an axiomatic definition of the Shapley-Shubik index Eq. (3), see, e.g., Owen (1995).
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3. Spectrum games We shall now define a class of voting games where certain coalitions are not allowed. This type of game can, for example, be applicable to voting situations where the players form an ideological spectrum (usually from left to right). In such cases it is hardly likely that a coalition member deviates too much in either direction, i.e. a voter on the left flank probably hesitates before joining a coalition containing only players from the right wing. To define which coalitions are to be allowed, we need the following definitions. Definition 1. Two voters are called ideological neighbors if there are no other voters who ideologically are in a position between them. Definition 2. A permutation of the n players is called a maximal chain if and only if the voter in position i (in the permutation) is ideological neighbor to a voter in position 1, 2, . . . , i 2 1, ;i [ N. As explained in Section 2, a permutation of the n players can be thought of as a way in which the grand coalition comprising all the players is built up. It follows from the above definitions that the set of maximal chains represents all the ways in which the grand coalition can be formed, such that at each step in the process we have a connected coalition, i.e. no ideological gaps between coalition members. Definition 3. A spectrum game [q; p1 , . . . , pn ] Sn with n players is a weighted voting game [q; p1 , . . . , pn ] where the players are N 5 h1, . . . , nj, and the allowable coalitions are all coalitions where the members constitute a connected interval on the positive integers. Note that the players numbers /labels and positions on the spectrum coincide. Hence the set of allowable permutations in a spectrum game is the set of maximal chains. The following two lemmas establish important facts about spectrum games. The lemmas are proven in Edelman’s paper (Edelman, 1997). Alternative proofs are given in Section 4. Lemma 1. In a spectrum game with n players there are 2 n21 maximal chains. Lemma 2. In a spectrum game with n players there are begin with player i, i [ N.
Sni 2211D maximal chains that
Next we introduce a suitable index of voting power in spectrum games. We follow Edelman (1997) and define the extension of the Shapley-Shubik index in the same vein as the ordinary Shapley-Shubik index Eq. (3), restricting calculations to the set of maximal chains. For each maximal chain, there is a point at which player i enters a connected coalition. The coalition consisting of i and his predecessors in the chain M is denoted M i . The extended Shapley-Shubik index is now defined as
1 w i* [v] 5 ] m
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
219
O [v(M ) 2 v(M \hij)],
(5)
i
i
i [ N,
M
where the summation is over all the m maximal chains M. w *i measures the i:th player’s expected influence under the assumption that all maximal chains are equally probable. As in the ordinary Shapley-Shubik index Eq. (3), there is exactly one player in each maximal chain who is in a pivotal position. Therefore
O w *[v] 5 1. n
(6)
i
k51
As shown by Edelman (1997), the power distribution of the spectrum game [k 1 1; 11 1, . . . , 1] n52k , measured by the extended Shapley-Shubik index Eq. (5) is S
w i* 5
Si 2k 1D / 2
5
k 11
,
k
1/2 ,
1 # i # k,
(7)
i 5 k 1 1.
Due to the symmetry with respect to the center, the power distribution for the players on the other half of the ideological scale follows immediately. Edelman shows that if n is moderately large, only the outermost players are weaker than the median player and, moreover, the quartile players tend to dominate the game. The problem of power distribution among voters in a spectrum game will be examined more closely in the next section.
4. The Markov-Po´ lya index Obviously, allowing only connected coalitions to be formed entails a fairly radical reduction in the number of allowable permutations. Therefore we may expect that some of the properties of the ordinary Shapley-Shubik index are altered, when we use the modified formula Eq. (5). In this section we focus on one of these properties. It is convenient to introduce here the concept of an initiator as the first member of a coalition, in other words the first player in a maximal chain or allowable permutation. With no restrictions imposed on the coalitions, each player appears as initiator in an equal number of permutations. However, in a spectrum game this is no longer the case. As is readily checked, in the extended Shapley-Shubik index Eq. (5) the players are no longer equally likely to be the initiator. Our idea here is to introduce a more general index of power, or class of indices, which makes it possible to vary the initiator distribution. To this end we use a probabilistic approach. Consider a sequence of n 2 1 binary (0,1) stochastic variables X1 , X2 , . . . , Xn21 which all have the same expectation E(Xk ) 5 1 / 2, i.e. equal probability on 0 and 1. It is easy to see that there is a one-to-one correspondence between the set of possible realizations of the sequence of stochastic variables and the set of maximal chains in a spectrum game on n players. If, in the left–right ideological spectrum, we interpret Xk 5 0 as the event that the maximal chain at the kth step moves to the left and Xk 5 1 that it moves to the
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right, then a particular realization x 1 , x 2 , . . . , x n 21 determines uniquely a maximal chain. The number of zeros (or ones) in the sequence implicitly determines the initiator of the maximal chain. It is now easy to prove the two lemmas of Section 3. Proof of Lemma 1. The number of maximal chains is the same as the number of realizations of X1 , . . . , Xn21 , i.e. 2 n21 . Proof of Lemma 2. The number of maximal chains starting in position i is the same as the number of realizations of X1 , . . . , Xn21 , in which there are exactly i 2 1 zeros, i.e. n21 . i21
S
D
The extended Shapley-Shubik index considers the maximal chains in a spectrum game as equally probable. This is equivalent to stating that the sequence X1 , X2 , . . . , Xn21 defined above represents a set of uncorrelated stochastic variables. It seems natural to ask: is it necessary for these stochastic variables to be uncorrelated? A possible modification of the extended Shapley-Shubik index is therefore to allow a correlational structure on the sequence of stochastic variables. It seems natural to assume that at each step the further development of the maximal chain depends on its past behavior. We therefore define the dependency among the random variables through conditional probabilities. An interesting approach is the ´ suggestion by Berg (1997) to use a Polya-Eggenberger model to describe the dependency structure among the random variables. (See also, e.g., Johnson et al., 1992.) Consider the following (idealized) urn sampling experiment. Initially, we have one white and one black ball in the urn. To describe outcomes of the sampling process, we use our previously defined set of binary random variables where Xk 5 1 is interpreted as the event: white ball at the kth draw. At each stage of the sampling process, we draw a ball at random and replace the ball drawn together with u balls of the same color. Hence at the kth stage we have a total of 2 1 (k 2 1)u balls in the urn, and the conditional probability of drawing a white ball in the kth stage, given a sequence x 1 , x 2 , . . . , x k 21 in previous draws, is given by 1 1 (x 1 1 ? ? ? 1 x k 21 )u P(Xk 5 1ux 1 , . . . , x k21 ) 5 ]]]]]]]. 2 1 (k 2 1)u
(8)
The probability of drawing a black ball is, of course, 1 1 (k 2 1 2 (x 1 1 ? ? ? 1 x k 21 ))u P(Xk 5 0ux 1 , . . . , x k21 ) 5 ]]]]]]]]]]. 2 1 (k 2 1)u
(9)
The probability of getting exactly k white balls in the urn experiment does not depend on the particular order of the balls drawn. Thus, the dependency structure has the effect that maximal chains originating from the same initiator are all equally probable. However, maximal chains originating from different positions are not equally likely to occur. ´ We call the resulting maximal chain probabilities Markov-Polya probabilities, due to
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
221
´ their close relation to the Markov-Polya distribution, see Johnson et al. (1992). It is convenient to introduce the following notation for generalized ascending factorials a [x,u ] 5 a(a 1 u ) ? ? ? (a 1 (x 2 1)u ), a [ R, x [ Z 1 , a [0,u ] ; 1.
(10)
n52k 11
´ Lemma 3. Consider the spectrum game [k 1 1; 1, . . . , 1] S . The Markov-Polya probability of an arbitrary maximal chain with the player i as the initiator is 1 [i 21,u ] 1 [n2i,u ] Pi (n; u ) 5 ]]]] , i [ N. 2 [n21,u ]
(11)
For this to represent a proper probability distribution defined on the set of all 2 n21 maximal chains, we add the restriction u . 2 1 /(n 2 2). Proof. The probability of the maximal chain corresponding to the realization x 1 , . . . , x n21 is P(X1 5 x 1 , . . . , Xn 21 5 x n 21 ) 5 P(Xn 21 5 x n 21 ux 1 , . . . , x n22 ) 3 P(Xn22 5 x n22 ux 1 , . . . , x n23 ) 3 ? ? ? 3 P(X2 5 x 2 ux 1 ) 3 P(X1 5 x 1 ), where the conditional probabilities are given by Eqs. (8) and (9). The fact that player i is the initiator tells us that the realization x 1 , . . . , x n 21 contains exactly i 2 1 zeros and n 2 i ones. And so Eq. (11) follows. The restriction on u follows from the fact that u can be interpreted as the adding of balls to a urn. That is, if u is negative this means that we after each stage withdraw balls from the urn. For Eq. (11) to represent a proper probability distribution we have to make sure that we always have a positive number of balls of both colors in the urn. At the last stage, if u is negative, we have at least 1 1 (n 2 2)u balls of each color and, hence, we have the restriction 1 1 1 (n 2 2)u . 0⇔u . 2 ]]. h n22 Clearly, there are a number of maximal chains that have the player j as initiator. Lemma 2 together with Lemma 3 reveals that the probability for a maximal chain to have player j as the initiator is [ j 21,u ] [n2j,u ] 1 n21 1 q j (n; u ) 5 j 2 1 ]]]] , 2 [n21,u ]
S D
j [ N,
(12)
´ which is recognized as Markov-Polya distribution probabilities. Remark. The probabilities Eq. (11) are useful also when there are no restrictions on the coalitions, as shown by Berg (1997). Consider Eq. (11) as a probability distribution on the coalitions (subsets of N), where i represents the size of the coalitions i;i 5 1, . . . ,n.
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
222
Now, both the Shapley-Shubik index Eq. (3) and the Banzhaf-Coleman index appears as expectations of the random variable [v(S) 2 v(S\hij)], i.e. Eu 50 [v(S)-v(S\hij)] (Banzhaf-Coleman index) Eu 51 [v(S)-v(S\hij)] (Shapley-Shubik index) We are now in a position to introduce a family of power indices based on the ´ ´ Markov-Polya model Eq. (11), and it is therefore natural to call it the Markov-Polya index. The definition is in the same vein as Edelman’s definition Eq. (5) of an extended Shapley-Shubik index. ´ Definition 4. The parametrized Markov-Polya index of power in a spectrum game with player set N 5 h1,2, . . . , nj, is defined by
O [v(M ) 2 v(M \hij)]P (n; u ), i
zi (u ) 5
i
M
(13)
M
i
with M as in Eq. (5), and where the summation is over all maximal chains M. PM (n; u ) is the probability Eq. (11) for the maximal chain M in a spectrum game on n players. The difference between the extended Shapley-Shubik index Eq. (5) and the Markov´ Polya index Eq. (13) is that the latter assigns different weights to maximal chains. Next we derive an explicit formula for the voting power distribution of a spectrum game. Theorem 1. Consider the spectrum game [k 1 1; 1, . . . , 1] n52k11 . Based on the S ´ Markov-Polya index Eq. (13), the power distribution of this game is 1 1 (i 2 1)u ]]]] qi (k 1 1;u ), 1 # i # k 2 1 ku zi (u ) 5 2 1 2ku ]]]qk11 (k 1 1;u ), i 5 k 1 1, 2 1 ku
5
(14)
with an obvious symmetry for the other half of the ideological scale. Furthermore, the ´ Markov-Polya index is normed, i.e.
O z (u ) 5 1, n
i
;u.
(15)
i51
Proof. For the player i; 1 # i # k, there are exactly k initiators for whom i is a possible pivot. These are the players i 1 1, . . . , i 1 k. Since all maximal chains with a specific initiator are equally probable, it is sufficient to determine the number of maximal chains where i is in a pivotal position, and i 1 j; j 5 1, . . . , k, is the initiator. Hence, given j, we are looking for the number of maximal chains in a game with k players that starts in position j, multiplied by the number of maximal chains in a game with k 1 1 players that starts in position i, multiplied by the probability for a maximal chain with the player i 1 j as the initiator. Lemma 2 (used twice) and Lemma 3 give
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
1 1 Si 2k 1D O Skj 22 11D ]]]]] 2 1 1 k 5S D ]]]] O Sk 2j 1D(1 1 iu ) i21 2 1 1 k 5S D ]]]] (2 1 (k 1 1)u ) i21 2 1 1 k 5S D ]]]] i21 2 [i 1j 21,u ] [n 2i 2j,u ]
k
zi (u )
223
5
[n21,u ]
j 51
[i,u ] [k 112i,u ] k 21 [n21,u ]
[ j,u ]
(1 1 (k 1 1 2 i)u )[k 212j,u ]
j 50
[i,u ] [k 112i,u ]
[k 21,u ]
[n21,u ]
[i,u ] [k 112i,u ] [k 11,u ]
1 1 (i 2 1)u 5 ]]]] qi (k 1 1;u ). 2 1 ku That Eq. (14) indeed defines a normalized power distribution on the player set N, follows from the fact that it is a partition of the total probability of the maximal chains given by the sum of the probabilities Eq. (12)
OS D n
i51
1 [i 21,u ] 1 [n2i,u ] n 2 1 ]]]] , i21 2 [n21,u ]
which, of course, must equal 1. h It is interesting to note that the power distribution Eq. (14) on the player set 1, . . . , ´ k 1 1 represents one half of a weighted Markov-Polya distribution with the weight factor (2 1 2(i 2 1)u ) /(2 1 ku ). The median player is counted twice. Remark. There is nothing that forces the sequence of random variables to follow a ´ Markov-Polya distribution. Any sequence of binary random variables will do. More generally if we let P *i (n) denote the probability for a maximal chain with initiator i, then it follows from the proof of Theorem 1 that the voting power distribution is
Si 2k 1DOSkj 22 11DP * (n), k
z *i 5
5 OS k
2
j51
i 1j
1#i #k
j 51
D
k21 j 2 1 P *k111j (n),
(16) i 5 k 1 1,
In the paper by Berg and Perlinger (1998) some results are derived when the variables of the sequence X1 , . . . , Xn21 follow a modified binomial distribution. h Let us now look at a numerical illustration.
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Example 1. Suppose that we have a spectrum game [4; 1, . . . , 1] 7S . The power ´ distribution of this game measured by the Markov-Polya index for a few values of u is Players
1
2
3
4
5
6
7
u 5 2 0.2 u 50 u 5 0.5 u 51 u 5 10
0.060 0.062 0.057 0.05 0.014
0.238 0.188 0.129 0.1 0.021
0.179 0.188 0.171 0.15 0.041
0.048 0.125 0.286 0.4 0.848
0.179 0.188 0.171 0.15 0.041
0.238 0.188 0.129 0.1 0.021
0.060 0.062 0.057 0.05 0.014
We see from Example 1 that the median player’s voting power increases with the parameter u. When u is increased, maximal chains which have a greater possibility for one-way expansion, i.e. maximal chains that originate from the outer players, will have a larger probability of occurring, compared, for example, with maximal chains originating from the median player. Due to the fact that there are more maximal chains originating from the center players, a larger value of u can thus be interpreted as a compensation for this phenomenon, and thereby making the players approximately equally probable to be initiators. The following corollary highlights these ideas. ´ index Corollary 2. Consider a spectrum game on n players. If we use the Markov-Polya Eq. (14) to determine the power distribution of this game, the following holds. • u 5 2 1 /(n 2 1) means that the probability for a maximal chain originating from i is directly proportional to the number of maximal chains originating from i. • u 5 0 means that all maximal chains are regarded as equally likely. (The case considered by Edelman, 1997) • u 5 1 / 2 Let Ci denote the number of coalitions in which i is a member. The initiator probability then equals the ratio Ci / o m[N Cm . • u 5 1 means that all players are regarded as equally likely to be the initiator. • u ‘‘large’’, (i.e. u → `) entails that one of the outermost players almost surely will be the initiator, i.e. they have initiator probability 1 / 2 each. This means that, asymptotically, in the simple majority case, the median player will have all the power, i.e. zk11 → 1 when u → `. Proof. If u 5 2 1 /(n 2 1), Eq. (11) gives that
S
D S DS2nn 2212D
1 n21 Pi n; 2 ]] 5 n21 i21
21
,
which by Lemma 2 is the number of maximal chains originating from i multiplied by a constant. If u 5 0, Eq. (11) reduces to Pi (n;u ) 5 1 / 2 n21 , i [ N, i.e. all maximal chains are equally likely. When u 5 1 / 2, the initiator probabilities can, by means of Eq. (12), be n 1 2 21 shown to be i(n 1 1 2 i) , i [ N. It can also be shown that Ci 5 i(n 1 1 2 i), 3
S
D
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
S
225
D
n12 which implies that o m [N Cm 5 . If u 5 1, Eq. (12) reduces to 1 /n,; j [ N, i.e. 3 all players are equally likely to be the initiator. If u → `, the asymptotic behavior of the initiator probabilities is easily seen, due to the lack of growing terms in Eq. (12) when j[ ⁄ h1, nj. h Corollary 2 shows that the initiator probability distribution Eq. (12) drastically changes when u is increased. When u is close to zero, the mass of Eq. (12)) is concentrated at the center of the ideological scale. As u increases, the initiator probabilities are more evenly spread among the players, and when u equals one, Eq. (12) represents a uniform distribution on N, the set of player positions. If u is increased even more, the distribution Eq. (12) becomes more concentrated at the edges of the scale, and in the limit, as u → `, it reduces to a two point distribution, putting the same probability on positions 1 and n. For certain values of u, Eq. (14) reduces to a rather neat and simple power distribution formula. It has, for instance, already been seen that for u 5 0, Eq. (14) reduces to Eq. (7). As displayed in the following corollary, attractive formulas can be obtained also for other values of the parameter. Corollary 3. Consider the game of Theorem 1. The voting power distributions measured ´ by the Markov-Polya index for the parameter values u 5 1 / 2 and u 5 1 are i(i 1 1)(k 1 2 2 i) ]]]]], k14 4 4 zi (1 / 2) 5 12 ]]]], (k 1 3)(k 1 4)
5
S
D
1#i #k (17) i 5 k 1 1,
i ]]]], 1 # i # k (k 1 1)(k 1 2) zi (1) 5 2 ]], i 5 k 1 1. k12
5
(18)
The power distribution for the other half of the scale, follows by symmetry. Proof. The corollary follows immediately from Eq. (14) and the proof of Corollary 2 where it was shown that qi (n, 1 / 2) 5 i(n 1 1 2 i)
Sn 13 2D
21
, i [N
and
qi (n, 1) 5 1 /n, i [ N. h
It is interesting to see how the two peaks of the power distribution Eq. (14) moves towards the median player, as u increases in the interval 0 # u # 1. For u 5 0 the peaks are close to the quartile players. For u 5 1 / 2 the peaks are close to the players 2(k 1 1) / 3 and 4(k 1 1) / 3. For u 5 1, finally, the two peaks become one, and the median player is dominating the game, as is displayed in Fig. 1.
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T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
´ Fig. 1. The power distribution of the game [10; 1, . . . , 1] S , on n 5 19 players, measured by the Markov-Polya index for parameter values u 5 0, 1 / 2 and 1.
5. Concluding remarks ´ The parametrized Markov-Polya index is useful and flexible, in the sense that different parameter values allow different interpretive possibilities. It is a valuable amendment to the extended Shapley-Shubik index for voters positioned on an ideological scale. In particular, when Edelman’s extension of the Shapley-Shubik index is supplemented with ´ a Markov-Polya distribution for coalition formation, the median voter stands to regain part of, or even all of, his voting power.
Acknowledgements The Swedish Council for Research in the Humanities and Social Sciences has provided financial support. The author is grateful to Sven Berg for valuable suggestions. Also, many helpful comments and suggestions by anonymous referees are gratefully acknowledged.
References Banzhaf, J.R., 1965. Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Rev. 19, 317–343. Berg, S., 1997. On voting power indices and a class of probability distributions, To appear in Group Decisions and Negotiations. ´ Berg, S., Perlinger, T., 1998. Polya sequences, connected coalitions and voting power indices, To appear in Homo Oeconomicus. Edelman, P., 1997. A note on voting. Math. Soc. Sci. 34, 37–50. Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Univariate Discrete Distributions, 2nd ed. John Wiley and Sons. Owen, G., 1995. Game Theory, 3rd ed. Academic Press. Shapley, L.S., Shubik, M., 1954. A method for evaluating the distribution of power in a committee system. Am. Pol. Sci. Rev. 48, 787–792.
Voting power in an ideological spectrum ´ The Markov-Polya index Thommy Perlinger* Lund University, Department of Statistics, POB 7008, 220 07 Lund, Sweden Received December 1997; received in revised form December 1998; accepted August 1999
Abstract Power distributions of voting games in which only certain coalitions are allowed, due to the fact that the players are spread across an ideological spectrum, are considered. The paper extends Edelman’s model [Edelman, P., 1997. A note on voting. Math. Soc. Sci. 34, 37–50] concerning ´ allowable coalitions. The Markov-Polya index is introduced as a parametrized family of power indices which has Edelman’s extension of the Shapley-Shubik index as a special case. Unlike the extended Shapley-Shubik index, it puts different weights on the allowable coalitions. For particular parameter values, simple explicit power formulas are derived. 2000 Elsevier Science B.V. All rights reserved. ´ distribution Keywords: Voting power index; Connected coalitions; Initiator; Markov-Polya
1. Introduction Game theoretic solutions to voting situations, in the form of common power indices, do not take into account that the voters are often spread across an ideological spectrum. Voting power indices in common use are, for example, the Banzhaf-Coleman index, (Banzhaf, 1965) which treats all coalitions as equally probable and the Shapley- Shubik index, (Shapley and Shubik, 1954) where all permutations (i.e. vote sequences) are treated as equally probable. In these two power indices, voters are assumed anonymous and treated symmetrically. When in a voting situation, a coalition comes into being, this procedure can be regarded as the formation of a series of subcoalitions, where one voter at a time joins the coalition. When ideological positions matter, the problem arises of how to model this *Tel.: 146-46-222-3653; fax.: 146-46-222-4220. E-mail address: [email protected] (T. Perlinger) 0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0165-4896( 99 )00044-X
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process in a reasonably realistic way. Should, for instance, a coalition that consists of two voters from opposite extremes be as likely as a coalition that consists of two voters who are ideological neighbors? Is it at all reasonable to believe that the former coalition will arise? These questions suggest that in ideological voting situations, all coalitions should not be treated as equally probable to occur. The problem then is: which are the least likely coalitions and how do we restrict them? An interesting approach to this kind of problem is suggested by Edelman (1997). His model of coalition formation is based on the concept of convex sets, and only a certain type of coalitions are allowed to occur. A specific application of the model is when the players are spread across an ideological spectrum, such as the usual left–right ideology scale. The idea is that a coalition is permissible if and only if there are no ideological gaps between members of the coalition; in other words, there can be no non-member who is ideologically in an intermediate position between any two coalition members. Edelman shows that the Shapley-Shubik index very naturally extends to accommodate these restricted games, which henceforth will be referred to as spectrum games. In voting situations modeled by spectrum games, one would expect the player in the median position to be the strongest in terms of voting power, since he is connected to both flanks and thus have more possibilities to form and join coalitions. This is not the case, however, when voting power in spectrum games is measured by the extended Shapley-Shubik index. Somewhat surprisingly, it is instead the players in quartile positions who are the strongest. Only the outer players in the most extreme positions on the ideology scale are weaker than the median player. The purpose of the present paper is to study voting power in spectrum games by combining Edelman’s ideas with those of Berg (1997). To do this, we first introduce the concept of an initiator as the first member of a coalition. Edelman’s model determines implicitly an initiator distribution, which favors (or rather dis-favors) the player in the median position. Taking the cue from Berg, we then introduce a model for coalition formation, in which each grand coalition is seen as the realization of a sequence of ´ Markov-Polya binary random variables. In this manner, we derive a parametrized family of Shapley-Shubik type indices, and in the process we also obtain a simple and flexible family of initiator distributions. Edelman’s model of voting is a special case, and other special cases have simple and appealing probabilistic interpretations. The rest of this paper is organized as follows. To establish notation, Section 2 gives a brief overview of n-person weighted voting games and voting power indices. The definition of spectrum games, i.e. Edelman’s model regarding allowable coalitions applied to voting games on an ideological spectrum, is given in Section 3 along with some basic results. In this section we also define Edelman’s extension of the ShapleyShubik index. In Section 4, which is the main part of the paper, we introduce the ´ Markov-Polya index. We discuss properties of the index, and we provide illustrations and interpretations. Section 5, finally, concludes the paper.
2. n-person weighted voting games and power indices A normalized n-person weighted voting game is a cooperative n-person game in which the players possess a varying number of votes ( p1 , . . . , pn ), and where a coalition
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S is either winning or losing. A coalition is winning if and only if the aggregated votes of the coalition members exceed a certain quota q. Such a game is represented by the notation [q; p1 , . . . , pn ] and the characteristic function is 1
v(S) 5
5
if
O p $q O p , q. i
;i [S
0
if
(1)
i
;i [S
When the influence of the individual players is measured in these games, it is common to use some power index. A power index determines the players’ power mainly on the basis of so called swing sets. A swing set for player i is a winning coalition where i is an influential member, i.e. by defecting he can turn the coalition over into a losing one. For such a coalition we thus have v(S) 2 v(S\hij) 5 1,
(2)
and we say that the player i is pivotal in the coalition S. The Shapley-Shubik index is of particular interest here. This index focuses on permutations on the set of players (vote sequences). A permutation is an arranged row consisting of all the n players, which could be interpreted as the way in which the grand coalition (i.e. the coalition consisting of all the players) is coming into being. This interpretation means that player i is expected to join the coalition formed by the players in front of him in the row. The Shapley-Shubik index then registers if player i is pivotal in this coalition. There are n! such arrangements, but many of them will give the same value because the value of a coalition does not depend on the particular order of the coalition forming process. Taking account of this, the Shapley-Shubik index for the game v is given by
O
(s 2 1)!(n 2 s)! ]]]]] [v(S) 2 v(S\hij)], i [ N. n! S #N,i [S
wi [v] 5
(3)
where uSu 5 s and N 5 h1,2, . . . , nj. This means that in terms of the Shapley-Shubik index, it is better for a player to hold the balance of power in small coalitions (which indicates that the player is strong), rather than being pivotal in coalitions with approximately half of the players (which the weaker players are capable of also). In each permutation there is exactly one player who is pivotal, which guarantees that the sum of the players’ Shapley-Shubik indices Eq. (3) equals 1, i.e.
O w [v] 5 1. n
i
(4)
k51
The Shapley-Shubik index Eq. (3) can be given a probabilistic interpretation. If all the n! permutations are regarded as equally probable, Eq. (3) represents the expected number of times player i will be in a pivotal position. Henceforth, this interpretation will be used. For an axiomatic definition of the Shapley-Shubik index Eq. (3), see, e.g., Owen (1995).
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3. Spectrum games We shall now define a class of voting games where certain coalitions are not allowed. This type of game can, for example, be applicable to voting situations where the players form an ideological spectrum (usually from left to right). In such cases it is hardly likely that a coalition member deviates too much in either direction, i.e. a voter on the left flank probably hesitates before joining a coalition containing only players from the right wing. To define which coalitions are to be allowed, we need the following definitions. Definition 1. Two voters are called ideological neighbors if there are no other voters who ideologically are in a position between them. Definition 2. A permutation of the n players is called a maximal chain if and only if the voter in position i (in the permutation) is ideological neighbor to a voter in position 1, 2, . . . , i 2 1, ;i [ N. As explained in Section 2, a permutation of the n players can be thought of as a way in which the grand coalition comprising all the players is built up. It follows from the above definitions that the set of maximal chains represents all the ways in which the grand coalition can be formed, such that at each step in the process we have a connected coalition, i.e. no ideological gaps between coalition members. Definition 3. A spectrum game [q; p1 , . . . , pn ] Sn with n players is a weighted voting game [q; p1 , . . . , pn ] where the players are N 5 h1, . . . , nj, and the allowable coalitions are all coalitions where the members constitute a connected interval on the positive integers. Note that the players numbers /labels and positions on the spectrum coincide. Hence the set of allowable permutations in a spectrum game is the set of maximal chains. The following two lemmas establish important facts about spectrum games. The lemmas are proven in Edelman’s paper (Edelman, 1997). Alternative proofs are given in Section 4. Lemma 1. In a spectrum game with n players there are 2 n21 maximal chains. Lemma 2. In a spectrum game with n players there are begin with player i, i [ N.
Sni 2211D maximal chains that
Next we introduce a suitable index of voting power in spectrum games. We follow Edelman (1997) and define the extension of the Shapley-Shubik index in the same vein as the ordinary Shapley-Shubik index Eq. (3), restricting calculations to the set of maximal chains. For each maximal chain, there is a point at which player i enters a connected coalition. The coalition consisting of i and his predecessors in the chain M is denoted M i . The extended Shapley-Shubik index is now defined as
1 w i* [v] 5 ] m
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219
O [v(M ) 2 v(M \hij)],
(5)
i
i
i [ N,
M
where the summation is over all the m maximal chains M. w *i measures the i:th player’s expected influence under the assumption that all maximal chains are equally probable. As in the ordinary Shapley-Shubik index Eq. (3), there is exactly one player in each maximal chain who is in a pivotal position. Therefore
O w *[v] 5 1. n
(6)
i
k51
As shown by Edelman (1997), the power distribution of the spectrum game [k 1 1; 11 1, . . . , 1] n52k , measured by the extended Shapley-Shubik index Eq. (5) is S
w i* 5
Si 2k 1D / 2
5
k 11
,
k
1/2 ,
1 # i # k,
(7)
i 5 k 1 1.
Due to the symmetry with respect to the center, the power distribution for the players on the other half of the ideological scale follows immediately. Edelman shows that if n is moderately large, only the outermost players are weaker than the median player and, moreover, the quartile players tend to dominate the game. The problem of power distribution among voters in a spectrum game will be examined more closely in the next section.
4. The Markov-Po´ lya index Obviously, allowing only connected coalitions to be formed entails a fairly radical reduction in the number of allowable permutations. Therefore we may expect that some of the properties of the ordinary Shapley-Shubik index are altered, when we use the modified formula Eq. (5). In this section we focus on one of these properties. It is convenient to introduce here the concept of an initiator as the first member of a coalition, in other words the first player in a maximal chain or allowable permutation. With no restrictions imposed on the coalitions, each player appears as initiator in an equal number of permutations. However, in a spectrum game this is no longer the case. As is readily checked, in the extended Shapley-Shubik index Eq. (5) the players are no longer equally likely to be the initiator. Our idea here is to introduce a more general index of power, or class of indices, which makes it possible to vary the initiator distribution. To this end we use a probabilistic approach. Consider a sequence of n 2 1 binary (0,1) stochastic variables X1 , X2 , . . . , Xn21 which all have the same expectation E(Xk ) 5 1 / 2, i.e. equal probability on 0 and 1. It is easy to see that there is a one-to-one correspondence between the set of possible realizations of the sequence of stochastic variables and the set of maximal chains in a spectrum game on n players. If, in the left–right ideological spectrum, we interpret Xk 5 0 as the event that the maximal chain at the kth step moves to the left and Xk 5 1 that it moves to the
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right, then a particular realization x 1 , x 2 , . . . , x n 21 determines uniquely a maximal chain. The number of zeros (or ones) in the sequence implicitly determines the initiator of the maximal chain. It is now easy to prove the two lemmas of Section 3. Proof of Lemma 1. The number of maximal chains is the same as the number of realizations of X1 , . . . , Xn21 , i.e. 2 n21 . Proof of Lemma 2. The number of maximal chains starting in position i is the same as the number of realizations of X1 , . . . , Xn21 , in which there are exactly i 2 1 zeros, i.e. n21 . i21
S
D
The extended Shapley-Shubik index considers the maximal chains in a spectrum game as equally probable. This is equivalent to stating that the sequence X1 , X2 , . . . , Xn21 defined above represents a set of uncorrelated stochastic variables. It seems natural to ask: is it necessary for these stochastic variables to be uncorrelated? A possible modification of the extended Shapley-Shubik index is therefore to allow a correlational structure on the sequence of stochastic variables. It seems natural to assume that at each step the further development of the maximal chain depends on its past behavior. We therefore define the dependency among the random variables through conditional probabilities. An interesting approach is the ´ suggestion by Berg (1997) to use a Polya-Eggenberger model to describe the dependency structure among the random variables. (See also, e.g., Johnson et al., 1992.) Consider the following (idealized) urn sampling experiment. Initially, we have one white and one black ball in the urn. To describe outcomes of the sampling process, we use our previously defined set of binary random variables where Xk 5 1 is interpreted as the event: white ball at the kth draw. At each stage of the sampling process, we draw a ball at random and replace the ball drawn together with u balls of the same color. Hence at the kth stage we have a total of 2 1 (k 2 1)u balls in the urn, and the conditional probability of drawing a white ball in the kth stage, given a sequence x 1 , x 2 , . . . , x k 21 in previous draws, is given by 1 1 (x 1 1 ? ? ? 1 x k 21 )u P(Xk 5 1ux 1 , . . . , x k21 ) 5 ]]]]]]]. 2 1 (k 2 1)u
(8)
The probability of drawing a black ball is, of course, 1 1 (k 2 1 2 (x 1 1 ? ? ? 1 x k 21 ))u P(Xk 5 0ux 1 , . . . , x k21 ) 5 ]]]]]]]]]]. 2 1 (k 2 1)u
(9)
The probability of getting exactly k white balls in the urn experiment does not depend on the particular order of the balls drawn. Thus, the dependency structure has the effect that maximal chains originating from the same initiator are all equally probable. However, maximal chains originating from different positions are not equally likely to occur. ´ We call the resulting maximal chain probabilities Markov-Polya probabilities, due to
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221
´ their close relation to the Markov-Polya distribution, see Johnson et al. (1992). It is convenient to introduce the following notation for generalized ascending factorials a [x,u ] 5 a(a 1 u ) ? ? ? (a 1 (x 2 1)u ), a [ R, x [ Z 1 , a [0,u ] ; 1.
(10)
n52k 11
´ Lemma 3. Consider the spectrum game [k 1 1; 1, . . . , 1] S . The Markov-Polya probability of an arbitrary maximal chain with the player i as the initiator is 1 [i 21,u ] 1 [n2i,u ] Pi (n; u ) 5 ]]]] , i [ N. 2 [n21,u ]
(11)
For this to represent a proper probability distribution defined on the set of all 2 n21 maximal chains, we add the restriction u . 2 1 /(n 2 2). Proof. The probability of the maximal chain corresponding to the realization x 1 , . . . , x n21 is P(X1 5 x 1 , . . . , Xn 21 5 x n 21 ) 5 P(Xn 21 5 x n 21 ux 1 , . . . , x n22 ) 3 P(Xn22 5 x n22 ux 1 , . . . , x n23 ) 3 ? ? ? 3 P(X2 5 x 2 ux 1 ) 3 P(X1 5 x 1 ), where the conditional probabilities are given by Eqs. (8) and (9). The fact that player i is the initiator tells us that the realization x 1 , . . . , x n 21 contains exactly i 2 1 zeros and n 2 i ones. And so Eq. (11) follows. The restriction on u follows from the fact that u can be interpreted as the adding of balls to a urn. That is, if u is negative this means that we after each stage withdraw balls from the urn. For Eq. (11) to represent a proper probability distribution we have to make sure that we always have a positive number of balls of both colors in the urn. At the last stage, if u is negative, we have at least 1 1 (n 2 2)u balls of each color and, hence, we have the restriction 1 1 1 (n 2 2)u . 0⇔u . 2 ]]. h n22 Clearly, there are a number of maximal chains that have the player j as initiator. Lemma 2 together with Lemma 3 reveals that the probability for a maximal chain to have player j as the initiator is [ j 21,u ] [n2j,u ] 1 n21 1 q j (n; u ) 5 j 2 1 ]]]] , 2 [n21,u ]
S D
j [ N,
(12)
´ which is recognized as Markov-Polya distribution probabilities. Remark. The probabilities Eq. (11) are useful also when there are no restrictions on the coalitions, as shown by Berg (1997). Consider Eq. (11) as a probability distribution on the coalitions (subsets of N), where i represents the size of the coalitions i;i 5 1, . . . ,n.
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
222
Now, both the Shapley-Shubik index Eq. (3) and the Banzhaf-Coleman index appears as expectations of the random variable [v(S) 2 v(S\hij)], i.e. Eu 50 [v(S)-v(S\hij)] (Banzhaf-Coleman index) Eu 51 [v(S)-v(S\hij)] (Shapley-Shubik index) We are now in a position to introduce a family of power indices based on the ´ ´ Markov-Polya model Eq. (11), and it is therefore natural to call it the Markov-Polya index. The definition is in the same vein as Edelman’s definition Eq. (5) of an extended Shapley-Shubik index. ´ Definition 4. The parametrized Markov-Polya index of power in a spectrum game with player set N 5 h1,2, . . . , nj, is defined by
O [v(M ) 2 v(M \hij)]P (n; u ), i
zi (u ) 5
i
M
(13)
M
i
with M as in Eq. (5), and where the summation is over all maximal chains M. PM (n; u ) is the probability Eq. (11) for the maximal chain M in a spectrum game on n players. The difference between the extended Shapley-Shubik index Eq. (5) and the Markov´ Polya index Eq. (13) is that the latter assigns different weights to maximal chains. Next we derive an explicit formula for the voting power distribution of a spectrum game. Theorem 1. Consider the spectrum game [k 1 1; 1, . . . , 1] n52k11 . Based on the S ´ Markov-Polya index Eq. (13), the power distribution of this game is 1 1 (i 2 1)u ]]]] qi (k 1 1;u ), 1 # i # k 2 1 ku zi (u ) 5 2 1 2ku ]]]qk11 (k 1 1;u ), i 5 k 1 1, 2 1 ku
5
(14)
with an obvious symmetry for the other half of the ideological scale. Furthermore, the ´ Markov-Polya index is normed, i.e.
O z (u ) 5 1, n
i
;u.
(15)
i51
Proof. For the player i; 1 # i # k, there are exactly k initiators for whom i is a possible pivot. These are the players i 1 1, . . . , i 1 k. Since all maximal chains with a specific initiator are equally probable, it is sufficient to determine the number of maximal chains where i is in a pivotal position, and i 1 j; j 5 1, . . . , k, is the initiator. Hence, given j, we are looking for the number of maximal chains in a game with k players that starts in position j, multiplied by the number of maximal chains in a game with k 1 1 players that starts in position i, multiplied by the probability for a maximal chain with the player i 1 j as the initiator. Lemma 2 (used twice) and Lemma 3 give
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
1 1 Si 2k 1D O Skj 22 11D ]]]]] 2 1 1 k 5S D ]]]] O Sk 2j 1D(1 1 iu ) i21 2 1 1 k 5S D ]]]] (2 1 (k 1 1)u ) i21 2 1 1 k 5S D ]]]] i21 2 [i 1j 21,u ] [n 2i 2j,u ]
k
zi (u )
223
5
[n21,u ]
j 51
[i,u ] [k 112i,u ] k 21 [n21,u ]
[ j,u ]
(1 1 (k 1 1 2 i)u )[k 212j,u ]
j 50
[i,u ] [k 112i,u ]
[k 21,u ]
[n21,u ]
[i,u ] [k 112i,u ] [k 11,u ]
1 1 (i 2 1)u 5 ]]]] qi (k 1 1;u ). 2 1 ku That Eq. (14) indeed defines a normalized power distribution on the player set N, follows from the fact that it is a partition of the total probability of the maximal chains given by the sum of the probabilities Eq. (12)
OS D n
i51
1 [i 21,u ] 1 [n2i,u ] n 2 1 ]]]] , i21 2 [n21,u ]
which, of course, must equal 1. h It is interesting to note that the power distribution Eq. (14) on the player set 1, . . . , ´ k 1 1 represents one half of a weighted Markov-Polya distribution with the weight factor (2 1 2(i 2 1)u ) /(2 1 ku ). The median player is counted twice. Remark. There is nothing that forces the sequence of random variables to follow a ´ Markov-Polya distribution. Any sequence of binary random variables will do. More generally if we let P *i (n) denote the probability for a maximal chain with initiator i, then it follows from the proof of Theorem 1 that the voting power distribution is
Si 2k 1DOSkj 22 11DP * (n), k
z *i 5
5 OS k
2
j51
i 1j
1#i #k
j 51
D
k21 j 2 1 P *k111j (n),
(16) i 5 k 1 1,
In the paper by Berg and Perlinger (1998) some results are derived when the variables of the sequence X1 , . . . , Xn21 follow a modified binomial distribution. h Let us now look at a numerical illustration.
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Example 1. Suppose that we have a spectrum game [4; 1, . . . , 1] 7S . The power ´ distribution of this game measured by the Markov-Polya index for a few values of u is Players
1
2
3
4
5
6
7
u 5 2 0.2 u 50 u 5 0.5 u 51 u 5 10
0.060 0.062 0.057 0.05 0.014
0.238 0.188 0.129 0.1 0.021
0.179 0.188 0.171 0.15 0.041
0.048 0.125 0.286 0.4 0.848
0.179 0.188 0.171 0.15 0.041
0.238 0.188 0.129 0.1 0.021
0.060 0.062 0.057 0.05 0.014
We see from Example 1 that the median player’s voting power increases with the parameter u. When u is increased, maximal chains which have a greater possibility for one-way expansion, i.e. maximal chains that originate from the outer players, will have a larger probability of occurring, compared, for example, with maximal chains originating from the median player. Due to the fact that there are more maximal chains originating from the center players, a larger value of u can thus be interpreted as a compensation for this phenomenon, and thereby making the players approximately equally probable to be initiators. The following corollary highlights these ideas. ´ index Corollary 2. Consider a spectrum game on n players. If we use the Markov-Polya Eq. (14) to determine the power distribution of this game, the following holds. • u 5 2 1 /(n 2 1) means that the probability for a maximal chain originating from i is directly proportional to the number of maximal chains originating from i. • u 5 0 means that all maximal chains are regarded as equally likely. (The case considered by Edelman, 1997) • u 5 1 / 2 Let Ci denote the number of coalitions in which i is a member. The initiator probability then equals the ratio Ci / o m[N Cm . • u 5 1 means that all players are regarded as equally likely to be the initiator. • u ‘‘large’’, (i.e. u → `) entails that one of the outermost players almost surely will be the initiator, i.e. they have initiator probability 1 / 2 each. This means that, asymptotically, in the simple majority case, the median player will have all the power, i.e. zk11 → 1 when u → `. Proof. If u 5 2 1 /(n 2 1), Eq. (11) gives that
S
D S DS2nn 2212D
1 n21 Pi n; 2 ]] 5 n21 i21
21
,
which by Lemma 2 is the number of maximal chains originating from i multiplied by a constant. If u 5 0, Eq. (11) reduces to Pi (n;u ) 5 1 / 2 n21 , i [ N, i.e. all maximal chains are equally likely. When u 5 1 / 2, the initiator probabilities can, by means of Eq. (12), be n 1 2 21 shown to be i(n 1 1 2 i) , i [ N. It can also be shown that Ci 5 i(n 1 1 2 i), 3
S
D
T. Perlinger / Mathematical Social Sciences 40 (2000) 215 – 226
S
225
D
n12 which implies that o m [N Cm 5 . If u 5 1, Eq. (12) reduces to 1 /n,; j [ N, i.e. 3 all players are equally likely to be the initiator. If u → `, the asymptotic behavior of the initiator probabilities is easily seen, due to the lack of growing terms in Eq. (12) when j[ ⁄ h1, nj. h Corollary 2 shows that the initiator probability distribution Eq. (12) drastically changes when u is increased. When u is close to zero, the mass of Eq. (12)) is concentrated at the center of the ideological scale. As u increases, the initiator probabilities are more evenly spread among the players, and when u equals one, Eq. (12) represents a uniform distribution on N, the set of player positions. If u is increased even more, the distribution Eq. (12) becomes more concentrated at the edges of the scale, and in the limit, as u → `, it reduces to a two point distribution, putting the same probability on positions 1 and n. For certain values of u, Eq. (14) reduces to a rather neat and simple power distribution formula. It has, for instance, already been seen that for u 5 0, Eq. (14) reduces to Eq. (7). As displayed in the following corollary, attractive formulas can be obtained also for other values of the parameter. Corollary 3. Consider the game of Theorem 1. The voting power distributions measured ´ by the Markov-Polya index for the parameter values u 5 1 / 2 and u 5 1 are i(i 1 1)(k 1 2 2 i) ]]]]], k14 4 4 zi (1 / 2) 5 12 ]]]], (k 1 3)(k 1 4)
5
S
D
1#i #k (17) i 5 k 1 1,
i ]]]], 1 # i # k (k 1 1)(k 1 2) zi (1) 5 2 ]], i 5 k 1 1. k12
5
(18)
The power distribution for the other half of the scale, follows by symmetry. Proof. The corollary follows immediately from Eq. (14) and the proof of Corollary 2 where it was shown that qi (n, 1 / 2) 5 i(n 1 1 2 i)
Sn 13 2D
21
, i [N
and
qi (n, 1) 5 1 /n, i [ N. h
It is interesting to see how the two peaks of the power distribution Eq. (14) moves towards the median player, as u increases in the interval 0 # u # 1. For u 5 0 the peaks are close to the quartile players. For u 5 1 / 2 the peaks are close to the players 2(k 1 1) / 3 and 4(k 1 1) / 3. For u 5 1, finally, the two peaks become one, and the median player is dominating the game, as is displayed in Fig. 1.
226
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´ Fig. 1. The power distribution of the game [10; 1, . . . , 1] S , on n 5 19 players, measured by the Markov-Polya index for parameter values u 5 0, 1 / 2 and 1.
5. Concluding remarks ´ The parametrized Markov-Polya index is useful and flexible, in the sense that different parameter values allow different interpretive possibilities. It is a valuable amendment to the extended Shapley-Shubik index for voters positioned on an ideological scale. In particular, when Edelman’s extension of the Shapley-Shubik index is supplemented with ´ a Markov-Polya distribution for coalition formation, the median voter stands to regain part of, or even all of, his voting power.
Acknowledgements The Swedish Council for Research in the Humanities and Social Sciences has provided financial support. The author is grateful to Sven Berg for valuable suggestions. Also, many helpful comments and suggestions by anonymous referees are gratefully acknowledged.
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