Public Choice 115: 109–137, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
109
Paradoxes of voting power in Dutch politics ∗ ADRIAN VAN DEEMEN1 & AGNIESZKA RUSINOWSKA2 1 University of Nijmegen, NSM, P.O. Box 9108, 6500 HK Nijmegen, The Netherlands; E-mail:
[email protected]; 2 Tilburg University, Department of Philosophy, P.O. Box 90153 5000 LE Tilburg, The Netherlands; E-mail:
[email protected]; Warsaw School of Economics, Institute of Econometrics, Al. Niepodleglosci 162, 02-554 Warsaw, Poland; E-mail:
[email protected]
Accepted 22 May 2002 Abstract. In this paper we first evaluate thirteen seat distributions in the Second Chamber of the Dutch parliament by means of several indices of voting power. Subsequently, we search for the occurrence of the paradox of redistribution, the paradox of new members, and the paradox of large size for each power index. The indices used are the Shapley-Shubik index, the normalized Banzhaf index, the Penrose-Banzhaf index, the Holler index, and the DeeganPackel index.
1. Introduction It is well-known that the number of parliamentary seats for a party in a multiparty system is an inaccurate measure for the effective voting strength of that party. To illustrate this point in a simple way, consider an imaginary parliament consisting of three parties a, b and c each having, respectively, 74, 2 and 74 seats. At first sight, parties a and c, each with 74 seats, seem to be far more powerful than party b with only two seats. However, this is only at first sight. Suppose that the absolute majority rule is in use so that to form a majority coalition, at least 76 seats are necessary. The majority coalitions then are {a, b}, {a, c}, {b, c} and {a, b, c}, and we see that party b is in as many winning coalitions as the two other parties. More important, it can make as many coalitions losing or winning as the other parties. This strongly indicates that party b is not inferior to the other parties in terms of voting strength. In order to measure voting strength in a more accurate way, a number of power indices have been proposed in the course of time. The most important power indices to be found in the literature are the Shapley-Shubik index (Shapley and Shubik, 1954), the normalized Banzhaf index (Dubey and Shapley, 1979), the Penrose-Banzhaf index (Banzhaf 1965, Penrose, 1946), ∗ We would like to thank an anonymous referee for his/her many valuable comments and
suggestions for improvements. The research of Agnieszka Rusinowska for this paper was carried out during a stay as a Marie Curie Fellow at the University of Tilburg.
110 the Deegan-Packel index (Deegan and Packel, 1978, 1982), and the Holler index (Holler, 1982). An extensive discussion and comparison of these power indices (except the Holler index) is given in Felsenthal and Machover (1998). Though the game-theoretic origin of some of these indices is subject to dispute (Felsenthal and Machover, 1998), each index can be applied to situations that can be modelled in terms of game theory. In particular, power indices are appropriate for calculating the power of players in (weighted) voting games. So far, empirical applications of power indices can be found especially in the field of decision-making in the European Union. However, the power index approach can be applied equally well to national legislatures and parliaments. In this paper we use this approach to evaluate seat distributions in the Second Chamber of the Dutch parliament. In general, a parliament can be modelled as a weighted voting game in which each political party is a corporate player. The number of seats of a party assigned to that party after an election is the weight of that player. Power indices then can be applied in order to arrive at more accurate representations of power distributions in a parliament. In this paper, we apply this approach to the seat distribution arising from thirteen elections in Dutch politics. The elections were held in 1956, 1959, 1963, 1967, 1971, 1972, 1977, 1981, 1982, 1986, 1989, 1994 and 1998. Each of the thirteen resulting seat distributions will be re-calculated in terms of the Shapley-Shubik index, the normalized Banzhaf and the Penrose-Banzhaf index, the Deegan-Packel index and the Holler index. In the course of time, a number of unexpected and counterintuitive properties and effects of power indices have been discovered. These surprising properties are called paradoxes of voting power. They resemble the voting paradoxes in social choice theory as for example the Condorcet paradox. See Nurmi (1999) for a review and discussion of the paradoxes in social choice theory and related fields. Well-known paradoxes of voting power are, among others, the paradox of redistribution (Fischer and Schotter, 1978, Schotter, 1981), the paradox of new members (Brams, 1975, Brams and Affuso, 1976) and the paradox of large size (Brams, 1975). For a theoretical analysis and discussion of these and other paradoxes see Felsenthal and Machover (1998). Power indices have become an important tool in both the theoretical and the empirical study of voting and collective decision making. The theoretical possibility of paradoxes of voting power has long been noticed, but it is useful to assess their empirical relevance by investigating how frequent their occurrence is (according to several power indices) in data about parliamentary seats in a multiparty system. Many theoretically possible paradoxes are empirically infrequent (most notably the Condorcet paradox), and hence the theoretical discovery of a class of paradoxes always raises the question of whether these
111 paradoxes are just a theoretical artifact or whether they are likely to occur in the real world. The main objective of this paper is to find empirical evidence of the occurrence of the mentioned paradoxes of voting power in the Second Chamber of the Dutch parliamentary system. In particular, we calculate the frequency of the occurrence of a modified version of the paradox of redistribution. Further, we discuss the occurrence of a form of the paradox of new members. Though the version of the paradox as presented in this paper will not completely meet the conditions of the paradox as formulated by Brams (1975) and Brams and Affuso (1976), it will be very much in the spirit of that paradox. It is the same version as used by Brams and Affuso (1976) in their analysis of the expansion of the European Union in 1973. Finally, we study the paradox of large size. We give evidence that this paradox could have occurred by means of hypothetical re-constructions based on the data set. Applied to the Second Chamber of the Dutch parliament, the paradoxes of voting power to be studied in this paper translate into the following research questions: 1. Is it possible that a party receives more seats in comparison with a previous election, but gets less power? 2. Is it possible that a party receives less seats in comparison with a previous election, but gets more power? 3. Is is possible that a party receives as many seats as before, but gets more or less power? 4. Is it possible that an incumbent party has less or as many seats but more power after the entrance of new parties? 5. Is it possible that the power index of a merger of political parties is less than the sum of the power of the constituent parties? The first three questions are a translation of a version of the paradox of redistribution, the fourth of the paradox of new members, and the last one of the paradox of large size. It is also important to know to what extent the several indices agree about the existence of the respective paradoxes. In order to find this out we construct an agreement index that measures the amount of agreement between each pair of power indices. The paper is organized as follows. Section 2 gives the conceptual framework for analyzing the Dutch situation. We present the basic concepts concerning voting games and the definitions of power indices used in this paper. Further we present and discuss the paradoxes of redistribution, of new members and of large size. In this section, we also motivate why we use
112 slightly different definitions of these paradoxes in our empirical research. Section 3 is the empirical section. First, it contains a description of the parliamentary situations after the thirteen elections in terms of the theory of voting games. Subsequently, the three paradoxes of voting power are empirically investigated. Section 4 presents and discusses the agreement between the several indices about the existence of the paradoxes. In the final section, we give some conclusions and discussion.
2. Theoretical background 2.1. Voting games In this section we only present the basic concepts of the theory of voting games. More extensive treatments of voting games can be found, among others, in Shapley (1962), Owen (1995: Ch. XVI), Van Deemen (1997: Ch. 5), and Felsenthal and Machover (1998: Ch. 2). A voting game is a cooperative game in which only two types of coalitions can be formed, to wit, winning coalitions and losing ones. A winning coalition takes it all while a losing coalition receives nothing. Since winning is the essence of politics, voting games are extremely suitable for analyzing political situations. Let N = {1, 2, . . . , n} be a nonempty set of players. It is supposed that N is finite. Any subset of N is called a coalition. A voting game G is an ordered pair G = (N, W), where W is a set of coalitions. A member of W is called a winning coalition. W is supposed to satisfy the following assumptions: 1. W = ∅ 2. ∅ ∈ /W 3. If S ⊂ T and S ∈ W, then T ∈ W. The first assumption states that there is a winning coalition. The second one says that a winning coalition must contain members. The third one is a monotonicity assumption. It states that any coalition containing a winning sub-coalition must be winning. Note that the assumptions imply that N is a winning coalition. The set of all coalitions not in W is denoted by L. Clearly, ∅ ∈ L. The complement of a coalition S is the set of all players not in S. A voting game G = (N, W) is called
113 − proper if the complement of any S ∈ W is losing, − strong if the complement of any S ∈ L is winning, − decisive if G is both proper and strong. A decisive voting game is the counterpart of a constant-sum game in the theory of games in coalition or characteristic function form. A coalition is minimal winning if it is winning and if any member of it is needed to make it winning. In other words, the removal of any of its members will make it a losing coalition. The set of all minimal winning coalitions is denoted by Wm . A coalition is called blocking if it is losing and if its complement is also losing. Though a blocking coalition cannot enforce a decision, it can prevent the enforcement of any decision. A veto-player is a player who is in every minimal winning coalition. Because of the monotonicity assumption, a veto-player must be in every winning coalition. A voting game is called weak if it contains a veto-player. Clearly, a weak game can never be strong, since the complement of any losing coalition containing the veto-player must be losing as well. If one player, say i, forms the only minimal winning coalition, then i is called a dictator. Note the difference between a veto-player and a dictator. A dictator can win on her own, she can enforce any decision without help of the other players. In contrast, a vetoplayer is needed to win, but cannot win on her own. She only can block the decision-making process, not enforce a decision. A dummy is a player who is a member of no minimal winning coalition. A dummy is a powerless player. She cannot make a winning coalition losing or a losing coalition winning. An important class of voting games is the class of weighted voting games. In this kind of voting game, a weight is assigned to each player indicating her or his voting strength in that game. A coalition is winning if and only if the sum of the weights of its members exceeds a certain threshold or quota. Formally, a weighted voting game can be represented by the n + 1-tuple coalition to win, [q; w1 , w2 , ..., wn ], where q denotes the quota needed for a and where wi is the weight assigned to player i. Let w(S) = i∈S wi . w(S) is called the size of S. A coalition S in a weighted voting game is winning if its size is at least as large as q. That is, S ∈ W ⇔ w(S) ≥ q. A coalition S is of minimum size if 1. S ∈ W 2. w(S) ≤ w(T) for every T ∈ W. Clearly, a minimum size coalition is a minimal winning coalition. The famous
114 minimum size principle formulated by Riker (1962, Riker and Ordeshook, 1973) states that in political situations only minimum size coalitions will be formed. The situation after each election in the Second Chamber of the Dutch parliamentary system will be modelled as a weighted voting game. The players in the game are the political parties that received a number of seats after an election. The weights assigned to the parties are the number of seats received. 2.2. Power indices In order to measure voting power more accurately in the Second Chamber of the Dutch parliamentary system, we will use five power indices, namely, the Shapley-Shubik index (Shapley and Shubik, 1954), the normalized Banzhaf index (Dubey and Shapley, 1979), the Penrose-Banzhaf index (Penrose, 1946, Banzhaf, 1965), the Deegan-Packel index (1978, 1982) and the Holler index (1982). We only present the basic concepts and definitions. For more extensive studies of these indices including the axiomatics and combinatorial formulae, consider among others Felsenthal and Machover(1998), Lucas (1982), Owen (1995) and Straffin (1982). A power index is a function φ that associates a nonnegative real number φi to each player i ∈ N in a voting game (N, W). Probably the best known index in this respect is the Shapley-Shubik index. Consider a situation in which n players vote for a bill and suppose that they have to vote in a specific order. A player is called pivotal for this order if it turns the coalition of players preceding in that order into a winning coalition. Let αi denote the number of orders or permutations in which player i is pivotal. The Shapley-Shubik index for player i in voting game G = (N, W) is Shi (G) =
αi n!
Here, n of course is the number of players. The Shapley-Shubik index is based on the assumption that members of a committee vote in order. By choosing vote orders of the members randomly, it calculates how often a given individual is pivotal. The index for a player is simply equal to the fraction of times that that player is pivotal (Shapley and Shubik, 1954). The more orders there are in which a player is pivotal, the more power it has. According to Felsenthal and Machover (1998, especially Chapter 6), the Shapley-Shubik index belongs to the class of so-called Ppower measures. Hereby, P-power stands for power as a prize. According to the notion of P-power, each coalition gets a prize by the simple act of winning, whereby winning is determined by the decision rule in use. The P-power of a voter in a committee is the expected relative share in some prize. P-power as
115 such is based on office-seeking behavior. That is, expected shares in the prize are only based on winning. Players who are not in a winning coalition receive nothing. The normalized Banzhaf index and the Penrose-Banzhaf index both depend on the number of combinations instead of the number of permutations. Basic is the notion of a swing player. A player i in a coalition S is called swing in S if its defection turns S into a losing coalition. The normalized Banzhaf index for the voting game G = (N, W) is ηi nBzi (G) = k∈N ηk Here, ηi is the number of winning coalitions in which player i is swing. The Penrose-Banzhaf index, also called the absolute Banzhaf index, is defined by ηi ηi PBzi (G) = = n−1 ki 2 for each i = 1, ..., n. In this formula, ki denotes the total number of coalitions containing player i. In particular, we have: PBzi (G) k∈N PBzk (G)
nBzi (G) =
Consider the number of coalitions that a player can turn into a winning or losing one. Both the normalized Banzhaf and the Penrose-Banzhaf index are based on this number. The normalized Banzhaf index of a player is simply the ratio of this number to the total number of coalitions with a swing player, while the Penrose-Banzhaf index of a player is the ratio of this number to the number of possible coalitions divided by two. The more a player can turn losing coalitions into winning ones, the more power it has. According to Felsenthal and Machover (1998, Chapter 3), both indices belong to the class of I-power measures. Here, I-power stands for power as influence which is based on the policy-seeking viewpoint. According to this viewpoint, "a member’s voting power is the degree to which that member’s vote is able to influence the outcome of a division: whether the bill in question will pass or fail (Felsenthal and Machover, 1998, p. 36. Their emphasis)." Both the Deegan-Packel index and the Holler index are based on the notion of minimal winning coalitions. Let G = (N, W) be a voting game and define Mi = {S ∈ Wm : i ∈ S}. Thus Mi is the set of minimal winning coalitions of which i is a member. Further, let mi denote the number of coalitions in Mi and m the total number of minimal winning coalitions. The Deegan-Packel index for a player i in a voting game G = (N, W) is defined by 1 1 . DPi (G) = m S∈M s i
116 Hereby, s is the number of players in coalition S. The Holler index, also called the public good power index, is defined by Hi (G) =
mi
k∈N mk
for each i ∈ N. There is a nice difference in interpretation between the Deegan-Packel index and the Holler index. The Deegan-Packel index assumes that the players in a minimal winning coalition equally share the spoils of coalition formation. In contrast, the Holler index assumes that each player in a minimal winning coalition gets the whole bribe. In this sense, the payoff of forming a minimal winning coalition has the character of a public good for its members. The bribe as such is indivisible and no member of the winning coalition can be excluded from receiving the same amount. According to Felsenthal and Machover (1998 Chapter 6), both measures belong to the P-power class. It can be proved that for all power indices φ presented above it is true that φi = 0 if and only if i is a dummy. The proof of this proposition is easy. Let us write ‘iff’ for ‘if and only if’. 1. First, consider the Shapley-Shubik index. Now, player i is a dummy iff i is pivotal in no permutation. Hence αi = 0 and therefore Shi = 0 iff i is a dummy. 2. Now consider both the normalized Banzhaf index and the PenroseBanzhaf index. Basic in these measures is the concept of swing player. Obviously, a player i is a dummy iff i is swing in no coalition. Hence, ηi = 0 iff i is a dummy and therefore nBzi = 0 and PBzi = 0 iff i is a dummy. 3. Finally, consider the Deegan-Packel and the Holler index. Both indices are based on the concept of minimal winning coalitions. Clearly, a dummy is in no minimal winning coalition and hence both DPi = 0 and Hi = 0. This property is useful in detecting dummies in weighted voting games. According to the property, it simply suffices to look for players with a zero power index. However, it should be noticed that this property is true for the indices considered in this paper. Other power indices not discussed here may violate it. 2.3. Paradoxes of voting power By a paradox of voting power we mean an unexpected and counterintuitive feature of a power index. In this subsection, three of these paradoxes will be
117 described. Only the original definition of the paradoxes will be presented. In the empirical section of this paper, we slightly modify these definitions in order to allow the empirical investigation. The paradox of redistribution was introduced by Fischer and Schotter (1978). See also Schotter (1981) and Felsenthal and Machover (1998). A power index shows the redistribution paradox if either a party’s voting weight decreases and at the same time its power index increases, or if a party gains in terms of voting weight, but loses in voting power. To be more precise, consider two weighted voting games G = [q; w1 , ..., wn ] and G = [q; w1 , ..., wn ], where ni=1 wi = ni=1 wi . Voter i is said to be a donor if wi < wi , voter i is said to be a recipient if wi > wi . A power index φ displays the redistribution paradox if there is a donor i such that φi (G ) > φi (G) or if there is a recipient j such that φj (G ) < φj (G). The paradox of new members was first presented in Brams (1975) and, subsequently, in Brams and Affuso (1976). This paradox appears when a new party joins and there are incumbent parties which have more voting power in this new situation than in the old one, despite the fact that their weights constitute a smaller proportion of the total weight. The formal description is as follows. Consider a weighted voting game G = [q; w1 , ..., wn ] and assume that a new player (n + 1) joins the assembly. The new game then is G = [q ; w1 , ..., wn , wn+1 ]. A power index φ displays the paradox of new members if for some i ∈ N, φi (G ) > φi (G). Another paradox presented by Brams (1975) is the paradox of large size. This paradox occurs, in the context of parliamentary games, when the power index of a union of parties is less than the sum of the power indices of the separate parties of that union. Let I ⊆ N be a set of players who decide to unite. This union will be called index φ displays the player U. A power paradox of large size if wU = i∈I wi and φU < i∈I φi .
3. Paradoxes of voting power in Dutch politics 3.1. Power indices for the Second Chamber of the Dutch Parliament We study thirteen seat distributions in the Second Chamber of the Dutch Parliament. Each seat distribution is the result of a national election. The elections took place in 1956, 1959, 1963, 1967, 1971, 1972, 1977, 1981, 1982, 1986, 1989, 1994, and in 1998. The total number of seats for the Second Chamber in The Netherlands in each period mentioned was and still is equal to 150. Table 1 shows the seat distribution after the thirteen elections. For decision-making (passing a bill etc.) in the Second Chamber, an absolute majority is necessary. Only for constitutional reform a qualified majority
Party/Year CPN PSP PPR GL SP PvdA D66 DS70 KVP ARP CHU CDA VVD SGP GPV RPF RKPN EVP BP AOV U55 NMP CP CD Total
118
Table 1. Seat distribution matrix for the Second Chamber, The Netherlands 1956
1959
1963
1967
1971
1972
1977
1981
1982
1986
7
3 2
4 4
5 4
6 2 2
7 2 7
2 1 3
3 3 3
3 3 2
1 2
50
48
43
37 7 42 15 12
39 11 8 35 13 10
43 6 6 27 14 7
49 15 13
49 14 12
50 13 13
13 3
19 3
16 3 1
17 3 1
16 3 2
22 3 2
1989
1994
1998
6
11 5 45 14
29 38 3 2 3
53 8 1
44 17
47 6
52 9
49 12
5 2 37 24
49 28 3 1
48 26 3 1 2
45 36 3 1 2
54 27 3 1 1
54 22 3 2 1
34 31 2 2 3
1 1 3
7
1
3
1 6 1
2 1 150
150
150
150
150
150
150
150
150
150
1 150
3 150
150
CPN = Communist Party, PSP = Pacifist Socialist Party, PPR = Radical Party, SP = Socialist Party, GL = Green Left, PvdA = Labor Party, D66 = Democrats 66, DS70 = Democratic Socialist Party, KVP =Catholic People’s Party, ARP = Anti -Revolutionary Party, CHU = Christian Historical Union, CDA = Christian Democrats, VVD = People’s Party for Freedom and Democracy, SGP = Political Reformed Party, GPV = Reformed Political League, RPF = Reform Political Federation, RKPN = Roman Catholic Party Netherlands, EVP = People’s Gospel Party, BP = Farmer’s Party, AOV = General Elder People’s Party, U55 = Union 55+, NMP = Dutch Retailer’s Party, CP = Centrum Party, CD = Centrum Democrats.
119 of two-third is needed. However, constitutional reform issues occur rarely in Dutch politics. Furthermore, Dutch politics is characterized by a very strong party discipline. Each political party in the Second Chamber has a so-called ‘fraction leader’ who carefully watches the party discipline. Deviations from the party policy line are hardly allowed and may even result in the push off of the deviating party member. Consequently, any party in the Second Chamber almost always vote in one block and can thus be considered as one corporate actor. Considering these two aspects of Dutch politics, it is safe to model the Second Chamber as a weighted voting game with a threshold of 76 and with the number of seats assigned to a party after an election as the weight of that party. Using Table 1, it can be checked that the weighted voting games representing the situation after the election of 1956, 1959, 1963, 1967, 1971, 1972, 1981, 1989, 1994 and 1998 are proper but not strong. This means that in all of these legislative situations, blocking coalitions may occur. In contrast, the weighted voting games after the elections of 1977, 1982 and 1986 are decisive. Blocking coalitions cannot occur during these parliamentary periods. For each seat distribution arising after an election we calculated the Shapley-Shubik, the normalized Banzhaf and the Penrose-Banzhaf, the Deegan-Packel and the Holler power index vectors. Table 2 and Table 3 show, respectively, the results for the Shapley-Shubik index and for the normalized Banzhaf index. Some interesting things can already be said given these power index matrices. First, as shown in Section 2.2, the power indices all assign a zero to dummy players. Using this fact, it can be seen that dummy parties are not frequently met in Dutch politics. We observe the occurrence of dummy parties in only three of the thirteen seat distributions; to wit, in 1977, 1982 and 1986. In these seat distributions dummy players abound. In each of these distributions there are only three parties with effective voting power. Another fact is that the weighted majority games with dummies are decisive. The remaining games all are proper but not strong. However, the only reason for this is that coalitions of size exactly 75 may occur. To give an example, consider the seat distribution of the election of 1998. In this case, we have wGL + wSP + wPvdA + wD66 = 75. Since the threshold is 76, the coalition {GL, SP, PvdA, D66} ∈ L. The complement of this coalition must have a size of 75 as well and hence is also loosing. 3.2. The Paradox of Redistribution The definition of the paradox of redistribution will be modified in the following way. Let wt and φt denote, respectively, a voting weight in year
120
Table 2. The Shapley-Shubik Power index matrix for the Second Chamber Party/Year
1956
1959
1963
1967
1971
1972
1977
1981
1982
1986
CPN PSP PPR GL SP PvdA D66 DS70 KVP ARP CHU CDA VVD SGP GPV RPF RKPN EVP BP AOV Unie NMP CP CD Total
0.0071
0.0107 0.0107
0.0214 0.0214
0.0273 0.0221
0.0336 0.0125 0.0125
0.04 0.0107 0.04
0 0 0
0.0238 0.0238 0.0238
0 0 0
0 0
0.3071
0.2917
0.2548
0.2543 0.04 0.313 0.0959 0.0753
0.29 0.0664 0.0479 0.2438 0.0799 0.0598
0.3485 0.0349 0.0349 0.179 0.078 0.04
0.2905 0.1405 0.1238
0.3012 0.1155 0.1155
0.3286 0.104 0.104
0.1238 0.0071
0.144 0.0107
0.1302 0.0159 0.004
0.1122 0.0154 0.0047
0.1046 0.0178 0.0125
0.1429 0.0176 0.0107
1989
1994
1998
0.0385
0.0496 0.0234 0.3472 0.0496
0.2246 0.2603 0.0163 0.0127 0.0163
0.3333 0 0
0.2679 0.0377
0.3333 0
0.3333 0
0.2563 0.0385
0.0494 0.0185 0.2737 0.081
0.3333 0.3333 0 0
0.3393 0.2361 0.0238 0.006 0.0179
0.3333 0.3333 0 0 0
0.3333 0.3333 0 0 0
0.3718 0.2563 0.0194 0.0087 0.0052
0.2288 0.1898 0.0185 0.0185 0.0277
0.0054 0 0.0159
0.04
0.0061
0.0176
0 0.0583 0.0081
0.0125 0 0.9999
1
1.0002
1.0002
0.9999
1.0002
0.9999
1.0001
0.9999
0.9999
0.0052 0.9999
0.0277 1
1
Table 3. The normalized Banzhaf power index matrix for the Second Chamber Party/Year
1956
1959
1963
1967
1971
1972
1977
1981
1982
1986
CPN Psp PPR GL SP PvdA D66 DS70 KVP ARP CHU CDA VVD SGP GPV RPF RKPN EVP BP AOV Unie NMP CP CD Total
0.0089
0.013 0.013
0.0262 0.0262
0.0321 0.0261
0.0381 0.013 0.013
0.0397 0.0118 0.0397
0 0 0
0.0241 0.0241 0.0241
0 0 0
0 0
0.292
0.2739
0.2333
0.2205 0.0459 0.2847 0.108 0.0863
0.2657 0.0747 0.0544 0.214 0.0897 0.0672
0.3503 0.0341 0.0341 0.1619 0.0852 0.0397
0.2743 0.1504 0.1327
0.2826 0.1261 0.1261
0.3023 0.1119 0.1119
0.1327 0.0089
0.1522 0.013
0.1433 0.0199 0.0052
0.1268 0.0183 0.0054
0.1183 0.0195 0.013
0.1484 0.0187 0.0118
1989
1994
1998
0.0447
0.0526 0.0252 0.3364 0.0526
0.2403 0.2494 0.016 0.0114 0.016
0.3333 0 0
0.2649 0.0356
0.3333 0
0.3333 0
0.2565 0.0447
0.0482 0.0191 0.2746 0.0713
0.3333 0.3333 0 0
0.3222 0.258 0.0241 0.0057 0.0172
0.3333 0.3333 0 0 0
0.3333 0.3333 0 0 0
0.3459 0.2565 0.0259 0.0118 0.0071
0.2269 0.1983 0.0191 0.0191 0.0286
0.0061 0 0.0199
0.0459
0.0064
0.0187
0 0.0572 0.0009
0.013 0 0.9999
1.0001
1
1
1.0002
0.9999
1
0.9999
0.9999
0.0071 1.0002
0.0286 1
0.9999
121
0.9999
122 t and a power index of a party in year t. The occurrence of the following variations of the paradox of redistribution will be investigated: (P1): wt > wt+k and φt ≤ φt+k (P2): wt < wt+k and φt ≥ φt+k (P3): wt = wt+k and φt = φt+k Clearly, this definition of the redistribution paradox differs from the original formulation of Felsenthal and Machover (1998). First, the equality sign for the power indices φt and φt+k in paradoxes P1 and P2 is used. Second, situations in which a party’s weight does not change, but its power index does are also considered to be paradoxical (paradox P3). Finally, the original definition of the paradox requires that the number of parties is fixed in the succeeding parliamentary periods. Our definition drops this condition of fixing the number of parties. As will be discussed in the next section, it is exactly the dropping of this condition that blurs the difference between the redistribution paradoxes and the paradox of new members. However, as will be discussed in that section as well, the paradox of new members will satisfy an additional assumption which makes it different from the redistribution paradoxes. The essential feature of the redistribution paradox in the context of parliamentary situations is that a party may receive more seats but may have less effective power, or that it may receive less seats but more effective power. In other words, the paradox occurs according to our modified definition when a party wins an election but looses voting power or when it looses an election but wins in voting power. The following method for calculating the overall frequency of the paradox is used. First the frequency of the paradox for each relevant party is calculated. A party is relevant when it appears in two or more parliamentary periods. For the party calculations, the following notation is used: nPi is the number of cases in which paradox (Pi) appears for a power index of a party, i=1,2,3. k is the number of elections in which a party participates. l is the number of cases analyzed for a party. To analyze all possible cases in which the redistribution paradox might occur for a party, we have to check each pair of elections in which a party participates. To do this, let l be equal to the number of two-element combinations among k elements. Hence, k(k − 1) k! k = = l= 2!(k − 2)! 2 2 Furthermore, let fPi be the frequency of the paradox Pi, i = 1, 2, 3, for a power index of a
123 party, and fP be the frequency of the redistribution paradox for a power index of a party. Clearly, nP2 nP3 nP1 , fP2 = , fP3 = fP1 = l l l nP1 + nP2 + nP3 fP = l After having calculated the frequency of the paradox for each party, the overall frequency of the paradox for the Dutch Second Chamber can be calculated. For this, the following notation will be used: TNPi is the total number of cases in which the paradox (Pi) appeared, i=1,2,3. L is the total number of cases analyzed for all parties. FPi is the frequency of the paradox (Pi), i=1,2,3. FP is the frequency of the redistribution paradox. Clearly, we have: FP1 =
TNP1 , L
FP2 =
TNP2 , L
FP3 =
TNP3 L
TNP1 + TNP2 + TNP3 L Now we are able to count the frequency of the occurrence of the paradox. Tables 4 and 5 give the number of cases and frequencies of P1–P3 for, respectively, the Shapley-Shubik index and the normalized Banzhaf index. We do not present here such detailed tables for the other indices. Instead we present a summarizing table that includes the overall results for the PenroseBanzhaf index, the Holler index and the Deegan-Packel index (Table 6). What conclusions can be drawn from the results presented in Table 4 and 5? Looking at the frequency of the redistribution paradox, we see that for almost every party the paradox of redistribution occurs. Only for the SP and DS70 the frequency fP is equal to zero for every power index. In contrast, for the SGP the redistribution paradox occurs frequently; the fP for this party is greater than 0.9 for each power index. Clearly, this is caused in particular by P3, which is very likely to appear for parties with a constant number of seats. Furthermore, the frequency of the paradox according to the ShapleyShubik index does not differ much from the frequency as measured with the normalized Banzhaf index. In spite of the different interpretations in terms of policy-seeking (e.g. the normalized Banzhaf index) and office-seeking (e.g. the Shapley-Shubik index) as discussed in Felsenthal and Machover (1998), the empirical differences are minimal. According to Table 6, the Holler index appears to be the most sensitive for the occurrence of the paradox. Both the Deegan-Packel index and the FP =
124 Table 4. Redistribution paradoxes in the SC for the Shapley-Shubik index Party
nP1
nP2
nP3
nP
k
l
CPN PSP PPR GL SP PvdA D66 DS70 KVP ARP CHU CDA VVID SGP GPV RPF BP CP/CD
6 3 2 1 0 16 3 0 2 0 0 2 3 9 0 2 0 0
1 4 1 0 0 12 11 0 0 4 1 3 12 1 0 1 0 0
4 4 3 0 0 2 1 0 1 3 2 1 2 63 22 3 2 1
11 11 6 1 0 30 15 0 3 7 3 6 17 74 22 6 2 1
9 9 6 3 2 13 10 3 6 6 6 7 13 13 11 6 5 3
36 36 15 3 1 78 45 3 15 15 15 21 78 78 55 15 10 3
Total
49
51
114
215
522
freqP1
freqP2
freq123
TfreqP
0.16667 0.08333 0.13333 0.33333 0 0.20513 0.06667 0 0.13333 0 0 0.09524 0.03846 0.11538 0 0.13333 0 0 FRQP1 0.09387
0.02778 0.11111 0.06667 0 0 0.15385 0.24444 0 0 0.26667 0.06667 0.14286 0.15385 0.01282 0 0.06667 0 0 FRQP2 0.0977
0.11111 0.11111 0.2 0 0 0.02564 0.02222 0 0.06667 0.2 0.13333 0.04762 0.02564 0.80769 0.4 0.2 0.2 0.33333 FRQP3 0.21839
0.30556 0.30556 0.4 0.33333 0 0.38462 0.33333 0 0.2 0.46667 0.2 0.28571 0.21795 0.94872 0.4 0.4 0.2 0.33333 FRQP 0.41188
nP1 = number of cases distriubtion paradox 1; nP2 = number of cases distribution paradox 2; nP3 = number of cases distribution paradox 3; nP = total number of distribution paradoxes; k = number of elections; l = number of comparisons; freqP1 = frequency paradox 1 = nP1/l; freqP2 = frequency paradox 2 = nP2/l; freqP3 = frequency paradox 3 = nP3/l; Tfreq = total frequency redistribution paradoxes nP/l
Holler index are seemingly more sensitive for the paradox than the ShapleyShubik index and the normalized Banzhaf and Penrose-Banzhaf indices. We conclude that the (modified) paradox of redistribution can be observed frequently in Dutch politics. The overall frequency of this paradox (FRQP) is larger than 0.4 for every power index. 3.3. The paradox of new members The next paradox of voting power to be observed is the paradox of new members. It occurs when an old party gains in voting power but not in seats after the entry of a new party. Also the original definition of this paradox will be modified. In the original definition, the weights of the old parties are held
125 Table 5. Redistribution paradoxes in the SC for the normalized Banzhaf index Party
nP1
nP2
nP3
nP
k
l
CPN -PSP PPR GL SP PvdA D66 DS70 KVP ARP CHU CDA VVD SGP GPV RPF BP CP/CD
5 1 2 1 0 18 3 0 2 0 0 3 2 7 0 2 0 0
1 4 1 0 0 12 12 0 0 4 1 2 14 1 0 1 0 0
4 4 3 0 0 2 1 0 1 3 2 1 2 63 21 3 2 1
10 9 6 1 0 32 16 0 3 7 3 6 18 71 21 6 2 1
9 9 6 3 2 13 10 3 6 6 6 7 13 13 11 6 5 3
36 36 15 3 1 78 45 3 15 15 15 21 78 78 55 15 10 3
Total
46
53
113
212
522
freqP1
freqP2
freqP3
TfreqP
0.13889 0.02778 0.13333 0.33333 0 0.23077 0.06667 0 0.13333 0 0 0.14286 0.02564 0.08974 0 0.13333 0 0 FRQP1 0.08812
0.02778 0.11111 0.06667 0 0 0.15385 0.26667 0 0 0.26667 0.06667 0.09524 0.17949 0.01282 0 0.06667 0 0 FRQP2 0.10153
0.11111 0.11111 0.2 0 0 0.02564 0.02222 0 0.06667 0.2 0.13333 0.04762 0.02564 0.80769 0.38182 0.2 0.2 0.33333 FRQP3 0.21648
0.27778 0.25 0.4 0.33333 0 0.41026 0.35556 0 0.2 0.46667 0.2 0.28571 0.23077 0.91026 0.38182 0.4 0.2 0.33333 FRQP 0.40613
nP1 = number of cases distriubtion paradox 1; nP2 = number of cases distribution paradox 2; nP3 = number of cases distribution paradox 3; nP = total number of distribution paradoxes; k = number of elections; l = number of comparisons; freqP1 = frequency paradox 1 = nP1/l; freqP2 = frequency paradox 2 = nP2/l; freqP3 = frequency paradox 3 = nP3/l; Tfreq = total frequency redistribution paradoxes nP/l Table 6. The redistribution paradox in the Second Chamber nP1 nP2 nP3 Shapley-Shubik index norm. Banzhaf index Penrose-Banzhaf index Holler index Deegan-Packel index
49 46 51 48 47
51 53 61 77 72
114 113 113 115 115
nP 214 212 225 240 234
L FRQP1 FRQP2 FRQP3 FRQP 522 522 522 522 522
0.0939 0.0881 0.0977 0.092 0.09
0.0977 0.1015 0.1169 0.1475 0.1379
0.2184 0.2165 0.2165 0.2203 0.2203
0.41 0.4061 0.431 0.4598 0.4483
nPi = total number of cases of the redistribution paradox i for i = 1, 2, 3; NP = overall number of cases of the redistribution paradox; FRQPi = the frequency of the redistribution paradox i for i = 1, 2, 3; FRQP = overall frequency of the paradox of redistribution
126 constant. Here, we drop this condition. The formal definition is: (P4): wt ≥ wt+k and φt < φt+k To be in line as much as possible with the original definition, we impose the following assumption: The set of parties participating in an election in year t must be a subset of the set of parties in the election in year t + k. Notice that according to the definition the paradox only occurs when there is a rise in the power of an incumbent party with less or equal weight after the entrance of new parties. Clearly, paradox P4 looks like a combination of the redistribution paradoxes P1 and P3. However, the additional assumption to P4 makes it special. In the redistribution paradoxes, the set of parties at time step t need not be a subset of the set of parties at time step t + k. It can be easily derived from the definitions that nP4 ≤ nP1 + nP3, where nPi means the number of cases in which paradox (Pi) for i = 1, 3, 4 occurs. The version of the paradox as presented here is similar to the case of the paradox of new members that occurred with the expansion of the European Community from six to nine members in 1973. This case is analyzed in Brams and Affuso (1976: 41–42). We use the following counting method. First, all cases of entrances of new parties in the thirteen elections have to be identified. Subsequently, we try to discover the effects of these entrances on the power indices of the incumbent parties. From Table 1 we identify 16 cases of such entrances: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
1956–1959 (PSP appeared) 1956–1963 (PSP, BP and GPV) 1956–1967 (PSP, BP, GPV and D66) 1956–1971 (PSP, PPR, BP, GPV, D66, DS70 and NMP) 1956–1972 (PSP, PPR, BP, GPV, D66, DS70 and RKPN) 1959–1963 (BP and GPV appeared) 1959–1967 (BP, GPV and D66) 1959–1971 (PPR, BP, GPV, D66, DS70 and NMP) 1959–1972 (PPR, BP, GPV, D66, DS70 and RKPN) 1963–1967 (D66 appeared) 1963–1971 (PPR, D66, DS70 and NMP) 1963–1972 (PPR, D66, DS70 and RKPN) 1967–1971 (PPR, DS70 and NMP appeared) 1967–1972 (PPR, DS70 and RKPN)
127 15. 1981–1982 (EVP and CP appeared) 16. 1989–1994 (SP, AOV and Unie appeared) Then we determine a list of parties for which at least one power index increases after a new party joins. Let us adopt the following symbols: nSP4 – number of cases in which the paradox of new members appeared for the Shapley-Shubik index of a party. nBP4 – number of cases in which the paradox of new members appeared for the normalized Banzhaf index of a party. nnBP4 – number of cases in which the paradox of new members appeared for the Penrose-Banzhaf index of a party. nHP4 – number of cases in which the paradox of new members appeared for the Holler index of a party. nDP4 – number of cases in which the paradox of new members appeared for the Deegan-Packel index of a party m – number of cases analyzed for a party. More specific, m is the number of cases among sixteen comparisons of elections in which a party is an "old" member participating in the elections compared. Let: fSP4 be the frequency of the paradox of new members for the Shapley-Shubik index of a party, fBP4 be the frequency of the paradox of new members for the normalized Banzhaf index of a party, fnBP4 be the frequency of the paradox of new members for the absolute Banzhaf or Penrose-Banzhaf index of a party, fHP4 be the frequency of the paradox of new members for the Holler index of a party, and fDP4 be the frequency of the paradox of new members for the Deegan-Packel index of a party. Then, we have fSP4 =
nSP4 , m
fBP4 =
nBP4 m
fnBP4 =
nnBP4 m
nDP4 nHP4 , fDP4 = . m m The results are presented in Table 7. In each case analyzed, we observed that at least one incumbent party gained in terms of power after a new party entered the legislative scene. According to Table 7, the SGP, which is a small party with an almost constant number of seats (cf. Table 1), has most frequently gained in power after a new party entered the Second Chamber. The paradox occurred most frequently fHP4 =
128
Table 7. The paradox of new members for the Second Chamber in The Netherlands Party
nSP4
nBP4
nPBP4
nHP4
nDPP4
m
fSP4
SGP CPN PvdA KVP GPV PSP GL BP CDA ARP Total
13 5 5 3 2 2 1 1 0 0 32
11 5 5 3 2 0 1 0 1 0 28
12 5 8 3 2 2 1 0 0 0 33
10 5 1 1 1 1 0 0 1 1 21
11 5 0 1 1 3 0 0 1 1 23
16 15 16 14 7 10 1 5 2 14
0.8125 0.3333 0.3125 0.2143 0.2857 0.2 1 0.2 0 0
fBP4
fABP4
fHP4
fDPP4
0.6875 0.3333 0.3125 0.2143 0.2857 0 1 0 0.5 0
0.75 0.3333 0.5 0.2143 0.2857 0.2 1 0 0 0
0.625 0.3333 0.0625 0.0714 0.1429 0.1 0 0 0.5 0.0714
0.6875 0.3333 0 0.0714 0.1429 0.3 0 0 0.5 0.0714
P4 = the paradox of new members; nSP4 = the number of P4 for the Shapley-Shubik Index; nBP4 = the number of P4 for the normalized Banzhaf Index; nPBP4 = the number of P4 for the Penrose-Banzhaf Index; nHP4 = the number of P4 for the Holler index; nDPP4 = the number of P4 for the Deegan-Packel Index; m = number of cases analyzed for a party
129 for this party. This suggests a relation between constancy in seats of a party and the probability that such a party will meet the paradox. Another striking fact is that both the Holler index and the Deegan-Packel index are less sensitive for the paradox than the Shapley-Shubik and the Banzhaf index. The Penrose-Banzhaf index is the most sensitive according to our data. The general conclusion is that the paradox of new members occurs frequently in the Second Chamber of the Dutch parliament. An additional note should be made here. We do not assert here that an incumbent party gains in voting power because of the entrance of a new party. Since we dropped the condition of constant weights, the causal relation between the entrance of a new party and the rise of the voting power of an incumbent party cannot be determined empirically. The rise also might have been caused by the variation of the seat distribution itself. It is impossible to separate the effects of the two causes (entrance of new members and variation of the distribution). The thing we can say for sure is that we have observed the rise of voting power of some incumbent parties with less or equal weight after the entrance of new parties. 3.4. The paradox of large size This paradox occurs when the power index of a union of two or more parties is less than the sum of the power indices of the constituent members of that union. More rigorously, let I ⊆ N and suppose that the parties in I form a union U. Now, consider U to be one party. Then the paradox of large size occurs if (P5): wU =
i∈I wi
and φU
0.0262 = ShGPV∪RPF , nBzGPV + nBzRPF = 0.0274 > 0.0273 = nBzGPV∪RPF , HGPV + HRPF = 0.1617 > 0.1042 = HGPV∪RPF , DPGPV + DPRPF = 0.1384 > 0.0885 = DPGPV∪RPF . Thus, the paradox of large size occurs for four of the five power indices. It did not occur for the Penrose-Banzhaf measure. For this measure we found a sum index of 0.0468 and a union index of 0.0469. In addition, some doubts might be raised against the normalized Banzhaf index. The difference here is very small and might be due to rounding errors. Again, note that the differences are the largest for the Holler index and the Deegan-Packel index.
4. Agreements between power indices Besides comparing the frequency of the voting power paradoxes under the different indices, we also would like to check how frequently the power indices agree and disagree about the precise cases in which each paradox occurs. In order to investigate agreements and disagreements between the different indices, we use the following method. If for a party, paradox (Pi) (i = 1,2,3,4) appears for at least one of two different power indices φ, φ , then we define for a given party: nAPi(φ − φ ) – number of the precise cases in which power indices (of a party) φ and φ agree about the occurrence of paradox (Pi) nDAPi(φ − φ ) – number of the precise cases in which power indices (of a party) φ and φ disagree about the occurrence of paradox (Pi) fAPi(φ − φ ) – agreement index for a party between φ and φ (frequency of agreements for a party about the precise case in which (Pi) occurs between φ and φ ) nAPi(φ − φ ) , fAPi(φ − φ ) = nAPi(φ − φ ) + nDAPi(φ − φ ) where nAPi(φ − φ ) + nDAPi(φ − φ ) = 0.
133 After having calculated the agreement indices for separate parties, the overall frequency of agreements for the Dutch Second Chamber can be calculated. We introduce the following notation: TNAPi(φ − φ ) – total number of the precise cases in which power indices φ and φ agree about the occurrence of paradox (Pi) TNDAPi(φ − φ ) – total number of the precise cases in which power indices φ and φ disagree about the occurrence of paradox (Pi) FAPi(φ − φ ) – agreement index between φ and φ (frequency of agreements about the precise case in which (Pi) occurs between φ and φ )
FAPi(φ − φ ) =
TNAPi(φ − φ ) , TNAPi(φ − φ ) + TNDAPi(φ − φ )
where TNAPi(φ − φ ) + TNDAPi(φ − φ ) = 0. The agreement indices for each pair of power indices φ and φ where φ, φ ∈ {Sh, nBz, PBz, DP, H} and φ = φ will be calculated for the occurrence of the paradoxes (P1), (P2), (P3) and (P4). The pairwise agreement indices for paradox (P5) will be left out. There are only a few cases for this paradox and the relevant information about agreement is contained in Table 8 and 9. So, in total 40 agreement indices have to be calculated. The agreement indices are given in Table 10. Clearly, every agreement index ρ gives rise to a disagreement index δ by means of the formula δ = 1 − ρ. We do not give the disagreement indices here. When analyzing Table 10, we conclude that the agreement indices are high (from about 0.7 to 1) for the following pairs of power indices: DP − H (the agreement index is always greater than 0.8), Sh − nBz (in each case it is greater than 0.79), Sh − PBz (greater than 0.7) and nBz − PBz (greater than 0.75). For the remaining pairs, the agreement indices concerning paradoxes (P1), (P2) and (P4) are relatively low, ranging from 0.36 to 0.55. For each pair of the power indices, the agreement index concerning the occurrence of paradox (P3) is always greater than 0.96 and thus very high. In short, the conclusion is that we can distinguish two classes of indices. The first class contains the Shapley-Shubik, the normalized Banzhaf and the Penrose-Banzhaf index while the second contains the Deegan-Packel and the Holler index. The indices in each class show a large amount of agreement in detecting the same voting power paradoxes. However, there is little agreement between the two classes.
134 Table 10. Agreement indices P1=
P1
P2
P3
P4
FAPi(Sh-nBz FAPi(Sh-PBz FAPi(Sh-DP) FAPi(Sh-H) FAPi(nBz-PB) FAPi(nBz-DP) FAPi(nBz-H) FAPi(PBz-DP) FAPi(PBz-H) FAPi(DP-H)
0.7925 0.7544 0.5238 0.5156 0.7636 0.55 0.541 0.4203 0.4143 0.9792
0.9259 U.7077 0.4302 0.4066 0.7538 0.4706 0.4396 0.3854 0.3663 0.8861
0.9739 0.9739 0.9913 0.9913 0.9652 0.9826 0.9826 0.9826 0.9826 1
0.8182 0.8571 0.5135 0.4595 0.7941 0.5294 0.5455 0.5 0.4595 0.84
FAPi(Sh-nBz) = agreement index between the Shapley-Shubik index and the normalized Banzhaf index about paradox (Pi), i=1,2,3,4; FAPi(Sh-PBz) = agreement index between the Shapley-Shubik index and the Penrose-Banzhaf index about paradox (Pi); FAPi(Sh-DP) = agreement index between the Shapley-Shubik index and the Deegan-Packel index about (Pi); FAPi(Sh-H) = agreement index between the Shapley-Shubik index and the Holler index about paradox (Pi); FAPi(nBz-PBz) = agreement index between the normalized Banzhaf index and the Penrose-Banzhaf index about (Pi); FAPi(nBz-DP) = agreement index between the normalized Banzhaf index and the Deegan-Packel index about (Pi); FAPi(nBz-H) = agreement index between the normalized Bazhaf index and the Holler index about (Pi); FAPi(PBz-DP) = agreement index between the Penrose-Banzhaf index and the Deegan-Packel index about (Pi); FAPi(PBz-H) = agreement index between the Penrose-Banzhaf index and the Holler index about (Pi); FAPi(DP-H) = agreement index between the Deegan-Packel index and the Holler index about (Pi)
5. Conclusion and discussion Returning to our research questions as formulated in the introduction of this paper, we can say that we observed the following empirical and reconstructed facts in Dutch politics: 1. A political party receives more seats than in a previous election, but gets less power. 2. A political party receives less seats than in a previous election, but gets more power. 3. A political party receives as many seats as in a previous election, but gets more or less power. 4. An incumbent party receives less than or as many seats as in a previous election, but gains in power after the entrance of new parties.
135 5. The power index of a merger of parties is less than the sum of the power indices of the constituent parties. The first three facts are related to a modified version of the paradox of redistribution. The three facts together were observed frequently for all the five power indices used in this paper, to wit, the Shapley-Shubik index, the normalized Banzhaf index, the Penrose-Banzhaf index, the Holler index and the Deegan-Packel index. The Holler index and the Deegan-Packel index appeared to be more sensitive for this paradox than the Shapley-Shubik and the normalized Banzhaf index. These observations are curious indeed. They allow us to conclude that in Dutch politics it is empirically possible that a party wins an election but looses in voting power, that a party looses an election but wins in voting power, or that a party does not win or loose an election but receives in spite of that less or more voting power. The fourth fact is a translation of a version of the paradox of new members. Also this fact occurred frequently for all five power indices. But now, the Holler index and the Deegan-Packel index appear to be less sensitive than the Shapley-Shubik and the normalized Banzhaf index. Unfortunately, we are not able to say whether the increase of the power index of an incumbent party really was caused by the entrance of a new party or merely by the variation of the seat distribution. Very likely, both causes will be operational, but it is impossible to quantify their separate contributions to the increase. The only fact we can state for sure is that we observed increases of the voting power indices of incumbent parties after the entrance of new ones. This validates the conclusion that in Dutch politics the entrance of new parties may be beneficial for incumbent parties. In such cases, it is possible that incumbent parties loose an election or stay equal, but gain in power. We studied two mergers of political parties that actually occurred in Dutch politics before 1998. The first was the merger of the Catholic People’s Party (KVP), the Anti Revolutionary Party (ARP) and the Christian Historical Union (CHU) into the Christian Democrats (CDA) in 1977. The second was the merger of the Communist Party (CPN), the Pacifist Party (PSP) and the Radicals (PPR) into Green Left (GL) in 1989. With respect to the CDA we found the paradox of large size for the Holler index and the Deegan-Packel index for four of the six reconstructed parliamentary situations. We observed the paradox only twice for the Penrose-Banzhaf index. We did not observe the paradox for the Shapley-Shubik and the normalized Banzhaf index. With respect to Green Left, the paradox of large size occurred only once in the nine reconstructed parliamentary situations for the Shapley-Shubik index, the normalized Banzhaf index, and the Penrose-Banzhaf index. It occurred six out of nine times for the Holler index and the Deegan-Packel index. The
136 larger sensitivity of the last two indices for this paradox can be explained by the fact that these indices are based on the number of minimal winning coalitions. Since the larger the party, the less the probability for that party of being a member of a minimal winning coalition, the Holler index and the Deegan-Packel index will in general be lower for a merger of parties. Recently, the Reformed Political League (GPV) and the Reform Political Federation (RPF) merged into the Christian Union. By studying the effects of a hypothetical merger of these parties in 1998 and re-calculating the index for the union and comparing it with the sum of the power indices of the GPV and the RPF in that election, we were able to detect the paradox for four of the five power indices. It did not appear for the Penrose-Banzhaf measure. Of course, we admit that the evidence of the paradox of large size is obtained by means of hypothetical reconstructions. However, we think that in spite of this it is safe to conclude that it is not always beneficial for political parties to form a union. It might lead to a loss in voting power. This will hardly occur according to the Shapley-Shubik index and the normalized Banzhaf index; it will occur far more frequently according to the Holler index and the Deegan-Packel index. From an empirical point of view, there is hardly any difference between the Shapley-Shubik index and the normalized Banzhaf index. Both indices result in an almost identical power index matrix. Moreover, both are almost equally sensitive for the several paradoxes studied in this work. This fact is affirmed by their agreement index. The real differences are to be found between the Shapley-Shubik and normalized Banzhaf index on the one hand and the Holler and Deegan-Packel index on the other.
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