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Measuring voting power:
The paradox of new members vs. the null
player axiom∗
L´aszl´o ´
A. K´oczy†
Abstract
Qualified majority voting is used when decisions are made by voters
of different sizes. In such situations the voters’ influence on decision
making is far from obvious; power measures are used for an indication
of the decision making ability. Several power measures have been pro-
posed and characterised by simple axioms to help the choice between
them. Unfortunately the power measures also feature a number of
so-called paradoxes of voting power. In this paper we show that the
Paradox of New Members follows from the Null Player Axiom. As
a corollary of this result it follows that there does not exist a power
measure that satisfies the axiom, while not exhibiting the Paradox.
Keywords and phrases: a priori voting power, paradox of new
members, null player axiom.
JEL classification: C71, D72.
1 Introduction
Power measures or, more appropriately: a priory measures of voting power
give an indication of a voter’s ability to change decisions. Voting bodies are
everywhere from faculty councils to the UN Security Council, from national
parliaments to shareholders’ meetings. The most discussed case is, however,
∗The author thanks the funding of the OTKA (Hungarian Fund for Scientific Research)
for the project “The Strong the Weak and the Cunning: Power and Strategy in Voting
Games” (NF-72610) and of the European Commission under a Marie Curie Reintegration
Grant (PERG-GA-2008-230879).
†Keleti Faculty of Economics, Budapest Tech, Budapest, Tavaszmez˝o u. 15–17.,
koczy.laszlo@kgk.bmf.hu and Department of Economics, Maastricht University
1
the EU Council of Ministers that uses qualified majority voting, rather com-
plicated procedures to determine whether a subset of EU member countries
is able to pass a decision or not. A key element of these procedures is the
voting weight that is determined by EU treaties for all countries. How should
these weights be determined is a topic of ongoing discussions partly because
the implications of a particular set of weights is not totally clear.
To illustrate the difficulties consider one of the simplest possible voting
bodies (such as a shareholders’ meeting) with only three decision makers
respectively having 49, 49, and 2 votes (such as shares). A decision can
be passed with plain majority, that is, if a coalition supporting the motion
has at least 51 votes in total. It is easy to verify that despite the dramatic
differences in weight (size) the three voters have equal influence on decision
making as any pair of voters has majority, plus of course the grand coalition
including all voters has majority, too. The problem is general: when voters
have weights, these weights are difficult to translate into shares of voting
power.
To complicate the arguments consider now a very similar problem, where
three parties have the above number of representatives in a legislative body,
such as a national parliament. When the MPs are allowed to vote freely and
their preferences are largely independent of the parties, the probability that
the voter that turns a losing coalition into a winning one belongs to a given
party is proportional to the shares of seats the party has K´oczy (2008a).
In order to measure voting power several approaches have been proposed.
The first measured voting power by directly applying the Shapley value to
simple cooperative games (Shapley and Shubik, 1954). The most common
alternative is the Banzhaf measure and index1(Banzhaf, 1965; Coleman,
1971; Penrose, 1946) which, although predates the Shapley-Shubik index,
the first two discoveries have been largely forgotten and were only connected
to the mainstream literature later. Since then several alternatives have been
introduced: the Johnston-index (Johnston, 1978), the Deegan-Packel index
(Deegan and Packel, 1978), the Public Good Index (Holler and Packel, 1983)
or the partition value (Neyman and Tauman, 1979) so the question naturally
arises: which one of these should be used?
The key to answering this question is to study the properties of the various
indices. As we have seen in the above example the institutional design might
imply one or another index. In general, however it is rather difficult to
make a direct choice. On the other hand if the various indices and measures
are characterised by a handful of simple, elementary properties or axioms
making a choice between these axioms is often easier. In the best case a set
1It is common to refer to power measures normalised to 1 as power indices.
of axioms fully characterise an index, that is, there is a unique index that
satisfies the given axioms. While in theory the idea is simple and appealing,
many of the axioms used in these characterisations are rather technical and
show little relevance to practical problems (Laruelle and Valenciano, 2005,
pp37-38). It seems other properties, that perhaps do not help towards a
full characterisation, but ones that have clearer practical implications can
be equally useful to make a choice (Felsenthal and Machover, 1998, 2004).
Unfortunately as researchers have produced positive results also some rather
unattractive properties have been discovered. The so-called paradoxes of
voting power (Brams, 1975/2003) are three rather intuitive properties that
are nevertheless not satisfied by the best-known power indices, the Shapley-
Shubik and the Banzhaf index. Felsenthal and Machover (1995) argue that
“paradox” is perhaps a word too strong to describe these properties and argue
that these are merely ‘apparently strange pieces of behaviour’ (Felsenthal and
Machover, 1998, p. 221), but then the question arises whether we should be
guided by our intuition and consider the paradoxes a problem or whether we
should be comforted by the theoretical underpinnings of these indices and
revise our intuition.
In an earlier paper (K´oczy, 2008a) we have shown that there is a rather
intuitive index, the proportional index where none of the paradoxes arise.
On the other hand there is ample empirical evidence that suggests that in
practical matters this proportional index is used as a rule of thumb to assign
power2. We believe that the paradoxes and a number of other results stem
from this proportional index, but then it is somewhat difficult to decide what
“natural properties” should serve as the basis of evaluation of power indices
and which ones are prejudices rather than true requirements.
In this paper we show that the problem is general: an index that satisfies
the widely accepted axioms will necessarily exhibit some of the paradoxes.
In particular we show that the paradox of new members is in conflict with
the null player axiom.
The outline of this paper is then as follows. First we introduce voting
games and some of the well-known properties. Then we present the Brams’s
paradoxes and an axiomatisation of the Shapley-Shubik and Banzhaf indices,
including the null player axiom. Next we prove our main result. We end the
paper with some conclusions.
2See Diermeier and Merlo (2004); Gelman, Katz, and Bafumi (2004); Fr´echette, Kagel,
and Morelli (2005) and references therein
2 Voting games
Since Shapley and Shubik (1954) it is common to study voting situations as
cooperative games. A cooperative game is given by a pair (N, v) consisting
of a set of nplayers and a real valued function, the so-called characteristic
function3vdefined over the set of coalitions: a coalition is a subset of the
player set. Thus v: 2N−→ R. It is common to make a number of assump-
tions about the characteristic function. We assume that the empty set has
no value, therefore v(∅) = 0 and that the function is superadditive or
v(S∪T)≥v(S) + v(T) if S∩T=∅.(1)
For such games the total value of the players is maximised by the grand
coalition Nand hence the purpose of the game is to find an equilibrium allo-
cation of this payoff among the players. While strategies are not explicit in
cooperative games (they correspond to forming coalitions), we still deal with
the same intelligent, payoff-maximising agents as in noncoopertive games,
and in fact the cooperative concepts used here have been implemented as
equilibria of certain noncooperative games (Gul, 1989; P´erez-Castrillo and
Wettstein, 2001).
In the following we shall be interested in simple games: A game is simple
if the value of the coalition is either 0 or 1, that is, if v: 2N−→ {0,1}. As we
shall see these values can be seen as wins and losses, and correspondingly we
can talk about winning and losing coalitions. We denote the set of winning
coalitions by W, thus
W={S|S⊆N, v(S)=1}.
In a winning coalition we will be interested in critical players, that is, players
whose presence is essential for the success of the coalition. Formally the
player iis critical in coalition Sif S∈ W, but S\ {i} 6∈ W. A player that
is never critical is called null . Among winning coalitions, minimal winning
coalitions containing only critical players deserve special attention. The set
of such coalitions is denoted by M, where
M={S|S∈ W,∀i∈S:S\ {i} 6∈ W } .
We assume that the grand coalition is always winning, that is, v(N)=1
and that the addition of new members to a coalition does not make it loosing,
3The name comes from the early literature of game theory. Von Neumann and Morgen-
stern in their seminal work (von Neumann and Morgenstern, 1944) assumed that when in
an nplayer game a subset Sof players forms a coalition, this coalition must play against a
natural opponent, the complementer coalition N\S. The value the coalition Scan obtain
in this game is its characteristic value.
formally if v(T) = 1 and T⊆Sthen v(S) = 1. While this assumption is
a standard one, in some cases the addition of a new member to a winning
coalition may also result an infeasible (winning) coalition. In such cases
the coalition, though has the power to make decisions, cannot. We will use
infeasible coalitions to formulate one of the paradoxes below, but this idea is
used in the literature of games over convex geometries (Bilbao and Edelman,
2000; Bilbao, Jim´enez, and L´opez, 1998) and in the models of strategic power
indices, where players have the ability to block coalitions (K´oczy, 2008b).
The set of feasible coalitions is denoted by F. Unless otherwise stated we
assume that F= 2N.
Aweighted voting game G = (N, (wi)i∈N, q) consists of a collection Nof
nvoters having w1, w2, . . . , wn>0 votes such that w=Pn
i=1 wi, and a quota
q,w≥q > w/2, or the number of votes required to pass a bill. For more on
weighted voting games see Straffin (1994). It is clear that there is a unique
mapping from weighted voting games to voting games, and therefore the first
is a subset of the latter. Given (N, (wi)i∈N, q) we define the corresponding
voting game (N, v) by
v(S) = (1 if Pi∈Swi≥q
0 otherwise. (2)
2.1 Power indices
Now that the games have been defined we can move on to defining the ways
to establish the power of the individual players. In this respect there are two
approaches depending on the goal of the study. On the one hand we can look
at the players’ share of power. In this case we calculate power indices, that
is, normalised power measures. This is the approach we take here. When the
focus is on engineering the voting situation, determining the voting weights
and quotas or, more generally, the set of winning coalitions, the likelihood
that any coalition is winning is important too. For it is good to know if one
can change the decision in a certain percentage of cases, but such cases rarely
occur the power is certainly more limited. Put it differently, power is often
money – it is sufficient to think of lobbying to see this. If so, it is one thing
the get a good slice of the cake and another to have a large cake, that is,
much lobbying.
Since our focus is not on the institutional design, we present power indices.
Apower index is a function kthat assigns to each weighted voting game
a non-negative vector in RN
+.
The Shapley-Shubik index (Shapley and Shubik, 1954) is an application
of the Shapley value (Shapley, 1953) to measure voting power, motivated by
a story where parties throw their support at a motion in some order until a
winning coalition is reached. The last, pivotal party gets all the credit; the
Shapley-Shubik index is then the proportion of orderings where it is pivotal
Φi=# times iis pivotal
n!.
There is also an explicit formula to express the Shapley-Shubik index:
Φi=X
S3i
(s−1)!(n−s)!
n!(v(S)−v(S\ {i})) (3)
The Banzhaf measure (Penrose, 1946; Banzhaf, 1965) is the probability
that a party is critical for a coalition, that is, the probability that it can turn
winning coalitions into losing ones.
ψi=# times iis critical
2n−1.
Or explicitly:
ψi=1
2n−1X
S3i
(v(S)−v(S\ {i})) (4)
The Banzhaf index βColeman (1971) is the normalised Banzhaf measure,
where the total power is scaled to 1 –already in the spirit of the Shapley-
Shubik index.
The two indices can give substantially different implications, despite the
fact that the main difference is in the probabilities they attach to the forma-
tion of particular coalitions or to the fact that a given player is critical for
some coalition. While in the Banzhaf index all such instances of criticality
happen with equal probability, for the Shapley-Shubik index the probability
depends on the size of the coalition (when a player is critical in medium sized
coalitions, this is taken with a smaller weight into account).
There are a few variants of the (normalised) Banzhaf index. In the John-
ston index γ(Johnston, 1978) the credit a critical player gets is inversely
proportional to the number of critical players in the coalition. In effect,
coalitions of different sizes have the same contribution to the distribution of
power or the probability that a given coalition is the one making the deci-
sion is the same for all coalitions. The Deegan-Packel index ρ(Deegan and
Packel, 1978) is a further modification that only considers minimal winning
coalitions, motivated by the idea that only minimal winning coalitions should
form so that the benefits from winning should be least divided (Riker, 1962).
Finally the Holler-Packel or Public Good Index h(Holler and Packel, 1983)
modifies the Deegan-Packel index: here the benefit of forming a winning
coalition is given to each and every player in the coalition. With the normal-
isation in simple games the index is nothing but a normalised Banzhaf index,
where only minimal coalitions are taken into account. Finally the partition
index (Neyman and Tauman, 1979) is motivated by decision making with
multiple alternatives that then results in a partition of the voters that con-
sists of possibly more than two coalitions. The probability that a coalition
forms is then the probability that a partition containing this coalition forms.
The partition index clearly favours smaller winning coalitions.
The proportional index αis the trivial power index given by αi=wi
w.
This measure is popularly known in political science as Gamson’s Law: ‘Any
participant will expect others to demand from a coalition a share of the pay-
off proportional to the amount of resources which they contribute’ Gamson
(1961).
2.2 Axioms
In the following we present the full characterisations of Dubey (1975) and
Dubey and Shapley (1979) for the Shapley-Shubik and the Banzhaf index
respectively.
Before we move to the different axioms we need to introduce some addi-
tional terminology. The permutation πof the players is a bijective mapping of
the player set. The permutation of a game πv is given by (πv)(S) = v(π(S)).
Definition 1 (Anonymity Axiom).For all simple games vany permutation
πof N, and any i∈N,
ki(πv) = kπ(i)(v).(5)
Definition 2 (Null Player Axiom).For any simple game vand any i∈N,
if iis a null player in game vthen
ki(πv) = 0.(6)
For two simple games vand wover the player set Nlet
(u∨w)(S) = max {v(S), w(S)}and (u∧w)(S) = min {v(S), w(S)}.
Of these two conditions the latter is perhaps the more interesting one as
it gives a formula for a combination game, for instance a game where a
coalition must be winning in both chambers of the parliament or when there
are multiple criteria to determine the winning coalitions as in the case of the
EU Council of Ministers, for instance.
Definition 3 (Transfer Axiom).For any simple games v, w such that v∨w
is a simple superadditive game, too the transfer axiom states that
ki(v) + kj(w) = k(v∧w) + k(v∨w) (7)
Finally there are two axioms that express a notion of efficiency for both
the Banzhaf and the Shapley-Shubik case.
Definition 4 (Shapley Total Power Axiom).For any simple game v
X
i∈N
Φi(v) = 1 (8)
Definition 5 (Banzhaf Total Power Axiom).For any simple game v
X
i∈N
ψi(v) = 1
2n−1X
i∈NX
S3i
(v(S)−v(S\ {i})) (9)
With these definitions the Shapley-Shubik index and Banzhaf measure
can be characterised as follows:
Theorem 6 (Dubey (1975)).If a power index ksatisfies Anonimity, Null
Player, Transfer and Shapley Total Power Axioms, then k= Φ.
That is the Shapley-Shubik index is the unique power index satisfying
the above axioms.
Theorem 7 (Dubey and Shapley (1979)).If a power measure ksatisfies
Anonimity, Null Player, Transfer and Banzhaf Total Power Axioms, then
k=ψ.
The Banzhaf measure is the unique power measure satisfying the above
axioms.
While there are many other axiomatisations of these (and other indices),
the null player axiom is one of the central properties. In fact, the proportional
index is mostly criticised for not satisfying this property (for players with
nontrivial weights). These motivate our interest in the Null Player Axiom.
2.3 Paradoxes
Brams (1975/2003) lists three natural properties that any power index should
satisfy (the list is extended by Felsenthal and Machover (1998)), but the best-
know indices satisfy none of these. These disappointing negative results are
known as paradoxes. In the following we list them in their positive form as
“properties.”
Given a game (N, v) by the merger of players iand jwe mean a modified
game (Nij , vij ) with one less players Nij =N\i, j∪{ij}and winning coalitions
Wij =S∈2Nij |S∈ W,or ij ∈Sand S\ {ij}∪{i, j} ∈ W
When the game is defined as a weighted voting game the combined player
ij has a weight wij =wi+wj.
Definition 8 (Property of (large) size).Let (N, v) be a voting game and k
a power index. Define (Nij , vij ) by the merger of players iand j. The game
satisfies the property of large size if ki(v) + kj(v)≤kij (v0).
Definition 9 (Property of new members).Now define (N+, v+) as an exten-
sion of (N, v) by parties n+1, . . . , m such that W \W+=∅. The property or
new members is satisfied if ki(N, v)≥ki(N+, v+), that is, the introduction
of new members should not increase a party’s power.
Starting from a voting game (N, v) consider the game (N , vij), which only
differs in the fact that players iand jrefuse to cooperate, thus
Wij ={S∈ W |{i, j} 6⊂ S}
Definition 10 (Property of quarrelling members Riker (1962)).If two par-
ties refuse to vote together, this should not increase their total power. For-
mally kh(N, v)≥kh(N, vij ) for h∈i, j.
Note that the game defined here is not a proper voting game, since nec-
essarily N6∈ Wij , so the grand coalition is not a feasible winning coalition.
2.4 Axioms and paradoxes
The properties above turn into paradoxes when we find that none of the
well-known power indices satisfy all these. In fact the Shapley-Shubik and
Banzhaf indices fail all three (Brams, 1975/2003). This result is well-known,
and we leave it to the reader to find examples of games where the para-
doxes appear. Unfortunately the paradoxes do not only appear in made-up
examples, but van Deemen and Rusinowska (2003) found numerous real life
instances in Dutch politics.
In fact, we show that the Null Player Axiom implies the Paradox of New
Members and therefore any index based on the aforementioned axiom will be
unreliable in circumstances where the player set is likely to expand. Unfor-
tunately, yet again, such examples are common, in fact, the most common
application of power measures is the EU Council of Ministers that is expected
to expand further in the coming years, moreover the recent surge of interest
in power indices is largely due to the “problems” caused by the extensions.
Theorem 11. The Null Player Axiom implies the Paradox of New Members.
Proof. Consider a power index kthat satisfies the Null Player Axiom. For
such an index all players that do not contribute to any of the winning coali-
tions receive a value of 0. Now consider an extension of the game and we
show that for all games there is an extension such that a null player becomes
non-null.
On the other hand the Paradox of New Members implies that there exist
games that fail the Property of New Members. We therefore restrict our
attention to weighted voting games of the form (N, (wi)i∈N, q). Consider an
arbitrary game with a null player that we denote by i. Instead of minimal
winning coalitions consider maximal losing coalitions, in the following sense
L(i)∈arg max
S3i
S6∈W X
j∈S
wj,(10)
That is, losing coalitions that lose with the smallest margin. Then consider
the extension (N∪ {k},(wi)i∈N∪{k}, q0) where wkand q0are determined as
wk= (q0−q) +
q−X
j∈L(i)
wj
(11)
Which can be reorganised as follows:
q0=wk+X
j∈L(i)
wj.(12)
For wksufficiently large (or, rather, not too small) the game is a proper
simple game. On the other hand observe that the coalition L(i)∪ {k}is a
minimal winning coalition in the extended game, moreover, per definition,
i∈L(i)∪ {k}. In a minimal winning coalition all players are critical, so i
is critical in L(i)∪ {k}and a player that is critical in any coalition is not
null. Therefore iis not null in the extended game (N∪ {k},(wi)i∈N∪{k}, q0)
and therefore its power increases as a result of a new member. Therefore the
index exhibits the Paradox of New Members.
In the more general case considering winning coalitions, but no voting
weights the argument is not so easy to present formally and would rely on
Table 1: The 1958 Council of Ministers and two possible expansions.
Original (%) Expansion I (%) Expansion II (%)
Country wiΦiβiwiΦ0
iβ0
iwiΦ00
iβ00
i
Germany 4 23.3 23.8 4 22.4 21.6 4 21.9 22.0
France 4 23.3 23.8 4 22.4 21.6 4 21.9 22.0
Italy 4 23.3 23.8 4 22.4 21.6 4 21.9 22.0
Netherlands 2 15.0 14.3 2 10.7 11.8 2 13.6 12.2
Belgium 2 15.0 14.3 2 10.7 11.8 2 13.6 12.2
Luxembourg 1 0 0 1 5.7 5.9 1 3.6 4.9
new member 1 5.7 5.9 1 3.6 4.9
Total 17 100.0 100.0 18 100.0 100.0 18 100.0 100.0
Quota 12 12 13
some additional assumptions, but it is clear that if a new member is very
weak, the coalition {i, k}is losing, while if it is very powerful kcan be a
dictator. Assuming a “smooth” change in the size of k(and the corresponding
update of W) at one point the coalition {i, k}becomes winning, while k, in
itself is not yet, and then the player iis critical.
To illustrate that such extensions are not completely artificial, consider
the EU Council of Ministers. While in the early days of –what is now called–
the EU most decisions were made unanimously, already in the 6-member
Community the rules of qualified majority voting to make decisions were
laid down. The weights were specified as follows: 4 votes for each of the
large members (Germany, France, Italy), 2 for the medium-sized members
(Belgium, Netherlands) and 1 for the smallest country, Luxembourg. The
Shapley-Shubik and Banzhaf indices are presented in Table 1. Observe that
Luxembourg is a null player.
Now consider an extension by a small new member, who receives the
same number of votes as Luxembourg.4We look at two scenarios depending
on the quota in the new union. Observe that in either of these scenarios
Luxembourg is not any more a null player, its power has increased. While
this is a hypothetical extension, since Denmark, Ireland and the UK have
joined the EU none of the players are null.
4This was a realistic scenario in the late 60s: Soon after the formation of the EC,
Denmark, Ireland, Norway and the UK have applied. The application of the UK was
vetoed by France, Norway withdrew due to a referendum, while the remaining two due to
the veto on the UK. Had any of the smaller countries joined, the result would have been
close to one of the above alternatives.
3 Conclusion
In democracies decisions are commonly made by (qualified majority) voting,
be those decisions in a national parliament, at a shareholder meeting or uni-
versity senate. When the voters are of different sizes voting weights can be
used to express the differences and the voting rules can then be expressed as
qualified majority voting. As soon as the voting situation is asymmetric it
is not so straightforward to see the actual influence of a single voter in the
voting process not to mention complex voting situations such as in the UN
Security Council, where the permanent members have veto rights (Owen,
1995, Chapter XII), or the International Monetary Fund where some coun-
tries vote directly some via voting coalitions that themselves share power in
varying ways (Reynaud, Lange, Gatarek, and Thimann, 2007). The topic
has entered even the popular press when the extension of the EU made an
update to voting in the Council of Ministers necessary. What is then the
power of the individual member states? The Council uses currently a very
complicated voting mechanism with three criteria, so it is hardly surprising
that one needs powerful tools to evaluate this and similar voting situations.
The theory of a priory measures of voting power has developed much
since Shapley and Shubik, and it does have a number of answers to such
and similar questions. Unfortunately more than one answer has been given
and a selection is far from obvious. The commonly used indices have been
axiomatised by a number of elementary properties hoping that based on
the attractiveness of simple properties a choice can be made. On the other
hand a number of shortcomings of current theories have surfaced and been
summarised as “paradoxes”. The word paradox may be too strong, the
behaviour these properties describe is certainly odd. In this paper we have
shown that the Paradox of New Members is a direct consequence of the Null
Player Axiom therefore of one insists on the latter, the first is not at all
paradoxical.
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