Who has the power in the EU?
Jason Barryand Francesco Passarelliz
July 2, 2007

Abstract
The European members have reached an agreement on how toreform the EU’s institutions. This has has strong implications forthe balance of power among member states. Building on the workof Shapley (1977) and Owen (1972), we present a measure of powerthat is based on players’preferences and number of votes. We apply
this measure to the Council of Ministers to see who wields power now
and who is likely to wield power with the future voting scheme. We
also provide a rationale to explain why the negotiations for the newConstitution have been so di¢ cult. Further, we show how a country’s
power can change based on the preferences of the agenda setter, which,in this case, is the European Commission.
JEL Classi…cation: C71, D72, D78, H11
Keywords: Power Indices, European Union, Principal Components
The authors would like to thank Massimo Bordignon, Sandro Brusco, Mario Gilli,
Manfred Holler, Annamaria Lusardi and Stefan Napel for their comments and suggestions
on earlier drafts. Finally we appreciate the helpful discussions with Alberto Alesina, Pierpaolo Battigalli, Gérard Roland and Guido Tabellini. Any errors are the sole responsibilityof the authors.
yRutgers University, Newark. Dept. of Economics, Newark, NJ 07102. email: jmbarr@andromeda.rutgers.edu.
zCorresponding Author. University of Teramo and Bocconi University. email:francesco.passarelli@unibocconi.it

11

Introduction

The European Union (EU) is facing the challenge of an enlargement thathas almost doubled the number of members. The Treaty of Nice (December
2000) represents the …rst attempt to give European decision-making mechanisms deeper democratic foundations and greater e¢ ciency. The widespread
belief is that this attempt failed primarily because the representatives of thenational governments were reluctant to change the current institutional architecture that grants more power to the member states (through the Councilof Ministers) than to the European Parliament (directly elected by the citizens) and to the European Commission (a centralized institution with powerof initiative).
In March 2002, a Constitutional Convention began working on the enormous undertaking of constructing a decision-making system that remainse¢ cient and meets the principles of legitimacy and acceptability. After theintergovernmental negotiations and adjustments, the Convention’s proposals
were endorsed by the Bruxelles Summit on June 2004. In October, the headsof states signed in Rome the Constitutional Treaty. However, the rati…cation
process was stopped in June 2005 by two “no”votes in the French and Dutch
referenda. In June 2007, after a two year “re‡ection phase”, the project ofwriting a Constitution was abandoned, and the new institutional rules were
included in the reformed New European Treaties (NET). There will be afull-time President for the European Council, a Vice-President in charge of
foreign a¤airs, a new Commission, a more powerful EU Parliament, and a
new voting scheme for the Council of Ministers. The NET will be rati…ed by
2009, but the new voting mechanism will enter into force in 2014.1
During the negotiations the voting mechanism in the Council became
one of the most important issues. Initially, for example, Spain and Poland
have opposed the proposed changes, saying that it would radically modify
the power distribution among the member states. Arguably, all countries
have been evaluating the new system in terms of losses and gains in theirbargaining ability.
In this paper we analyze how the new scheme a¤ects the countries’political power in the Council. We take a coalitional game perspective; therefore
we do not decribe the structure of the legislative game that takes place in
1
In the event of a dispute, Member States may invoke the Treaty of Nice and demand
the postponement of an undesired decision until 2017.

2

the Council. However, we move away from the typical hypothesis of symmetry, that belongs to the classical coalitional power analysis. In fact, in orderto give a more realistic description of the legislative bargaining within theCouncil, we take countries’political preferences and agenda setter distortions
into account.2

1.1

Constitutional rules and European bargaining

Despite the relative narrowness of its budget, the EU has already acquired
a wide set of competencies. The bene…t from participating in the Union
comes from the coordination and centralization of several policy areas, suchas a single currency, internal and external trade, competition, international
relations, and social protection. The literature on political economy, from…scal federalism to contract theory, o¤ers contributions on what the European
Union should do and how it should be done (Alesina, et al., 2002, Alesina
and Perotti, 2004, Berglof, et al., 2003).
The distribution of the EU bene…ts through negotiations and lobbying
are part of the daily life of the EU institutions. Part of this bargaining gametakes place at intergovernmental level: every Council meeting, including the
preparatory work, can be considered a non-cooperative game played by thedelegations of the states within the rules of the Treaty.
However, in the constitutional phase of reforming the Treaty, it is important to abstract away from the political interests present in particularvoting environments and concentrate on the rules of the game, and on theirability to generate equitable opportunities to in‡uence political decisions.
As such, this analysis can be conducted within the theoretical framework ofcooperative game theory. Thanks to the seminal work of Shapley and Shubik (1954), the concept of the Shapley Value (Shapley, 1953) is taken as anindex of the a priori power of the members in a committee. In summary,the Shapley-Shubik index is a measure of the relative frequency with which amember country can determine the outcome of a particular vote if all possible
2
We concentrate on the Council because there is a broad consensus that it has a prevailing role in the EU institutional architecture. Napel and Widgrén (2006) using a Nashbargaining rationale, show that, despite codecision, a quali…ed majority in the Council
versus a simple majority in the Parliament gives the former more power than the latter.
Moreover the Council is the intra-state legisltive arena. Noury and Roland (2002) showthat the coalitions within the European Parliament follow “party group”dynamics, rather
than “national” ones.

3

coalitions of a …xed number of member states were equally likely to occur;and it is, in general, some function of the number of votes and the majority
threshold.3
Conventional wisdom holds that France and Germany, for example, arethe big “players” in the EU arena. But why is the same power not conventionally assigned to Italy and United Kingdom, who have the same weightin the Council? Moreover, and in general, how are the preferences of thecountries likely to a¤ect outcomes? What will happen now that enlargement
has taken place? Here, we expand upon the de…nition and measurement of
power proposed by Shapley and Shubik. We discuss and estimate a spatial
measure of the Shapley Value, which comes directly from the preferences ofthe member states. Our approach is able to address these questions.
This paper is novel in several respects. First, from a theoretical point ofview, we present a simple analytic extension of the work of Owen and Shapley(1989). By directly incorporating the preferences of the players we generatea probabilistic-based power index. We argue that when the preferences ofthe players are stable and predictable, the “asymmetric”spatial perspective
better captures the e¤ects of long-lasting voting schemes. Second, we extend
this theory in a new direction by modelling the e¤ect that an agenda-setter
can have on the outcome of the game. The interaction of the preferencesof the players (e.g., the EU member countries) and the preferences of theissue-setter (e.g., the EU Commission) can substantially alter the power distribution among players. To the best of our knowledge, this approach has
not been done elsewhere.
From an empirical point of view this paper is new in the following ways.
First we directly measure the “political preferences” of the older and newer
EU countries themselves. By analyzing EU-based polling data, we can geta measure of the extent to which member countries are relatively “pro” or
3In our analysis, the Shapley-Shubik index (and Normalized Banhalf Index) is a simple
linear function of the number of votes of the nations. For example, for the 15 EU countries(pre-enlargement), with votes distributed according to the ’Pre-Nice’ scenario, we canpredict the Shapley Value from the votes by the simple OLS regression function:^

SS =0:002+ 0:012 V otes:(0:0009) (:0014)

R2 = :998; n = 15: Standard Errors below estimates.

4

“con” in regard to relinquishing decision making to the EU Council. We
then apply these measured preferences to computing power indices for the
EU countries. Finally, we measure how the distribution of power depends
upon the preferences of the agenda setter.
By directly using these preferences we …nd some interesting and novel
results:
When considering the political positions of the countries, the numberof votes is not necessarily a good predictor of power. For example,
decreasing the quali…ed majority threshold (from the current scheme
to the NET) tends to shift the power to countries with moderate preferences. The Franco-German axis emerges from the centrality of theirpreferences and their size. Little power rests upon the Northern “Euroskeptics”or the Mediterranean “Euroenthusiasts.”
The Euroskeptics, such as the United Kingdom or Denmark, become
prominent if unanimous decisions, like the EU budget or taxation, haveto be taken.
Under the system agreed at Nice, after the enlargement the older leaders tend to lose most. Having a certain degree of Euroenthusiasm will
put Spain, for example, in a favorable position. Euroenthusiasm willin turn favor the newcomers. If the NET does not come into force, theEastern countries are likely to exercise a very strong political in‡uenceon the Council.
In comparing the Nice arrangement to the NET, the Nice rules willallow the Eastern countries to collect almost 40% of total power, despite
less than one fourth of population. The reapportionment proposedby the NET favors moderate positions and restores the power of the
populous members, such as Germany and France. Spain emerges as a
big player. The power shifts back to the Western members.
A distorted pro-Europe Commission can cause the power to shift tocountries located on the Euroskeptic side of the political space. This
shift tends to be more important when voters have less vague expectations about the agenda setter’s preferences, when countries are highly
dispersed on the political space, and when the majority threshold increases. The largest amount of power redistribution due to the agenda
5setter distortion occurs in the post-enlargement scenario with the Nicerules. The reallocation of power in favor of the large old members dueto the NET scenario is partially o¤set when the Commission is pro-EU.

1.2

Literature review

Although “power” in political science is a “penumbral” concept (Shapley,
1977, p. 5), cooperative game theory has proved useful when investigating the in‡uence that a voting system gives to the voters. Applications tonational and international legislative bodies have gained legal importancein evaluating reapportionments of votes. The literature on applications of
power indices to the European Council of Ministers is rich and widespread.4
This is partially due to the frequent enlargements of the EU, which provide
new voting distributions to evaluate. This literature consists in computing
or re…ning the standard Shapley-Shubik (1954) or Banzhaf (1965) indices for
the EU members; thus, one usual assumption is that the countries cannot be
distinguished by their attitudes toward the EU. However, ignoring the “policy positions”of the European governments could yield an overestimation of
the power of the national governments with extreme preferences. Moreover,a priori power indices cannot take into account the “location” of the EU
Commission, which plays the role of agenda-setter for the Council.
The theory of spatial indices can provide a strong analytical backgroundwhen ideological di¤erences among players are crucial. Owen (1972) suggests a scheme of coalition formation that considers the ideological distance
between voters in a political space. Building on Owen’s intuition, Shapley(1977) and Owen and Shapley (1989) provide a “non-symmetric”generalization of the Shapley-Shubik index in which each player’s power depends, inaddition to the voting rules, on her location in a political space. This generalized spatial index emphasizes the role of ideology in coalition formation.
In this scheme the coalitions are inspired by policy issues. Given a policy
4

A very good survey on power indices is in Stra¢ n (1994). General criticism of theimplicit behavioral assumptions behind power index analysis is in Coleman (1971).
Holler and Owen (2001) and Felsenthal and Machover (1998) contain very good surveys
and detailed references to the applications to the EU. On the application of this literature
to other institutional contexts, see the survey by Benoit and Kornhauser (2002).
A general debate on power indices with some focus on the EU is in Power Indices and theEuropean Union, a special issue of the Journal of Theoretical Politics, July 1999, Volume
11, No. 3.

6issue, the players can be ordered by the level of support for that issue. The
support de…nes the ordering within a coalition; only the “ideologically consistent”orderings are considered. Thus the probability of a coalition emerging
is related to the number of policy issues it is inspired from.
Owen and Shapley consider all the policy issues as equiprobable, assuming
the absence of any information about the issue generating mechanisms. We
argue that in the presence of an agenda setter some policy issues can be
more likely than others. As a consequence, ceteris paribus, the countries
which tend to be in pivotal positions in policy areas preferred by the agenda
setter will have more power. In other words, the agenda setter alters theprobability of the issues and distorts the distribution of the power.
In this context, the theoretical framework o¤ers an interesting perspective
for analyzing the political games that take place among the states’representatives within the Council of Ministers of the EU. It accommodates the most
relevant criticisms raised about the application of power indices to the Council. The problem then becomes how to de…ne a political space for EU mattersand how to place countries in it. As far as we know, the only empirical workthat tries to answer this question in regards to the EU is Passarelli and Barr
(2007), which computes a probabilitic value for a single dimensional policy
space. Here we use principal component analysis (PCA) to extract the preferences of the countries toward the EU, then we build a two-dimensional
space (in the spirit of Rabinowitz and MacDonald (1986), who use PCA for
the U.S. presidential elections).
1.2.1

Cooperative versus non-cooperative political games

The cooperative spatial approach for solving political games is not without controversy and it is worthy of some comments and warnings. A noncooperative view is frequently more suitable than the cooperative approach
to illustrate contingent contexts of policy decision making, although also
strategic or extensive form games can have practical di¢ culties with complex institutional environments and intransitivity of majorities.5 Cooperative
games have the advantage of simplicity. However the nature of the coalitional
agreements can be very complex and issues such as enforcement, representa5

See Pearsson and Tabellini (2000) for a complete survey of models of political economy.
An essencial reference for the public choice theory is Mueller (2003).

7

tion, procedures, etc. emerge.6 In addition, the cooperative solution concepts
often rely on hypotheses about coalition formation and bargaining that could
appear too strong.7
These di¢ culties arise mostly when cooperative games are used to describe speci…c short-run political situations, where the contingent aspects ofthe game and the strategic attitudes of the players are known and predictable.
But these problems are less severe in a long-run constitutional perspective,
in which the analysis concerns a voting mechanism that will be applied to alarge set of policy decisions and to a variety of di¤erent contingencies. In thisperspective the attention can be restricted on a game in which the primary
aim of the players is “... to succeed in forming certain decisive coalitions.”“... and there are no other motives which require a quantitative description.”(Von Newmann and Morgenstern, 1944, p. 420, 424)8 This is how Von
Newmann and Morgenstern present simple games. Then simple coalitional
games perfectly …t the analysis of a voting system in which we abstract awayfrom contingent information.
Shapley (1953) o¤ers a solution for symmetric games in which the coalition formation can be described as a random process. The Shapley solution
is based on the symmetry of players, that is a situation in which the players“...do not o¤er to any player any possibility which is not equally open to
any other player” (Von Newmann and Morgenstern, 1944, p. 224). In thisperspective, any ordering of players is equally likely. The Shapley-Shubik
index is the Shapley value for simple voting games.
The ideological perspective removes the symmetry assumption, allowing
for orderings that are not equally likely. In some sense, it frames the analysis
in the direction of higher realism. However, it cannot be asked too much:
6

Snyder et al. (2005) compare the standard power indices with the equilibrium of anon-cooperative game à la Baron-Ferejohn with weighted votes. They …nd that the indices
are not good predictors of the expected payo¤s of the players.
7
Sloss (1973) shows the relationships between the core and the existence of a Condorcet
winner: games with an empty core present the Arrowian problem of cyclical majorities.
In general, the core is empty for constant-sum games and it is very likely to be empty
for spatial coalitional games with many players and a limited number of dimensions. The
core is the most intuitive solution concept of cooperative games. However, even if it isempty, other solutions can reasonably be adopted, such as V-sets, kernels, bargaining sets
and symmetric values.
See Owen (1995) for a survey of cooperative solution concepts. A complete reference on
the di¢ culties of the cooperative spatial approach is Ordeshook (1997).
8
Reprint of the third edition, 2004.

8

the ideological power analysis cannot explain how some speci…c decisions
are made and implemented and how much value a single decision produces.
Moreover, a non symmetric generalization of the Shapley solution is notunique. 9 Summing up, when the players have not identical attitudes, theideological power analysis o¤ers at least a measure of how likely each player
is to be in a favorable position to in‡uence on her own advantage the futuremajority decisions. This perspective can provide useful evaluations of the“players’prospects”of having to play with di¤erent voting schemes.
The rest of the paper is organized as follows. Section 2 discusses thetheoretical measures of power. Then in section 3 we present the results ofour empirical analysis of the EU countries’preferences and power measures.
Next in section 4 we show how changing the agenda-setter preferences a¤ects
the distribution of power. Section 5 concludes. Several appendices containtechnical information for the interested reader.

2

The theory of voting power

Consider a set N = f1; 2; :::; ng of players and denote by 2N the collection ofsubsets (coalitions) of N . A game is a real-valued function v : 2N ! < thatmeasures the worth of each coalition. Let G be the collection of all games on
N . For a given player i, let piT : Ti N n i be a probability
distribution overPithe collection of coalitions not containing i, with Ti N ni pT = 1 for all i.
De…nition 1 A value i for i on any collection of games T
abilistic value if for every v 2 T :X(v)=piT [v(Ti [ i) v(Ti )]i

G is a prob-(1)

Ti N niE(Ti ) piT [v(Ti [i) v(Ti )] is player i’s expected worth from joining thecoalition Ti . From a probabilistic viewpoint, the value of the game for player
i is a measure of her prospects from playing the game; it is calculated bysumming up the expected values of participating in all the possible coalitions.
If v takes only the values 0 and 1, the game is said to be “simple” andif v(S) = 1 (with S
N ), then S is a winning coalition, otherwise S is a
9

Excellent surveys on asymmetric Shapley solutions and on values for non-transferable
utility games are Monderer and Samet (2002) and McLean (2002), respectively.

9losing one. For a given simple game v and a coalition Ti N n i, the player
i is called the pivot if v(Ti ) = 0 and v(Ti [ i) = 1.
In other words, being in the pivotal position allows player i to change theworth of the coalition. Arguably, in political situations, casting the vote thatturns one coalition from losing to winning is a valuable position, worthy tobe rewarded by the other voters already in the coalition. This gives rise to
the question: How much do the voting rules in‡uence each player’s relative
frequency to cast the swing vote?
The pivot is the member who casts the “last”vote needed for the passage
of a bill. The ordering of the support to the bill is then relevant, but in
Shapley-Shubik’s perspective it is taken at the most abstract level since no
information about the members’preferences are available, and only the rulesof the game are relevant. Thus, if we interchange the players, the value ofthe game for an individual in a particular position will be the same as theone assessed by any other player in that position. This is the basic idea ofthe symmetric approach which inspires the Shapley-Shubik index.10
10“Abstract games are played by roles ... rather than by players external to the game”(Shapley, 1953, p. 308). Thus the individuals cannot be distinguished by their level ofenthusiasm or lack thereof when they participate in a certain election, and the bills cannot
be characterized by a measure of their “acceptability” (see “Axiom 1” in Shapley (1953),
p. 309).
In a political game, this justi…es the abstract idea of voters, who subjectively believe that
all the coalitions are equally likely to be of any size and that all the coalitions Ti N n i
of size t (with t = jTi j) are equally likely. In sum, for any player i, we get the Shapley
Shubik index if we substitute the following in (1),

piT =

1
n

n

1
t

1

(2)

(Weber, 1988. p. 103).
Retaining symmetry, we get another famous measure of power due to Banzhaf (1965)
when the players believe that each coalition Ti N n i has equal probability
1
(3)
2n 1
It is clear that both the Shapley-Shubik and the Banzhaf indices can be helpful methods
to determine the distribution of the power if the “names” of the players do not matter.
piT =

10

2.1

Preferences and power

The symmetry is a desirable characteristic if we do not have information
about the stable attitudes and di¤erences among the players. In many situations, however, we have knowledge about the preferences of the players andwe can use it to evaluate the most likely outcomes of the game. Ultimately,
the value of the game for any player and the solution of the bargaining problem should depend on the personal characteristics of each participant, as far
as we have information about those characteristics.
This is particularly true in political games when the voters can be assumed
to vote according to their political pro…les over an m-dimensional politicalspace. In the simplest case of one dimension, for example, we can think ofcongressmen being distributed on a left-right wing axis. We can add politicaldimensions if we detail the political preferences (e.g., pro/con federalizationof political areas, pro/con strict budget policies, etc.).
We expect that ideologically similar players will tend to behave similarly
in coalition formation. This means that coalitions with ideologically similar
players are more likely to emerge than coalitions that have distant voters.
In a probabilistic perspective, we should relate the ideological positions of
the players to the randomization scheme that assigns a probability to eachcoalition in 2N .
Suppose each voter i has an ideal point (or location) Pi 2 <m in an mdimensional Euclidean space, where each dimension captures the ideologicalparameters of the (political) game. Let<m be the set of all the issues
to vote on. Each issue is a vector U 2 .
Suppose we have a function fi (U ) such that fi : ! < exists for eachplayer i = 1; :::; n and measures player’s i level of enthusiasm toward theissue U . Thus, we can induce an ordering on N through the fi (U )’s. Moreprecisely,
j i i¤ fj (U ) fi (U ) 0:
(4)
Equation (4) says that if player j is more enthusiastic than player i when theissue U is proposed, she will vote “yes”before player i.
Let U be randomly chosen from a probability distribution p(U ) :!
[0; 1]. Since U is a random vector, we can de…ne the random variable Yij =fi (U ) fj (U ). Note that if Yij 0 player j will participate before i.
Consider the coalition Ti N n i and let ATUi denote the subset of such11

that j 2 Ti i¤ Yij

0. Then the probability of observing Ti is given by
Z
Z
i
pT =
p(U )dU:
(5)
T

AUi

Equation (5) says that the probability of a coalition in which some players
j enter before i is given by the probability of observing all the issues U for
which players j are more enthusiastic than i.
In particular, via (5) we de…ne a probability distribution over the set ofthe possible coalitions not containing i, piT : 2N ni ! [0; 1]. This is useful tocharacterize a probabilistic value in a spatial context.
De…nition 2 A value i for i on any collection of games T
G is a probabilistic spatial value if for every v 2 T , i is de…ned by (1) and piT is de…nedby (5).
For a given simple game v, player i’s value i (v), speci…ed by (1) and (5),can be interpreted as the probability of being in a pivotal position, out of all
the possible coalitions that the random issue U can inspire. Of course, since
the pivotal argument is useful for evaluating the voting rules from an a priori
perspective, we require that the locations capture long run policy attitudesof the voters.11
2.1.1

The Owen-Shapley scheme

Owen and Shapley (1989) propose to restrict U to lie on the unit-sphere
Hm 1 . This is equivalent to imposing
hU; U i = 1(6)

for all U 2 . Moreover they introduce a special formulation for the fi ’s,whose nice characteristics will become clear soon:fi (U ) = hU; Pi i
11(7)

As pointed out above, our analysis is at the contitutional level. The use of thisapproach to describe speci…c voting contexts is questionnable, since additional information about the policy outcome would need to be embodied in the characteristic function.
Moreover that speci…c outcome should correspond to a point in the policy space, and moredetailed information about the preferences of the players with respect to that speci…c pointshould be included. Finally, also assumptions about the transferability of utility or theability of the voting game to generate e¢ cient outcomes should be made.

12

Finally, Owen and Shapley assume that U is chosen from Hm 1 by a uniform probability distribution and that v is a simple game, then they conclude
that piT is the Lebesgue-measure of what we have called ATUi Hm 1 .
The Owen-Shapley approach is is not the only way to give the Shapley
Value an asymmetric generalization. What makes this approach intuitively
appealing is that orderings of players are generated, or “inspired”by random
political issues: players come close in a certain ordering because they sharesimilar views on the policy issue that generates that ordering.12
Combining the (6) and (7) into the spatial context depicted above we
get a probabilistic characterization of the Owen-Shapley spatial value. In
section 2.2 below we provide an example of a game with three players in a
two dimensional political space.
Now we will show that the probabilistic value de…ned by Owen-Shapley’s
randomization mechanism may represent an equitable distribution scheme of
the full yield of the game; in other words, the vector (v) = ( 1 (v); :::: n (v))
can represent the payo¤s of the players from participating in the game v. We
know that this corresponds to satisfying the so called e¢ P
ciency axiom (seeAxiom 2, Shapley(1953, p. 309)); i.e. if for every v 2 T ,i (v) = v(N ).
i2N

Weber (1988, p. 113) demonstrates that a probabilistic group value =( 1 ; :::: P
satis…es the e¢ ciency axiomn ) on a collection T of
Pasymmetric
P gamesiiti¤: (a)pN ni = 1 and (b)pT ni =
pT for every nonempty T N .13
i2N

i2T

t2T
=

Proposition 1: The probabilistic spatial value de…ned by (1), (5) and by(7) satis…es the e¢ ciency axiom.
Proof. We have to show that both (a) and (b) are satis…ed.
RR(a) Consider that from (5) piN ni =p(U )dU . Remember that for
N

AU i
Ni
any i 2 N , A
the subset of Hm 1 in which i comes
It easy
TU is
S last.
Niito see that
AN
has
zero
Lebesgue-measure,andA=
Hm 1 ,
UU
i2N
i2N

then Pr

S

i2N
12

i
AN
U

=

P

i2N

piN ni = 1.

In addition, Winter (2002) oberves that if clusters of ideal points in the polical space
are considered a priori unions of players, then the Owen-Shapley method is related to thevalue suggested by Owen (1977) for games with coalition structures.
13
Probabilistic values that satisfy the e¢ ciency axiom are called quasivalues and can bedone a random-order description.

13

(b) Let P
Ti be any Rpossible
subset in N
From
R
R the
R randomization
Pn i.
i
t
schemepT ni =
p(U )dU andpT =p(U )dU . Thus,
S

i2T

t2T
=

T

i2T

AUi

in order to satisfy the (b) we must have (b:1):

S

S

t2T
=

i2T

AT
U

ATUi =S

ATU for every

t2T=

nonempty T N . The left hand of (b:1) is the set of all the U 2 Hm 1such that for all j 2 Ti and all t 2= Ti , Yij
0 and Yit 0. For every
T , call i the “least enthusiastic”player. Thus the right hand of (b:1) isthe set of all the U 2 Hm 1 such that for every player t 2= T and every
j 2 T we must have Yit 0 and Yij 0. It is easy to see that the twounions coincide for every T N .

2.2

An example

As an example, in …gure 1 we present graphically a 2-dimensional politicalspace with 3 voters, N = fa; b; cg ; who have ideal points Pa ; Pb and Pc . Below
we consider a simple political game and compute the probabilistic spatial
power index, adopting the Owen-Shapley ordering generating mechanism.
Pb
Pc

β

γ
Χ

γ

β
∆

αγ
Ε

Β

α

Α

β

α

Pa

Φ

Figure 1: An example of a spatial game in two dimensions.

For two dimensions, Hm 1 is the unit circle H1 , and the random vector
U can be any point on H1 . Due to this, every U 2 H1 can be identi…ed by
14

one and only one angle, 2 [0; 2 ), by the function U = (cos ; sin ). Exploiting this bi-univocal correspondence U $ , we can conveniently reduce
the number of dimensions by one. Hereafter, for two dimensional cases, wewill refer to the value of as the political issue.
In …gure 2 we have drawn the ordering functions fi ( ) : [0; 2 ) ! < forall the players i 2 N . Given the player i’s political pro…le, fi ( ) measures
her level of enthusiasm in supporting the bill “inspired”by the political issue. We can see that, for example, as long as the proposed bill lies within theinterval [0; A) voter a will say “yes”…rst; player b will come after, and c willbe the last one. In other words, the coalition Ta of players more enthusiastic
than a is empty if the political issue is within [0; A). Below we report all thepossible coalitions Ta of players more enthusiastic than a and the subsets ofrotation of U over H1 (or, equivalently, the movement of along [0; 2 )) thatgenerate those coalitions.
8899Ta = f?g >2 [0; A) [ (E; 2 ) >
>
>
>
>
>
>
<
<
=
=
Ta = fbg
2 [A; B)takes place if
Ta = fcg >2 ( ; E]
>
>
>
>
>
>
>
:
:
;
;
Ta = fb; cg
2 [B; ]In …gure 2 all the possible n! orderings are listed in the row at the topof the graph. Note the correspondence between the width of the angles ,and in …gure 1 and the subsets of [0; 2 ) on the horizontal axis of …gure 2.
Within this political space let’s now consider the game v1 = [3; 2; 1; 1] inwhich the quali…ed majority threshold is three votes out of four; player acan cast 2 votes, b and c each have 1 vote. What is player a’s prospect fromplaying this game? It’s easy to see that player a succeeds in being pivotal
only if at least one player has said “yes” before her. Thus her power in thisgame is given by the probability of observing any Ta fN n ag n f?g. This
probability is the player a’s power index for the game v1 .
Let p(U ) be the probability distribution of the random vector U over theunit circle. Considered that U = (cos ; sin ) is a one-to-one transformationfrom H1 to [0; 2 ) we can specify the density function p( ) for the random
R2variable . In particular, p( ) : [0; 2 ) ! [0; 1] and p( )d = 1.
0

Having speci…ed p( ) and the ordering generating mechanism induced byfi ( ), we can now compute the probability of any possible coalition Ta N na
by integrating p( ) over the subsets of [0; 2 ) in which that coalition occurs.

15

abc
β2

bac bcaγαcbaβcab acb
α
γ

abc
β1

fa

fb

0

Α

Β

Χ

∆

Ε

Φ

2π

ϑ

fc

β1 + β 2 = βFigure 2: Orderings for three players from 0 to 2 :For example pr fTa = fb; cgg =Rp( )d .

B

Returning to player a, her chance of being a pivot is:
a (v1 ) =

ZE

p( )d

A

Owen and Shapley (1989) suppose that the political issues have equal
probability of being on any point of the unit-circle. They justify this hypothesis by the absence of information about the circumstances that cana¤ect the proposed bill. With uniform probability, it is easy to see that thepower index for player i is given by the proportion of [0; 2 ) in which coalitions for which i is pivotal are generated (the shaded area in the …gure 3).
In our example above, voter a’s power index would be, a (v1 ) = (E A)=2 .
The spatial value de…ned by (1), (5) and by (7) is related to a point
X in the Euclidean political space that is dominated by a set of points
with minimum Lebesgue measure. In other words, X is the policy outcome
with the lowest probability to be beaten by any other alternative; i.e. it
is a “Copeland winner.” This is a nice characteristic of the Owen-Shapley
method, especially for games with and empty core, since it helps to predict
where policy outcomes are likely to be located. Unfortunately this result has
16

p(θ )

1
2π

0

Α

Β

Χ

∆

Ε

Φ

2π

ϑ

Figure 3: Power index with uniform probability distribution.

been proved only for simple majority games and for uniformly distributed
issues (Grofman et al.,1987 and Owen and Shapley, 1989).

2.3

The political windIn our spatial political games the random variable U captures the “blowingwind”that inspires the bill the voters are called to vote for. We can reasonably assume U as determined by random circumstances outside the controlof the players. However, in some cases a certain amount of knowledge about
the likelihood of these circumstances is available to the players and it can justify a non-homogeneous probability distribution over all the possible politicalissues that inspire a bill. Of course, we require that all the players share thesame knowledge consistently.14 If the issues that generate a certain coalition
are highly likely, the subjective probability that the players assign to thatcoalition will be high as well. The probability of the issues will in‡uence theplayer’s prospects from adopting a given voting system.
In …gure 4 we have a radically di¤erent story from …gure 3: the issues
that inspire coalitions for which a is pivotal are relatively unlikely. This canchange the players’ prospects substantially; the power of player a is very
low, despite the veto power and the relatively favorable political position. In
general, in spatial political games the power of the players will be determined,
not only by the votes and the relative positions, but also by the probability
14
Heuristically the conditions on which the e¢ ciency of the power index depends (seealso proposition 1 above) require that all voters share the same probability distribution
over the set of all possible coalitions.

17

distribution of the issues, and ultimately by all the known circumstances that
can in‡uence the political content of the bills.
p(θ )

0

Α

Β

Χ

∆

Ε

Φ

2π

ϑ

Figure 4: Power index with non-uniform probability distribution.

2.4

The agenda setter

The factors that can a¤ect the political nature of the bill include the preferences of the institutions that have prerogatives in setting the policies tobe voted on, the order in which policies are voted on, and the way the policies to vote on are split or grouped. Usually, however, the “agenda setter”(i.e., the institution that proposes new policies) does not directly vote in thecommittee. Despite this, its ability to a¤ect the voting outcome is positively
related to a series of variables, such as the degree of monopoly power insetting the agenda, the amount of information regarding voters’preferences,and the scope of implementing complex voting sessions (sequential referenda,
sophisticated voting, etc.). These topics have been deeply explored in theframework of non-cooperative game theory, starting from the pioneering workby Romer and Rosenthal (1978).
In this paper we look at the power of the agenda setter in a more abstract
light, where bills are generated at random. If there is an agenda setter witha certain monopoly in proposing the bills, the probability distribution overthe set of all possible bills will re‡ect the type of the agenda setter. The typeincludes any relevant information for the agenda setter’s decision making
such as payo¤s, institutional constraints, beliefs about the voters’preferences,
etc. We assume that the voters can infer the likelihood of each issue from aprobability distribution over the set of all types.
18

Let<m denote the space of all the possible types of agenda setter T .
We assume that the voters share common knowledge about the distributionq(T ) over , from which the types are drawn. Let U = s(T ) be the optimal
issues the agenda setter proposes as a function of his type. We assume that s
exists and that the voters have common knowledge of it. Thus the voters canuse their beliefs, q(T ), to compute a distribution, p(U ), that illustrates how
likely each issue is. We want to show that, under some general hypotheses,a higher probability assigned to a certain type implies a higher probabilityof the optimal bill being selected by that type.
Our investigation into the e¤ect of the agenda setter in this paper issomewhat general. Here the agenda setter is described as the only institutionthat a¤ects the direction of the blowing political wind. No speci…c analysis of
his preferences or strategies is introduced, such as his ability to promote social
rather private welfare or his independence from the players. More speci…cassumptions on the behavior and preferences of the agenda setter would be
too speci…c in this stage of analysis, and arguably incompatible with our apriori approach. Nonetheless, one can view this game as if there was a noncooperative pre-stage in which the agenda setter chooses his equilibrium purestrategy from . In the cooperative voting game presented below the votersanticipate how the agenda setter will play from the knowledge that they haveof his type. Thus, we are keeping the characteristics of the pre-stage at themost general level.
Proposition 2: If there exists a continuous joint distribution q(T ), over<m and a one-to-one function s : ! whose inverse is continuous,
then there exists a probability distribution, p(U ) : ! < and a one-to-one
function h : ! < such thatp(U ) = q(T ) h(T ):Proof. Since s :!

we can specify:
u1 = s1 (t1 ; :::; tm )::::::um = sm (t1 ; :::; tm );

19

(8)

with U = (u1 ; :::; um ) and T = (t1 ; :::; tm ). Moreover, since s is a one-toone transformation we can invert the m equations in (8) and we obtain:
t1 = g1 (u1 ; :::; um )::::::tm = gm (u1 ; :::; um ):(9)

Since the gi ’s are continuous, then for every i = 1; :::; m and j = 1; :::; meach partial derivative @gi =@uj exists at every point (u1 ; :::; um ) 2 . Thus
the jacobian, J, of the inverse transformation (9) can be constructed. Exploiting a common result of the probability theory, we know thatp(U ) =

q(g1 ; :::; gm ) jJj for U 2
0otherwise(10)

Proposition 2 is proved if we take J(T ) = h(T ).
Corollary 1: If q(T) increases (decreases) for some T, then p(s(T))
increases (decreases).
Proof. Just observe that jJj in (10) is always positive. Thus p and q are
positively related.
Corollary 2: For any subset A
,
Z
Z
Z
Z
p(U )dU =q(T ) jJ(T )j d(g1 ; :::; gm )
(11)
A

A

Proof. This proof is trivial.
The meaning of the proposition and the corollaries above is simple andcan be described by the following example. A Prime Minister (the agenda
setter) is going to start his mandatory period. In order to anticipate thepolitical content of the bills he will propose, the political groups in the Parliament (the voters) are likely to use their knowledge about, say, the electoralpromises of the Prime Minister, his political pro…le, his linkages with interest groups, etc. Then if, for example, the Prime Minister is perceived to
be strongly in favor of environmental protection, the groups will reasonably
expect bills with high expenditures in this area. As a consequence, coalitions that include pro-environment groups will be perceived as more likely
20than coalitions that exclude them. Moreover, the pro-environment parties
will tend to vote “yes” before the others. As a result, each group’s view ofbeing in a pivotal position will be anticipated accordingly. In other words,
the voters’ perception about the agenda setter’s attitudes can distort the
distribution of the power. Hence, a power index, even an a piori one, should
not disregard any available information about the type of agenda setter.

3

The European political game

The EU has recently enlarged to 27 countries and possibly more members
will occur. Thus, the number of the possible orderings of states is very high.
Nevertheless, the question of which possible orderings are more likely should
be deepened in order to shed light on possible political outcomes in the
future EU. In our spatial analysis we expect, for example, that orderings inwhich Slovenia (rather Euroenthusiastic) and Denmark (usually Euroskeptic)
occupy close positions will be rather unlikely; whereas, coalitions in which
France and Germany are in close and “central” positions will tend to occur
very often. One aim of our political analysis is to give structural valence tothese subjective perceptions.
We use factor analysis (principal components) to identify the politicalpreferences of the countries, and to provide their locations within the political space. We then use these preferences to measure power with the spatial
pivotal approach. In this section we adopt the Owen-Shapley ordering generating mechanism based on fi = hU; Pi i and on a uniform probability distribution of the issues over Hm 1 . We then compute the Owen-Shapley (spatial
O-S) values and compare them with Shapley-Shubik (S-S) and normalized
Banzhaf indices (NBI).
The data set that we employ to build up the political space comes fromthe Eurobarometer (EC, 2003). The Eurobarometer polls European citizensof their stance toward several policy issues, which range from domestic issues
such a crime and poverty to international issues such as foreign policy anddefense. We use three years of data that was collected for all 27 countries in
the Fall of 2001, the Fall of 2002 and the Spring of 2003. Data reveal thatthe preferences of the citizens are rather stable over time.15 This method ofbuilding the political space is based on the assumption that the way the Min15

We choose not to use more recent surveys, since the Eurobarometer questionnaires
after 2003 have slightly changed, making aggregation of data sets somehow arbitrary.

21

isters represent the preferences of the citizens is the same for each country
and is not a¤ected by di¤erences in the national electoral systems. Moreover, we assume that there is no con‡ict of interests between citizens andrepresentatives: we exclude any agency problem from our analysis at thisstage.
We employ principal component analysis to reduce the number of variables to two latent factors that capture much of the variance in the data.16
We compute the two main principal components for each year, then we calculate our Owen-Shapley spatial values and …nally we take averages to makethe results more robust. Though the preferences of the European countries
are relatively stable over time, the O-S method is rather sensitive to thepositions of the players in the political space; this is why we take averages
over the three years to o¤set this undesirable characteristic of O-S. Appendix A contains the list of issues from the Eurobarometer. In Appendix B, welist the rotated factors for Spring 2003 to illustrate how the political space
has been constructed. Information on the calculation of the Owen-Shapleyspatial values is provided in Appendix C.

3.1

The political space

The EU 15 Factor analysis captures well what is subjectively recognized about the attitudes of the countries toward EU policy issues. The …rst
two principal components account for over 70% of the variation in the data.
Thus we limit our analysis to the …rst two factors, bene…ting also from thegraphical representation of the two-dimensional political space. After applying (varimax) rotation and “scoring”the factors, a clear pattern emerges.
The …rst factor – denoted as the “inter-national stance”– measuresthe degree to which each country would like to have a strong EU on
the international scene (centralized foreign policy, common defence,common …ght against international crime, harmonized rules in justice,environment, etc.).
The second factor – denoted as the “intra-national stance” – represents the desired involvement of the EU in the“internal policies” ofnations, which include areas in which the EU has already acquired
strong responsibilities (agriculture, taxation, welfare, poverty, justice,16

See Lawley and Maxwell (1971) for more information on principal component analysis.

22

Greece2

Portugalintra-national stance

1

Italy
Spain
Luxembou
Ireland0

Austria

BelgiumUK

Germany
FranceFinlandNetherla-1
SwedenDenmark

-2
-2

-1

0
inter-national stance1

2

Figure 5: EU 15 stance toward EU (Spring 2003).

etc.). Member states with high intranational stance desire to relinquish
more responsibility in those policy domains to the EU.17
Figure 5 represents the political space which originates from our factor
analysis (using the Spring 2003 data). It includes the former 15 members
before the 2004 enlargement. It shows, for example, the UK’s Euroscepticism and the Franco-German closeness. As well, we can see that the “oldest”
members (Luxembourg, Netherland, Belgium, France, Germany, and Italy)
are more favorable to further developments of the EU’s presence on the international scene. The small and “older”members (Austria, Finland, Portugal)
are less in favor of a stronger EU in foreign policy and have di¤erentiatedattitudes toward the EU involvement in domestic policy domains.
Each of the plotted factors has a mean of zero and a standard deviationof one; thus we can think of each country’s factor as the number of standard deviations away from the mean stance. For example, Finland’s score
17

The names we give to the two latent factors should not be overemphasized. We
just observed that the way each factor was related to the policy domains allowed us tocharacterize the two dimension of the political space. Actually, the fact that the dimensions
have a meaning is attractive, but not crucial for the power measurement. Moreover, toa deeper analysis our grouping of policies in “inter-national” an “intra-national” is evenquestionnable. Details on this are provided in Appendix C.

23

2Poland1

Estonia

Cy prus
Slov enia

Slov akia

Latv ia
Hungary

Romania

intra-national stance

Lithuani
Czech
Bulgaria

GreeceMalta

0Denmark
IrelandGermany
Portugal

SwedenSpain
Belgium

Luxembou

-1

Italy

Netherla

Austria

FranceUK
Finland

-2
-2

-1

0
inter-national stance1

2

Figure 6: EU27 stance toward the EU (Spring 2003).

(for Spring 2003) for the inter-national stance is -1.6, which is 1.6 standard
deviations away from the average stance.
Observe that for some of the 26 surveyed policy issues the member states
decide with di¤erent procedures. For example, decisions on foreign policy,
defence, and most of taxation or welfare require unanimity. In the factor
analysis we do not treat these issues di¤erently. We implicitly assume thatthe political space derives from a uniform view the citizens look at the European policy making. The way di¤erent citizens look at similar issues ishighly correlated to their nationality. This justi…es our assumption. Other
motives of caution are due to the fact that unfortunately the Eurobarometer
does not include questions on the single market.
The EU 27 In regards to the 27 member countries, the …rst two principal components also account for roughly 70% of the variance in the data.
We again found a similar pattern: that …rst factor is the stance toward the
EU on inter-national issues, while the second factor is the stance toward the
EU on domestic issues. The ideal points are presented in …gure 6.
The newcomers from Eastern Europe tend to have generalized strong attitudes toward EU centralization in domestic policy domains (high intranational stance). A certain degree of diversity is associated to the inter-national
24

stance, probably due to mixed-feelings toward nationalism.
A rapid comparison of …gures 5 and 6 reveals that the “topology”of thecoalitions will change radically in the next few years, after enlargement. The“center” of the EU political space moves upwards, i.e., at least for intranational issues, the average propensity to centralize the decision making at
European level raises consistently. Some old members that could be considered relatively Euroenthusiasts become moderate, if not Euroskeptical after
enlargement.18 We expect that countries that were determinant (pivotal) forsome policy issues and irrelevant for some others will probably be in a very
di¤erent position after the new members will have joined. Below, we provide
quantitative evidence of these changes.

3.2

Measures of power

As discussed above, our objective is to evaluate the relative frequency thateach European member is pivotal within the Council of the Ministers, recognizing that the probabilities of the coalitions are constrained by the preferences of the players. We present three di¤erent voting games to highlight
how the interaction of voting rules and preferences can alter the distributionof power.
1. The …rst is the pre-enlargement situation: 15 members and vote allocation more favorable to the small and middle-size countries. We will referto this scenario as pre-Nice.
2. The second scenario takes into account the enlargement of the EU by
the 12 members and the re-weighting agreed at Nice. This is what we callthe post-Nice scenario. It has come into force as of November 1st 2004.19
18

Observe that here we are talking about the relative attitudes of the countries toward
the EU. The coordinates of the graphs change as a result of the factoral analysis. Intuitively, the origin of the graph somehow re‡ects the barycentre of the political space. The
position changes after the enlargement do not imply that the citizens change their mind
toward the EU because of the enlargement. It rather means that it is the centre of thespace to have shifted after enlargement.
Since our analysis is intended to generate all the possible orgerings of the players, thenany possible issue of the political space (on Hm 1 ) is considered. We do not have one
speci…c point of the space that represents the status quo.
19
Changes in votes allocation from pre-Nice situation to post-Nice can be seen by comparing column two of tables 1 and 2 below. In the Treaty of Nice the quali…ed majority
threshold was increased from 62 out of 87 to 250 out of 345.
The Treaty of Nice prescribes also that bills are passed by the Council with two quotas: a25

3. The third scenario is represented by the New European Treaties (NET):in which the old weighted voting system is substituted by a double majority
based on both population and number of countries.20
Pre-Nice –15 Members Table 1 shows the results for the Pre-Nicescenario. It reports the standard Shapley-Shubik index (S-S), the normalized
Banzhaf index (NBI) and the spatial index in the Owen-Shapley perspective(O-S spatial). If we look at the O-S spatial values we see that the number ofvotes is no longer a good predictor of power. Shifting from standard S-S andNBI indices to the spatial value yields a concentration of power. This is due
to zero-probability assigned to a large number of ideologically non-consistentcoalitions.
Since the quali…ed majority threshold is roughly 70%, we expect thatthose countries who tend not to be “highly enthusiastic or completely reluctant” in participating coalitions on random political issues will have morechances to be pivotal. In other words, the most powerful are the ones who
say “yes” not too early nor too late; they cast their vote broadly after theother more enthusiastic countries have already cast almost the two-third of
the votes, and before the more reluctant ones have cast theirs. For such countries the spatial O-S power index tends to be higher than the standard S-S
or NBI indices. Conversely, the power measure decreases for countries who
have extreme preferences (very strong or very little enthusiasm for Europe).
Austria, Belgium, Spain, Germany and Portugal gain substantial powerfrom occupying favorable positions in the ideological space.21 The traditional view of a strong Central Europe led by the Franco-German axis andsupported by Belgium and the Netherlands is con…rmed by the spatial apmajority of states, and at least 62% of the total population of the Union. These additional
conditions produce negligible e¤ects on winning coalitions. Then, we disregard in our
analysis these and other complex aspects of the EU decision making, such as amendments,
abstentions, etc.
20
The double majority sets two conditions for the passage of a bill: (a) more than 55%
of member states vote ”yes”; and (b) the population of the countries who have voted ”yes”
represents at least 65% of the total population of the EU.
For a limited number of issues, unanimity has been kept. Moreover, in the NET a sortof safegard clause has been introduced. For simplicity we focus here on double majority.
21
The presence of Portugal in the group of countries that accupy favorable positions
should not sound too surpring if one considers that Portugal has succeeded to enter Eurozone, to get large portions of cohesion funds and to yield the current presidency of theCommission.

26

proach. Little power rests upon the Northern countries. Denmark, is neverpivotal in any ideologically consistent coalition, whereas Finland and UK losea lot of power from being “too skeptical” and “too close” each other. Also
Greece and Italy lose power probably for the opposite reason of being “tooenthusiastic.”
In particular, for a given player, being close to another player can alternatively have two di¤erent consequences: (a) sharing the power with that
player that comes from occupying a certain portion of the space; (b) transferring a substantial part of her own power to the other player, who often
succeeds in being pivotal just before her or vice versa.
Country
Germany
Portugal
Spain
FranceAustria
Belgium
Netherlands
IrelandUK
SwedenGreeceItaly
Finland
Luxembourg
Denmark

Votes
10
5
8
10
4
5
5
3
10
4
5
10
3
2
3

S-S
0.117
0.055
0.095
0.117
0.045
0.055
0.055
0.035
0.117
0.045
0.055
0.117
0.035
0.021
0.035

NBI
0.112
0.059
0.092
0.112
0.048
0.059
0.059
0.036
0.112
0.048
0.059
0.112
0.036
0.023
0.036

O-S Spatial
0.142
0.141
0.118
0.114
0.092
0.083
0.076
0.059
0.048
0.047
0.045
0.025
0.009
0.003
0.000

Table 1: Power Values for Pre-Nice EU 15.

Post-Nice –27 Members The enlargement is taking place under therules of Nice. Table 2 shows the big changes in the distribution of power
after the enlargement to 27 members. Again the standard indices are linearlycorrelated to the votes: 69% of the power measured by the Shapley-Shubik
index will be allocated to the current 15 members, whereas the six founding
states will count for 31% of the power. However, once we shift to the spatial
approach, this broad idea of change becomes much more radical. Many
countries will lose a lot of their power as a consequence of their “unlucky
27

Country
Czech Rep
FranceGermany
Spain
GreeceBulgaria
Netherlands
Lithuania
Italy
PolandBelgium
Romania
Portugal
Slovakia
Hungary
Ireland
LatviaDenmark
SwedenUK
CyprusAustria
Finland
Slovenia
Luxembourg
Malta
Estonia

Votes
12
29
29
27
12
10
13
7
29
27
12
14
12
7
12
7
4
7
10
29
4
10
7
4
4
3
4

S-S
0.034
0.087
0.087
0.080
0.034
0.028
0.037
0.020
0.087
0.080
0.034
0.040
0.034
0.020
0.034
0.020
0.011
0.020
0.028
0.087
0.011
0.028
0.020
0.011
0.011
0.008
0.011

NBI
0.037
0.078
0.078
0.074
0.037
0.031
0.040
0.022
0.078
0.074
0.037
0.043
0.037
0.022
0.037
0.022
0.013
0.022
0.031
0.078
0.013
0.031
0.022
0.013
0.013
0.009
0.013

S-O Spatial
0.132
0.101
0.091
0.089
0.063
0.062
0.054
0.048
0.048
0.035
0.033
0.030
0.024
0.024
0.023
0.021
0.021
0.020
0.017
0.016
0.014
0.011
0.010
0.006
0.004
0.003
0.000

Table 2: Power Values for Post-Nice EU 27.
positions” in the political space (see …gure 6) and the unexpected result ispossibly the fact that these “losers” are more frequently current membersof the EU. UK’s spatial power decreases by more than 80% with respect to
S-S index. Italy succeeds in being pivotal only in 4.8% of the ideologicallyconsistent coalitions. Austrian power falls by 61%. In general those countries
with extreme preferences tend to lose in the political game: saying “yes”tooearly or too late is not a good idea when a 72% quali…ed majority has toform.
The Franco-German axis is weaker: by requiring a higher quali…ed majority, Nice system subtracts the two big “moderate” countries a portion of
the usual power. Some of the traditional allies, such as Belgium and Luxembourg, are less powerful. In the future, the axis will probably need a higher
28

support from Spain, which emerges from Nice as a very strong player, due
to a very good position in the political arena.
Di¤erently from the old members, the majority of new entrants could
pro…t from favorable positions in the ideological space. The twelve newcomers collect a total 39.8% of the spatial power, despite the 30.8% quota ofstandard S-S power and 31.3% of the votes. Too much enthusiasm penalizes
Poland. Moreover, accessing countries are very close to each other in thepolitical area and Czech Republic is in a very good position at the center ofthis Eastern-bloc. Thus we could predict an unexpected prominent role for
Czechs, whose 12 votes are enough to swing a large number of coalitions.
New Euroepan Treaties The double majority system included in the
new Treaties is favorable to the four most populous countries, in terms ofstandard S-S and NBI power. Note from table 3 that S-S and NBI are
substantially di¤erent: S-S is much more concentrated in the hands of thelargest six countries.22 This is due to a technical di¤erence between S-S
and NBI that becomes relevant in the case of double majority.23 A certain
degree of caution is then necessary when choosing one or the other index. For
example, in the debate on the winners and losers under NET the common
perception has been that not only the four largest countries, but also thesmallest six would have gained from double majority with respect to Nice
(Baldwin and Widgrén, 2004). Actually, this is true only if NBI evaluations
are used. S-S suggests that only the four largest members win, and all theothers lose.
22

In Table 3, the S-S and NBI values were calculated via the Monte Carlo method dueto the large computational requirements needed to solve them analytically.
23
As stated in section2, both S-S and NBI are symmetric indices. Di¤erences between
them can arise from di¤erent values in the probabilities of the swung coalitions. In the
Banzhaf’s perspective these probabilities are always the same and given by (3), whereas
in Shapley and Shubik they are given by (2).
NET prescribes that at least 15 members must be in the winning coalition. Intuitively,
the smallest countries tend to be pivotal thanks to this provision. Thus they are more
likely to swing coalitions that are “already”composed by 14 members. For such coalitions(3) is larger than (2). Thus, small countries’power under NET tend to be emphasized by
NBI evaluations.
On the contrary, larger countries also swing coalitions that are composed by a larger
number of players and do not reach the population threshold. The probability assigned
to such coalitions by S-S can be much larger than the one assigned by NBI. This explains
why the S-S index of largest countries is much higher.

29

Country
Germany
Spain
FranceBulgaria
Italy
Netherlands
Hungary
PolandLithuania
SwedenUK
GreeceMalta
Austria
Romania
IrelandFinland
Slovakia
Belgium
CyprusDenmark
Czech
PortugalSlovenia
LatviaEstonia
Luxembourg

Votes
82,19339,49059,521
8,170
57,844
15,983
10,02438,6493,696
8,883
59,832
10,565390
8,121
22,443
3,820
5,181
5,401
10,262
671
5,349
10,272
10,023
1,989
2,417
1,436
441

S-S
0.163
0.073
0.110
0.020
0.107
0.033
0.022
0.070
0.012
0.020
0.111
0.023
0.007
0.020
0.042
0.012
0.015
0.015
0.022
0.007
0.015
0.023
0.023
0.010
0.010
0.009
0.007

NBI
0.119
0.061
0.087
0.025
0.085
0.035
0.027
0.060
0.020
0.026
0.088
0.029
0.016
0.025
0.043
0.020
0.022
0.022
0.027
0.016
0.022
0.027
0.028
0.018
0.018
0.017
0.016

O-S Spatial
0.194
0.174
0.119
0.063
0.045
0.042
0.040
0.039
0.037
0.027
0.027
0.027
0.024
0.023
0.020
0.016
0.015
0.013
0.013
0.012
0.009
0.008
0.006
0.004
0.001
0.001
0.000

Table 3: Power Values for the New European Treaties, EU 27.
O-S power is more concentrated than the standard symmetric indices.
Moreover the concentration of the spatial power is higher under NET thanunder Nice. Germany emerges by far as the strongest member country. The
Franco-German axis is very powerful: 31.3% of the winning coalitions are
swung by one of the two countries. Actually, the NET voting system turns
in favor of moderate/slightly enthusiastic voters. Spain is the second big
player in the Council. During the intergovernmental negotiations, Spain
fought very hard for an increase of the population threshold, aiming at getting a strong blocking power. Not only does the O-S method capture theincreased blocking ability, but it also re‡ects the fact that a 65% majority
favors the moderate enthusiasm of Spain, which succeeds more often in cast-30

ing the “65th ” vote after the “64” votes of more enthusiast members. With
that threshold, too much enthusiasm can work against a country, such as
Italy, that, despite its population, loses almost two-thirds of its standard
S-S power. The same rationale explains the loss by Poland or the Czech
Republic. The new system is unfavorable to extremely skeptical countries:
UK and Denmark lose most, relative to their standard S-S power. However,except for Denmark, the new Treaties are less penalizing to big Euroskeptics,
compared to Nice.
The Eastern bloc is less powerful than under Nice. The double majority
empathizes the role of population in power apportionment, while the newcomers are small or middle-sized members. The ten Eastern countries represent
22% of the EU population and collect 22.6% of spatial power. For them, inaggregate, the political locations turn to be unfavorable (-0.7% with respect
to the standard S-S power). NET prevents huge shifts of power toward Eastern Europe that otherwise occur with Nice. The double majority distributes
the power more uniformly among the newcomers. The Czech Republic loses
its leadership.
Spatial values also explain why among the middle sized countries the
feelings toward the new voting system have been mixed. Despite that for
all of them NBI and S-S indices fall between 25-30%, some of them, such as
Netherlands or Austria, gain in ideological values compared to Nice.
Spain and Poland’s strong reluctance toward the new voting mechanism
is inspired by losses both in terms of standard S-S and of spatial power:
Nice favors Spanish and Polish pro-EU stance. The initial proposal of a
60% population threshold would have produced a signi…cant loss for thetwo countries. The agreement for the new voting system has been reachedthanks to a raising by 5 points in the two thresholds. If evaluated in termsof standard symmetric power, this adjustment does not explain why Spain
and Poland have accepted to endorse the new system, since both their S-S
and NBI values are lower under NET than Nice. The spatial power explains
much more: the raising of the thresholds allows Spain to be the second mostpowerful country in Europe and also improves the Polish position.
The new Treaties restrict the areas (such as taxation, welfare, foreign policy, budget, justice) in which the unanimity of member countries is required.
UK, Denmark or Finland, for example, are reluctant to abandon unanimity
in those areas. The spatial power approach provides arguments for explaining those countries’positions. In fact, if policy positions were disregarded,
unanimity would imply equal power distribution. Thus we would not be able31

to understand the strong opposition to quali…ed majority by selected countries. Once we adopt the spatial approach, we see that under unanimity thepower shifts in favor of either Euroskeptics or Euroenthusiasts. In particular,
unanimity allows countries that are against centralization in sensitive policy
domains to keep the highest amount of power. In the discussions regarding
taxation and EU budget, for example, UK has always played a crucial role.
24

4

The agenda setter and power

In section 2.4 we have argued that if there is an agenda setter, its most preferred bills should be more likely than others, putting the pivots for more
likely bills in better positions. Within the EU legislative system the Commission has the monopoly of the proposals for a large portion of issues, playing
the role of agenda setter. This can result in a substantial distortion of thepower distribution within the Council. 25
In the two-dimensional perspective developed in section 2.1.1:= H1 ,
the ordering generating function is fi ( ) = h ; Pi i, and the issue U is de…nedby 2 [0; 2 ). So far, in our application to the EU Council we have adopted
also the Owen-Shapley’s hypothesis of equal relevance of policy issues. In
this section we remove this hypothesis, and we introduce examples of nonuniform probability distribution over the set [0; 2 ), and estimate the impact
on power distribution.
Since hereafter the issues can have di¤erent probabilities, we are interested in “where”each country is pivotal: i.e., in which portions of the policy
issue space. In …gures 7 to 9 for Spring 2003 data and the three scenarios,
respectively, we have represented the sectors in the H1 circle in which eachcountry is pivotal. As soon as rotates from 0 to 2 , the pivotal role shifts
from one country to another one, accordingly.
24

Here we do not include spatial power evaluations for a unanimity game in the EU. A
table is available upon request. Also evaluations for the double threshold initially proposedby the Convention are available.
25
The agenda setting power of the Commission must not be overestimated: the EU
Treaty obliges the Commission to propose legislation when prompted by the Council orby the Parliament. The Council can amend the proposal by unanimity under the consultation procedure and by quali…ed majority (simple majority for the Parliament) underCodecision.

32

EU 15 Pre-Nice Pivotal Countries

UK

FranceAustria

Germany

Netherlands
Ireland
Portugal

Italy
FranceSwedenSwedenSpain

IrelandUK

Netherlands

GreeceFinland

GreeceLuxembourg
Spain
Belgium

Spain

UK
Austria

Portugal

Italy

Figure 7: Pre-Nice EU15. Sectors in which countries are pivotal (Spring 2003).

For example, in the pre-Nice context (see …gure 7) if the Commission
proposed a bill whose political contents are^

pro-international and pro-intranational ( = =4), the most likely
minimal winning coalition (ordered by level of enthusiasm) would be:{Greece, Italy, Spain, Belgium, Portugal, Germany, Luxembourg, Ireland, France, Netherlands}, and Netherlands is the pivotal country.
^

pro-intranational bill ( = =2), then a very likely outcome should be
the winning coalition {Greece, Portugal, Italy, Spain, Luxembourg, Ireland, Austria, Belgium, UK, Germany}, with Germany as the pivotal
country.
We know from section 2.4 that di¤erent expectations about the agenda
setter’s type will result in di¤erent probability distributions over the set of
the issues. In other words, the probability distribution over the set [0; 2 )of all the possible bills can be anticipated by looking at the Commission’s
preferences.
We assume a simple linear probability density function
8 h
9
i
2
b
b
>
>
+
(
)
=
if
0
<
>
>
>
>
<
=
2
b
b
b
=
if
< +
(12)
p( ) = (1= )
>
>
i
> h
>
>
>
:
( + b) = 2 if + b<2 ;33

EU 27 Nice Proposal Pivotal
Countries
Slova Fin
Lith

Pol

UK
Cze Hun

Cze

Ger

Neth
Est

Aus

Ger

Swe

Fra

Slove

Den
Neth

Den

Swe
Lux

Ita

Rom
Cze
Gre

Ita

Hun
Ita
UK

Gre
Spa

Spa

Bel
Pol

Slova Gre

Fra

Bel

Figure 8: Post-Nice EU27. Sectors in which countries are pivotal (Spring 2003).

^

is what the players expect as the most preferred issue by the Commission. Function (12) says that the probability density of a proposal will
increase linearly as approaches b. Moreover the probability density falls tozero with regards to the opposing issue b + .
We further assume that the Commission prefers to promote integration
with respect to the status quo. Thus the countries expect b to lie in 0; 2 .
We further suppose that the following three distributions re‡ect alternative
priors of the states about the Commission’s decision making.
^For a purely pro-international modal issue: = 0; we show the graphof this version of equation (12) in …gure 10(a) below.
Alternatively, if the Commission preferred issues that are pro-international
and pro-intranational, we would expect a density function with modein^

= =4; whose graph is represented in …gure 10(b).

If the most preferred issue is purely pro-intranational, thus
and the probability distribution will be as in 10(c).

^

=

=2

Of course, these are just examples of our ad hoc probability distribution. Nonetheless they can capture alternative behavior and preferences of
the agenda setter and get a reliable idea on how the power shifts from someplayers to others as a result of the action of the agenda setter. In more34

EU 27 New CT Plan Pivotal Countries
Slova

Cze
Ire

UK

Pol

Hun
Bul

Por

Slova

Lith
Fin

Ger
Cyp

Sw e

Fra

Rom

Aus

Den

Bel

Gre

Sw e

Aus
Bul

Sw e

Ita
Spa
Ita

Mal
Bul
UK
Spa
Spa

Cze
Fra
Pol

Gre

Figure 9: New CT Proposal EU27. Sectors in which countries are pivotal (Spring2003).
p (θ )p (θ )p (θ )

1

1

1

π

π

π

θˆ= 0

π
(a) pro - iternational modal issue

2π

θ

π
θˆ=
4

2π
π
4
(b) pro - iternational and pro - intranational modal issue
5

θ

2π
π
3
θˆ=
π
2
2
(c) pro - intranational modal issue

Figure 10: Pro-EU Agenda Setter with Di¤erent Modal Issues

sophisticated cases we could have density functions with di¤erent or multiple modal values, or non-linear relations between issues and probability. In
general, we could wonder how the voter can infer the distribution from thelimited information about the agenda setter’s type; but this is beyond thetasks of this paper.26
26

Note also that function (12) can re‡ect our assumptions only if

a problem since we have additionally supposed that35^

2 [0; =2].

^

2 [0; ]. This is notθPre-Nice 15 Members The spatial game predicts that with pro-European
proposals by the Commission and high majority thresholds the pivots are
more frequently on the Euroskeptic side of the political space. This can explain why in the pre-Nice scenario (quali…ed majority close to 70%, table 4)Austria, UK, Finland and Sweden gain additional value with respect to theOwen-Shapley spatial value. For the same reason (saying “yes” too early)
Country
Austria
Belgium
Denmark
FinlandFranceGermany
GreeceIreland
Italy
Luxembourg
Netherlands
Portugal
Spain
Sweden
UK

^=0
0.136
0.048
0.000
0.014
0.052
0.089
0.055
0.085
0.019
0.005
0.075
0.192
0.096
0.074
0.061

^= =4
0.129
0.035
0.000
0.010
0.106
0.145
0.040
0.069
0.016
0.003
0.104
0.154
0.064
0.068
0.058

^= =2
0.091
0.074
0.000
0.005
0.162
0.195
0.035
0.043
0.015
0.002
0.106
0.118
0.043
0.058
0.055

Table 4: Ideological Values with Di¤erent Types of Agenda Setter, pre-Nice EU
15

Italy, Belgium and Greece lose a quota of their power. In general, the expectations of a distorted pro-EU agenda setter, as depicted by (12), can resultin 9.8% redistribution of the total O-S power.27
Post-Nice 27 Members The EU 27 scenario resulting from Nice voting
rules is more complex. We have already remarked that the accession ofmany Euroenthusiasts will shift the “center” of the political space toward
Euroenthusiasm, making the current members relatively more Euroskeptic.
Thus, the countries who gain from pro-EU proposals are more frequently
27

Intuitively, the amount of power redistribution is negatively related to the variance ofthe probability distribution of the issues: when the agenda setters’proposals are easy to
predict, the probability of playing a pivotal role is concentrated.

36

current members (Germany, Ireland, UK, Austria, Finland, Italy). This
mitigates the risk of large concentration of O-S power on the so-called Eastern
bloc.
Country
Austria
Belgium
Bulgaria
Cyprus
Czech
Denmark
Estonia
FinlandFranceGermany
GreeceHungary
IrelandItaly
LatviaLithuania
Luxembourg
Malta
Netherlands
PolandPortugal
Romania
Slovakia
Slovenia
Spain
Sweden
UK

^=0
0.007
0.020
0.053
0.007
0.029
0.005
0.016
0.017
0.097
0.277
0.012
0.006
0.023
0.086
0.016
0.029
0.000
0.000
0.041
0.001
0.069
0.051
0.022
0.007
0.037
0.002
0.069

^= =4
0.015
0.012
0.065
0.013
0.034
0.019
0.028
0.018
0.069
0.279
0.012
0.008
0.046
0.068
0.021
0.035
0.000
0.002
0.035
0.002
0.063
0.037
0.014
0.012
0.035
0.005
0.056

^= =2
0.023
0.015
0.083
0.019
0.046
0.034
0.041
0.019
0.047
0.214
0.017
0.008
0.070
0.062
0.029
0.048
0.000
0.003
0.021
0.002
0.044
0.023
0.012
0.016
0.055
0.008
0.043

Table 5: Ideological Values with Di¤erent Types of Agenda Setter, Post-Nice EU
27

Due to the large number of members and to the relatively high majority
threshold agreed at Nice (around 72%) the pivotal role can rapidly pass fromone country to another one as the proposal changes. This makes more than
34% of the allocation of spatial power to depend upon the agenda setter’s
preferences, as given by (12). This con…rms what we would intuitively expect:raising the majority threshold reduces the chance to form winning coalitions,causing the identity of the pivot to be highly responsive to the Commission’s
37

proposal. The results are presented in table 5.
New Euroepan Treaties The NET reduces the quali…ed majority to 65%,
provided at least 15 members have voted “yes.”This lowering has at least twoe¤ects: it makes the pivotal role of moderate countries more likely; it reduces
the sensitivity of power distribution to the agenda setter’s preferences. When
the Commission turns from indi¤erence to pro-EU attitude “only”16.9% of
total O-S power is reallocated.
Country
Austria
Belgium
Bulgaria
Cyprus
Czech
Denmark
Estonia
FinlandFranceGermany
GreeceHungary
IrelandItaly
LatviaLithuania
Luxembourg
Malta
Netherlands
PolandPortugal
Romania
Slovakia
Slovenia
Spain
Sweden
UK

^=0
0.029
0.025
0.062
0.016
0.010
0.012
0.001
0.019
0.099
0.280
0.027
0.059
0.018
0.035
0.001
0.052
0.000
0.023
0.013
0.034
0.006
0.017
0.017
0.006
0.099
0.017
0.025

^ = =4
0.027
0.005
0.071
0.018
0.011
0.013
0.001
0.021
0.092
0.249
0.030
0.068
0.021
0.023
0.002
0.059
0.000
0.026
0.015
0.038
0.006
0.020
0.019
0.007
0.113
0.016
0.027

^ = =2
0.030
0.003
0.079
0.021
0.013
0.015
0.001
0.025
0.082
0.165
0.036
0.077
0.024
0.026
0.002
0.069
0.000
0.031
0.017
0.043
0.008
0.023
0.021
0.008
0.131
0.019
0.031

Table 6: Ideological Values with Di¤erent Types of Agenda Setter, NET, EU 27
Still Germany, Spain and France are very powerful, however Bulgaria andHungary are reinforced. The distortion produced by the agenda setter results
more frequently in favor of middle sized Eastern countries.
38

5

Conclusion

This paper has presented power indices based on the preferences of the players
and the agenda setter in a coalition form game. We have applied this index
to the case of the European Union, which is facing a large expansion.
Measuring power using simple Shapley-Shubik and Banzhaf indices do
not capture how political preferences a¤ect power. In short, when looking at
possible coalition formation, countries who are relatively “pro” a particularissue will be more likely to vote “yes”before those countries who are “con”.
Majorities formed by pro-countries are more likely than majorities formedby con-countries. Being able to swing a large number of coalitions puts thevoter in a powerful position only if those coalitions are also likely to occur.
More likely coalitions are those ones that include similarly minded voters.
Thus the power of countries will be determined not only by the number ofvotes they have and the number of votes needed for a majority, but also by
the attitudes that the countries hold. In cases where unanimity is needed,
for example, the most “con”countries hold the most sway. In games, wherea two-thirds majority is needed countries who are moderately “con” thenbecome more powerful since the likelihood of them joining the coalition “tooearly”is small.
The attitudes that create political coalitions depend also on the content
of the issue to vote on. If the issue is proposed by an agenda setter, theagenda setter’s preferences can distort the likelihood of the coalitions andultimately the distribution of power.
Using principal component analysis we are able to extract countries’attitudes toward the EU. These attitudes are then used to create what we call the
spatial Owen-Shapley index. Our results show, for example, that countries
with the greatest number of votes (and hence highest simple Shapley-Shubik
indices) do not necessarily have the greatest power after considering theirpreferences.
The spatial approach captures the current leadership of the Franco-German
axis and the political weakness of Northern Euroskeptics and Mediterranean
Euroenthusiasts. After enlargement the “positions”in the political space will
become even more relevant as a source of power, and, if the Nice rules are notchanged, the Western members will frequently occupy unfavorable positions.
The closeness of the new members will result in a strong Eastern politicalbloc.
The double majority included in the NET restores concentration of power39

in favor of big and politically moderate members; the Eastern bloc is less powerful. Lowering the majority threshold causes those countries with extreme
preferences to su¤er more.
In our spatial political games we have modeled the agenda setter’s preferences through an ad hoc distribution which assigns higher probability topro-European issues. We …nd that with higher majority thresholds the EU
Commission’s preferences can have a larger impact on the power distribution, i.e., a greater share of power can change hands. Under Nice the risk of
distortions of the European political game caused by the Commission’s proposals is high. This risk grows when the number of members increases andwhen there is little uncertainty about the Commission’s preferences. Within
the debate on the institutional reforms of the EU, our …ndings show the needfor a serious re‡ection about the strong concentration of the legislative prerogatives within the Council and about the recognition of a greater politicalrole to the Commission.
In summary, the measurement of power based on preferences leads to interesting and sometimes unexpected results, which can be used as a measureof the prospect of participating in political games. Nonetheless, a certain
degree of caution is necessary. First, power index analysis, even of the spatial kind, is reasonable only from a constitutional viewpoint, when the policy
issues to vote on are unknown and the random future contingencies tend too¤set each other in a “large number” perspective. Second, for low dimensional policy spaces, the spatial power indices seem to be quite responsive
to slight di¤erences in preference measurement. Third, the assumption ofexogeneity of preferences can be challenged if, for example, the voters canstrategically commit to political positions aiming at maximizing their power.
Finally, since we use the preferences of the citizens to evaluate the political location of their elected representatives, a principal-agent problem could
arise. For future work, we envisage the possibility of extending our modelwith non-cooperative stages which take these aspects into account, as well
as a more sophisticated role for the agenda setter.

40

6

Appendix A: Eurobarometer survey questions

The Eurobarometer survey covers the population of the EU member states.
The basic sample design consists of a number of sampling points that areproportional to the population size and density. In each country almost 1,000
face-to-face interviews are carried out. We use three Eurobarometer surveys(Fall 2001, Fall 2002 and Spring 2003). The part of the interview which isrelevant for our analysis is the one which concerns the opinions of the people
whether to centralize some policy domains, which is based on the following
question: “For each of the following area, do you think that decisions should
be made by the (NATIONALITY) government, or made jointly within theEuropean Union?”(EC 2003)28

7

Appendix B: Data analysis

7.1

Principal components

Factor models have been used in economic analysis and forecasting to reducethe dimensionality of large data sets. We apply the standard techniques offactor analysis/principal components using Stata 8.0. In our analysis we have
25 variables, xi; i = 1; :::; p(= 25); with a considerable degree of (positive)correlation among these variables. This leads to the use of factor analysis tocapture underlying, latent variables that can account for this high degree ofcorrelation among these variables. The aim of factor analysis is to account forthe covariances of the observed variables in terms of a much small numberof hypothetical variables, fr ; r = 1:; ; ; k; k << 25; such that the partial
correlation coe¢ cients between the original variables after eliminating thee¤ect of fr0 s are close zero. Each variable, xi is modeled as a linear functionof k common “factors”or latent variables:
k
X
xi =
(13)
ir fr + i ; i = 1; :::; p;r=1

where fr 2 f is the rth common factor, k is the number of factors being speci…ed, i is a residual source of variation a¤ecting only xi ; and the coe¢ cients28

We exluded the Terrorism variable since we did not have a response for this variable
for all 27 countries.

41

Issues
DefenseProtection of the environment
Currency
Humanitarian aid
Health and social welfare
Basic rules for broadcasting and press
Fight against poverty/social exclusion
The Fight Against Unemployment
Agriculture and …shing policy
The support of regions which are experiencing economic di¢ culties
Education
Scienti…c and technological researchInformation about the EU, its policies and institutions and bodies
Foreign policy towards countries outside the EU
Cultural policy
Immigration policy
Rules for political asylum
The …ght against organized crime
PoliceJustice
Accepting refugees
Juvenile crime prevention
Urban crime prevention
The …ght against drugs
The …ght against the trade in, and exploitation of, human beings
The …ght against international terrorism

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

Table 7: Eurobarometer survey questions.
ir 2

are called the factor loadings of xi on fr :The method of principal components minimizes
V (k) = min
;f

p
X
i=1

"

xi

k
X
r=1

ir fr

#!2

; i = 1; :::; p

The estimated factors, f^r ; are the eigenvectors corresponding to the k00largest eigenvalues of the matrix xx0 ; and ^ = ^f^f x^
f are the corresponding
factor loadings. If we denote r , r = 1; :::; k the eigenvalue for the rth factor,
then each P
factor explains r =p proportion of the total variance in the dataset (since pr=1 r = p). We have chosen k = 2 in order to measure power
42

values for each country in two dimensions; however, the two factors explain
over 70% of the variance in the data sets.
Furthermore in cases where k > 1; there are an in…nite number of choices
for : Thus to elicit clearer patterns in the data, we use the varimax rotation procedure, whereby we “rotate”the factors in such a way that the newloadings tend to be either relatively large or small in absolute size comparedwith the original ones. Since each factor is a vector of correlation coe¢ cients,
the most interpretable factor is one based upon correlation coe¢ cients which
are either close to one in absolute value or close to zero in absolute value(Lawley and Maxwell, 1971).
We can then inspect the rotated factor loadings to see which of the originalvariables are most highly correlated with each of the factors. Inspectionof the loadings shows that, in general, that the …rst factor is most highly
correlated with international issues, while the second factor is associated withdomestic/internal issues. Next the factors are then “scored”, which assigns
weights for the contribution of each variable to the factor. The mean of eachthe scored factor is zero and the standard deviation is 1: Each scored factor
for each country is plotted in …gures 5 and 6 above (Lawley and Maxwell,1971).

7.2

PCAs Spring 2003

Given our data set, principal component analysis gives the following tables,
where table 8 reports the results for EU 15 for Spring 2003, and table 9 reports
results for EU 27, Spring 2003. Each value for each variable is a correlationcoe¢ cient that measures by how much the variable is correlated with therespective factor. Looking at which variables are more highly correlatedwith each factor allows us to give an interpretation to each factor (i.e., theintra-national and inter-national factors).

8

Appendix C: Owen-Shapley spatial value
calculation

This section gives more description of how we calculate the Owen-Shapleyspatial value. For a given year, …rst we calculate the scored factors for EU
15 and EU 27, as described above. The scored factors are plotted in …gure43

Variable
defenceenvironment
currency
humanitarian aid
health and social welfare
media
poverty and social exclusion
unemployment
agriculture
regional aid
education
research
information
foreign policy
cultural policy
immigration
political asylumorganized crime
policejustice
accepting refugees
juvenile crimeurbancrime
drugs
exploitation of human beings

Factor 1
0.67
0.82
0.56
0.72
0.28
0.08
0.47
0.40
0.83
0.13
0.38
0.61
0.71
0.73
0.28
0.77
0.74
0.76
0.41
0.47
0.74
0.10
-0.15
0.72
0.94

Factor 2
0.41
0.36
0.38
0.44
0.89
0.64
0.61
0.74
0.14
0.32
0.80
0.65
0.25
0.47
0.69
0.59
0.60
-0.02
0.85
0.81
0.54
0.91
0.84
0.26
-0.03

Table 8: Rotated factors for EU 15 (Spring 2003).
5 and 6. For each country, we then calculate its relative “pro-con”value for10; 000 points from zero to 2 using the formula:yi = f1i cos + f2i sin ;2 [0; 2 ] ; i = 1; ::; nwhere yi is a country i0 s relative sentiment (ranking) on issue , and n is
the number of countries (either 15 or 27) and f1i and f2i are the two factors
for each country. We use 10; 000 values of in the [0; 2 ] range in order to
get many rankings, which will insure us that we catch all of the “switching
points,”i.e., the points along [0; 2 ] where the relative rankings of countries’
preferences change.
For each ; we rank yi ; i = 1; ::n; from highest to lowest (i.e., most pro
to most con), and record at what values of the rankings change. Then
we calculate the pivotal voters–given the particular rules of the game–and
44

Variable
defenceenvironment
currency
humanitarian aid
health and social welfare
media
poverty and social exclusion
unemployment
agriculture
regional aid
education
research
information
foreign policy
cultural policy
immigration
political asylumorganized crime
policejustice
accepting refugees
juvenile crimeurbancrime
drugs
exploitation of human beings

Factor 1
0.45
0.54
0.66
0.66
0.34
0.62
0.33
0.22
0.49
0.07
0.22
0.48
0.46
0.86
0.82
0.83
0.88
0.24
0.42
0.47
0.90
0.33
0.23
0.16
0.30

Factor 2
0.65
0.61
0.23
0.55
0.88
0.34
0.87
0.94
0.65
0.80
0.90
0.76
0.70
0.23
0.09
0.43
0.32
0.79
0.83
0.79
0.28
0.83
0.73
0.91
0.74

Table 9: Rotated factors EU 27 (Spring 2003).
record the interval length for which each country is pivotal. To calculate
the Owen-Shapley spatial value we add the length of the pivotal intervals foreach country. Lastly we normalize the results so the sum of power values isequal 1.
To calculate the power-values using a non-uniform distribution, we …rst
take the length of the intervals along [0; 2 ] for which each country is pivotal.
For each interval we take the integral using equation (12). We then sum theintegrals for each country and normalize them to get the power values.
We repeat each of the steps for the three years and take average of theOwen-Shapley spatial values. The calculations were done in Mathematica
3.0. The code is available upon request.

45

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