F O R M I N G COALITIONS A N D
MEASURING VOTING P O W E R
M A N F R E D J. H O L L E R
University of Munich
I N this paper I will present various concepts of coalition formation and power
measures and discuss them with respect to the view ‘that situations where
minor players possess greater potential for power are not anomalous, but
occur rather frequently in real-world situations’.’ We shall see that the analysis
of the various concepts of coalition formation will teach us much about the
implications of the so-called power indices and the character of a priori voting
power. A new index will be introduced which considers the coalition value of a
public good and takes into consideration the distinction between power and
luck. The values of this index will be calculated for the parties of the Finnish
Parliament (Eduskunta) for the period 1948-79.
1 . RIKER’S SIZE PRINCIPLE
Riker claimed that ‘parties seek to increase votes only up to the size of a
minimum Coalition’., This follows from the well-known ‘Size P r i n ~ i p l e ’ ~
which implies that, given a multi-member voting body u = ( d ; w 1 ,w,, . . ., w,)
a coalition so will be formed for which the sum of the corresponding voting
weights of its members
minimizes the difference A = so - d, given A > 0. Thereby, i = 1, . . ., n being
the votes, w , , w , , . . ., w, the corresponding voting weights, and d the decision
rule. The underlying idea of this solution is that payoffs for any winning
coalition S j are identical. If the coalition payoff is split between the members of
the winning coalition according to their respective voting weights, each
member’s share will be maximized through the minimizing of the coalition
partner(s)’ voting weight@). Hence, for a voting body u1 = (50; 40, 35, 25) the
minimum winning coalition (MWC) So = {35, 25) will be formed although
S , = (40, 35) and S , = (40,2 5 ) are also winning and minimal with respect to
the number of members-they do not contain any dummy member, i.e., a
member who is not necessary for the fulfilment of the decision rule d = 50.
’
E. W . Packel and J . Deegan, Jr., ‘An Axiomatic Family of Power Indices for Simple n-Person
Games’, Public Choice, 35 (1980). 229-39.
W. H. Riker, The Theory of Political Coalitions (New Haven, Conn., Yale University Press,
1962), p. 100.
Riker, Theory of Political Coalitions, pp. 32 8.
Political Studies, Voi. XXX, No. 2 (262-271)
M A N F R E D J . HOLLER
263
We can state that the voting power of the players who are not included in
coalition So has to be zero in accordance with Riker’s Size Principle.
2. T H E DEEGAN-PACKEL INDEX
A different concept of MWC was presented by Deegan and P a ~ k e lThey
.~
introduced an interesting ‘paradox’ as regards the measuring of power. For a
given game u on the player set N = { 1,2, . . ., n ) Deegan and Packel defined the
set of MWC’s M(u)as follows:
M(u)= { S
c NIu(S) = 1 and u ( T )= 0 for all T s j S J .
From this definition it follows that T is a nonwinning coalition. Every strict
subset of S is therefore nonwinning: v(S - {i}) = 0 , for i E S. The set S - {i} is
a strict subset of S , given that i E S .
On assuming that (1) only minimal winning coalitions will form; (2) each
such coalition has an equal probability of forming; (3) players in a (minimal)
winning coalition divide the spoils equally, Deegan and Packel’s measure of
power is
IM(u)l and IS1 are cardinalities of the corresponding sets. This measure
expresses what player i can expect to ‘get’ from a game u. If we apply this
power index to a voting body uz = (51; 35, 20, 15, 15, 15) the power
distribution is given by
cuz = (18/60, 9/60, 11/60, 11/60, 11/60),
This shows that a larger voting power is related to a player of voting weight 15
than to a player of voting weight 20. The Deegan-Packel index is not
monotonic in votes. We may ask whether this index is a plausible way of
attaching members to entities as implied in the term measurement.6 As long
as we do not consider voting power to be monotonic in seats the DeeganPackel index can be accepted. However, if we believe in a monotonic relation,
and obviously this is a tenet to representational democracy,’ we must question
the axioms fundamental to this index.’ Packel and Deegan contmented on this
‘paradoxical’ result by referring to sociologists (such as Caplow’), who argue
‘that situations where minor players possess greater potential for power are not
anomalous, but occur rather frequently in real-world situations’.
J . Deegan, Jr. and E. W . Packel, ‘A New Index of Power for Simple n-Person Games’,
International Journal of Game Theory, 7 (1979), 113-23.
Packel and Deegan, ‘An Axiomatic Family of Power Indices for Simple n-Person Games’,
p. 231.
H. Nurmi, ‘Measuring Power’, in M . J . Holler (ed.), Power, Voting and Voting Power
(Wurzburg. Physica, 1982).
H. Nurmi, ‘Power and Support: the Problem of Representational Democracy’, Munich Social
Science Review, 4 (1978), 5-24.
Packel and Deegan, ‘An Axiomatic Family of Power Indices for Simple n-Person Games’,
p. 236.
T. Caplow, Two Against One: Coalitions in Triads (Englewood Cliffs, N.J., Prentice-Hall,
1968).
’
264 F O R M I N G C O A L I T I O N S A N D M E A S U R I N G V O T I N G POWER
This might be so, but we can ask whether this argument is plausible for
Packel and Deegan’s conceptual framework. Firstly, we can recognize that
Riker’s Size Principle would clearly favour the coalition So = { 35,201 resulting
from the voting body u 2 . All other coalitions are either not winning or are
related to a sum of voting weights which is larger than so = 55. Secondly, we
must realize that the 15 per cent players in voting body u2 get the relatively
higher power index, as one minor player does not suffice to form a winning
coalition with the 35 per cent player. That is how a 15 per cent player gets into
three coalitions in accordance with M ( u ) , whereas the 20 per cent player is
member of only two coalitions. Thirdly, since, as assumed, ‘players in a
(minimal) winning coalition divide spoils equally’, there is actually no reason
why the 35 per cent player should prefer a minimum winning coalition
S , = (35, 15, 15) to a non-minimum winning coalition S , = {35,20, 15). Since
the 15 per cent player is a dummy in S , , we could expect the major part (if not
all) of the coalition’s payoff to be allocated to the 35 per cent player and 20 per
cent player. Indeed, according to the above division rule, it is more profitable
for the 35 per cent player to form a coalition with the 20 per cent player and
receive half of the coalition pie. Were he to co-operate with the two 15 per cent
players, he would receive only a one-third share. Since this also holds under the
given decision rule for the payoffs of the 20 per cent player, when comparing
his share from coalitions {35, 20) and (20, 15, 15, 15}, it seems doubtful that
any coalition other than (35, 20) is plausible under the conceptual framework
of Packel and Deegan. By the division rule, as assumed by Packel and Deegan,
Leiserson’s bargaining concept, which favours the coalitions with small
numbers of parties (players), gains plausibility and gives support to the
coalition (35, 20}.’O One way out of this trap is introduced by Packel and
Deegan themselves. They presented a second power index [{(u) which is also
based on their concept of the minimum winning coalition and the above
division rule but incorporates different occurrence probabilities of the
coalitions.’ If the occurrence probability of the coalition { 35,20} is 1 and that
of the others 0, [ i 5 ( u ) = 1/2 and [ [ o ( u ) = 1/2 for the voting body u = u , .
’
3 . PROBABILITIES O F O C C U R R E N C E
The underlying probability assumption might be considered rather crude,
nevertheless it bears some theoretical justification. We can deduce from
Leiserson’s analysis that a two-member winning coalition dominates all
coalitions which consist out of more than n = 2 members.’, Kalisch, Milnor,
Nash and Nering reported that coalitions of more than two players within
experimental n-person games are seldom formed except by being built up by
smaller c0a1itions.l~This supports, what we would like to coin, the ‘Principle
l o M. A. Leiserson, ‘Factions and Coalitions in One-Party Japan: An Interpretation Based on
the Theory of Games’, American Political Science Reoiew, 62 (1968), 77C87.
I ’ Packel and Deegan, ‘An Axiomatic Family of Power Indices for Simple n-Person Games’,
p. 231.
‘ I Leiserson, ‘Factions and Coalitions in One-Party Japan’.
l 3 G . K . Kalisch, J . W. Milnor, J. F. Nash and E. D. Nering, ‘Some Experimental n-Person
Games’, in R. M . Thrall, C. H. Coombs, and R. L. Davis (eds), Decision Processes (New York,
Wiley, 1954), pp. 301-27.
MANFRED J . HOLLER
265
of MWC in Number'. This principle obviously cannot be in favour of members
of small voting weights. It clearly says that the coalition (35, 20) will be the
only (stable) one for the voting body 0,. (Riker's Size P r i n ~ i p l e 'gives
~
also
support to coalition { 35, 20). This principle is, however, based on a different
division rule.)
Instead we can introduce a probability mix with respect to the coalitions as
defined by M ( u ) for u = 0,. The use of the well-known Banzhaf power index
actually implies a specific probability mix.I5 This has been shown by Straffin.I6
The Banzhaf power index bl of a player i is defined, as the number of swings for
player i divided by the total number of coalitions containing player i. A swing
occurs when the defection of player i changes a coalition from winning to
losing. For reasons of comparability, the Banzhaf index is often standardized
by the formula
i= 1
Given the voting body u l , the power distribution as measured by this index is
b , = (1/3, 1/3, 1/3), For the voting body u2 we can calculate b , = (44%, 20%,
12%, 12%, 12%).
In the context of the Banzhaf index it is interesting to note that a fourth
distinct concept of MWC is offered: 'for each minimal winning coalition there
is at least one member whose removal would make the coalition nonwinning'.' ' The power of an actor i is thereby seen as his ability to threaten the
other members of the minimal coalition, as implied by his ability to change the
coalition from winning to losing. Hence, if only one player (and not all as in the
Packel and Deegan concept) is critical to a coalition, it can be called minimum
winning coalition. Thus, MWCs of the fourth type can include dummies.
The Banzhaf index can be read as the measure of a player's probability to
change the outcome." From this we can deduce that a 20 per cent player is
more likely to effect the voting outcome than a 15 per cent player. This does
not exclude the possibility of a specific 15 per cent player being, and a 20 per
cent player not being, a member of a winning coalition, it just makes it less
likely. Given the voting body u 2 , there exists a MWC of the fourth type (as well
as of the Packel and Deegan type) which does not contain the 20 per cent
player but rather the 15 per cent players and the 35 per cent player. This does
not contradict the (positive) monotonic characteristic of the Banzhaf index
with respect to voting weights, it only shows that the likelihood of occurrence
is in general different to the realization.
Riker, Theory oj'Polirica1 Coalitions, p. 32.
J . F. Banzhaf, 'Weighted Voting Doesn't Work: A Mathematical Analysis', Rurgers Law
Review, 19 (1965), 317-43. Also J . F. Banzhaf, 'One Man, 3.312 Votes: A Mathematical Analysis
of the Electoral College', Villanova Law Review, 13 (1968). 3 0 4 3 2 .
l 6 P. D. Straffin, Jr., 'Homogeneity, Independence and Power Indices', Public Choice, 30 (1977).
107-1 8.
Nurmi, 'Power and Support', p. 18.
Straffin, 'Homogeneity, Independence and Power Indices', p. 112.
l4
Is
266
F O R M I N G C O A L I T I O N S A N D M E A S U R I N G V O T I N G POWER
4 . T H E D I V I S I O N RULE A N D T H E P U B L I C G O O D ASPECT
Real-world situations where minor players possess greater power are
compatible with all four concepts of the MWC and the deduced voting power
measures. However, Riker’s Size Principle and the Packel and Deegan concept
might allocate larger a priori power to players with relatively smaller voting
weights. This counter-intuitive outcome is due to the division rule which
underlies these indices. It is consequent if the voting weights are positively
related to a claim on coalition payoffs as assumed by the Size Principle. Yet it
needs, however, further elaboration if the coalition payoff is divided by the
number of members as implied in the Packel and Deegan concept. It cannot be
advantageous for any player to form coalitions with minor players instead of
with major players (measured in voting weights). However, when examined in
conjunction with the division rule, it may be disadvantageous. All indices, as
discussed so far, face the problem of distributing (or assigning) the value of a
priori coalitions among their members. There might be no adequate solution to
this problem, for the coalition value is a collective good. The private good
approach, as implied in the discussed indices, is inappropriate if voting is not
only a matter of allocating spoils. In a recent article Barry claimed that the
concepts of dividing the value of a coalition ‘violates the first principle of
political analysis, which is that public policy is a public good (or bad).’’’ For
illustration :
If the death penalty is reintroduced, that pleases those who favour it and displeases
those who do not. Similarly, a tax break is a good or bad for people according to their
situation. The gains are not confined to those who voted on the winning side nor are the
losses confined to those who are on the losing side.
5 . POWER A N D LUCK
If we consider the value of a coalition (not the voting power as discussed by
Brams” to be a collective good, any member of the voting body whose
preferences correspond with the outcome of the winning coalition can be
considered as member of the specific coalition. A member who is essential for
the specific coalition can exert power. A non-essential member (dummy voter)
is merely lucky: the outcome will correspond to his preferences although he
does nothing. Barry labels the difference between success (i.e., the coalition
outcome corresponds to the member’s preference) and luck ‘decisiveness’.2
Power is understood in Weberian tradition as the actor’s ability to overcome
resistance. According to this definition, an all-powerful actor, i.e., dictator,
might not be decisive. If he is very lucky, all the outcomes he wants will occur
even if he does nothing.” According to Barry’s definitions of power and
decisiveness it follows that the more powerful an actor, the better his ‘chance’
of being decisive. Power and decisiveness are closely related, however, as
l 9 B. Barry, ‘Is It Better to be Powerful or Lucky?’: Parts 1 and 2, Political Studies, 28 (1980),
183-94 and 338-52.
’O S. J . Brams, Game Theory and Politics (New York, Free Press, 1975), p. 178.
” Barry, ‘1s I t Better to be Powerful or Lucky?, p. 338.
Barry, ‘Is I t Better to be Powerful or Lucky?’, p. 350.
’’
MANFRED J. HOLLER
261
defined by Barry they cannot be clearly explained in terms of success, luck, and
d e c i s i v e n e s ~ Yet
. ~ ~ if power is seen as an ability, a capacity or potential, to
influence, bring about or preclude an outcome,24 it is identical to decisiveness
as defined above. Power measures thereby become invariant to changes in the
distribution of preferences. This seems adequate for measuring a priori voting
power, when information concerning the decision rule and the voting weight
distribution (but not concerning the preferences of the voters) is given.
6 . T H E STORY O F A N E W I N D E X
From this we obtain the following ‘story’ for an appropriate power measure:
(1) Any member of a minimum winning coalition is decisive for the coalition
value. The (undivided) coalition value therefore expresses his power within
the coalition.
(2) An individual nonessential member does, by definition, not influence the
winning of a coalition. He therefore has no power. It is sheer ‘luck’ when
an outcome corresponds with his preferences.
(3) Since a nonessential member is not decisive for the winning of his preferred
coalition, i.e., his preferred policy, he has no incentive to vote.
(4) Because of (3), only those winning coalitions will be purposefully formed
(‘not by sheer luck’), which win by means of the votes of their essential
members. If, e.g., A B C is a winning coalition, then it will form if either all
three members are essential, or if either AB, AC, B C , or any of the single
coalitions A , B , and C is a winning coalition. ABC will not purposefully
form, if, e.g., only A is essential but not sufficient for the formation of a
winning coalition. If, however, coalition ABC forms, it is due to luck and
not due to A’s power (or decisiveness). We shall call the set of essential
members a ‘decisive set (of a coalition)’. Only those coalitions will form
which have a winning coalition as decisive set, i.e., a ‘winning decisive set’.
( 5 ) Each winning decisive set corresponds with a specific coalition outcome
(policy). The outcome of two coalitions differ if their decisive sets are not
identical. If, e.g., A B forms the winning decisive set of the coalition A B C ,
the outcome of coalition A B C will be identical with the outcome of
coalition A B . If A D is the winning decisive set of coalition A C D , then the
outcome of coalition A C D will be identical with the outcome of coalition
A D , but different to the outcome of coalitions A B and A B C . It follows
that if we consider the coalition outcome (value) to be a public good, we
must refer to the various winning decisive sets of the potential coalitions
when measuring the a priori voting power within a specified voting body.
Our definition of the winning decisive set is identical with the definition of
the elements of the set of minimum winning coalitions M ( u), which underlies the Deegan-Packel index. (Our story above does not imply that only
these coalitions will form. It merely suggests that only these coalitions
should be considered for measuring a priori voting power.)
Barry, ‘Is It Better to be Powerful or Lucky?‘, p. 350.
See, for example, N . R . Miller, ‘Power in Game Forms’, in M . J . Holler (ed.), Power, Voting
and Voting Power.
23
24
1962
1958
1954
1951
1948
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats (7;)
h-Index
b-Index
4
28.0
20.0
28.57
1
26.5
17.9
28.57
1
27.0
17.9
28.57
2
25.0
20.0
27.59
4
26.5
20.0
30.08
27.0
20.0
28.57
4
25.5
17-9
28.57
4
26.5
17.9
28.57
1
24.0
18.0
24.14
2
23.5
18.3
23.31
1
19.0
13.3
17.37
1
2
19.0
20.0
14.29
2
21.5
21.4
21.43
2
21.5
21.4
21.43
4
24.0
18.0
24.14
16.5
20.0
14.29
3
14.0
14.3
7.14
3
12.0
14.3
7.14
3
14.5
12.0
6.9
3
16.0
11.7
10.59
3
5
7.0
20.0
14.29
5
7.5
14.3
7.14
5
6.5
14.3
7.14
5
7.0
12.0
6.9
5
7.0
11.7
8.05
14.3
7.14
6
6.5
14.3
7.14
6
4.0
12.0
6.9
6
6.5
11.7
6.36
5.0
6
0.0
0.0
6
2.5
8.3
2.97
1 .o
I1
1.5
8-0
3-45
11
TABLE1
The distribution of seats and power in the Finnish parliament, 1948-79
0.5
5.0
1.27
1
z
?J
m
s
0
-0
m
3
CI
Z
>
cn
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats (%)
h-Index
b-Index
Party
Seats ( % )
27.5
15.2
30.08
1
27.0
15.4
32.92
1
26.00
1
4
24.5
12.1
25.45
3
18.5
13.5
17.2I
2
18.5
13.0
17.07
2
20.0
11.1
12.92
3
23.5
1
27.5
21.2
31.7
1
26.0
21.2
28.69
19.5
11.1
17.08
4
18.0
4
2
20.5
21.2
21.88
2
18.0
15.4
16.39
4
17.5
13.0
16.26
16.39
3
17.0
13.0
14.63
3
17.5
12.0
17.08
2
17.5
15.4
18.0
4
3
13.0
12.1
6.7
5.0
6.25
5
5.0
11.1
5
6.0
12.1
6.7
7
9.0
11.5
10.66
7
9.0
12.0
8.94
5
10.3
5-42
8
5.0
4.5
4.02
5
6.0
15.4
9.02
5
5.0
10.9
6.5
6
9.1
6
4.5
11
3.8
0.82
6
3.5
12.0
3.25
8
4.5
8.5
5-42
7
3.0
4.0
3.5
9.1
3.13
6
7
1.67
6
2.0
1 .o
5.1
7
3.0
0.45
8
0.5
3.8
0.82
8
2.0
10.9
3.25
0.5
9
0.63
0.5
5.1
10
0.63
0.5
5.1
Liberaalinen Kansanpuolue, Liberal Party; 7 = SMP, Suomen Maaseudun Puolue, Finnish Rural Party; S=SKL, Suomen
Kristillinen Liitto, Christian League of Finland; 9 = SKYP, Suomen Kansan Yhtenaisyyden Puolue, Party of Finnish People’s
Unity: 10= SPK Suomen Perustuslaillinen Kansanpuolue, Finnish Constitutional People’s Party; 1 1 = SPSL, Tyovaen ja
Pienviljelijain Sosialidemokraattinen Liitto, Social Democratic Union of Workers and Small Farmers.
1 = SDP. Suomen Sosialidemokraattinen Puolue, Social Democratic Party of Finland: 2 = SKDL, Suomen Kansan
Demokraatinen Liitto, Democratic League of the People of Finland; 3 = Kok, Kansallinen Kokoomus, National Coalition Party;
4 = Kepu, Keskustapuolue, Centre Party; 5 = RKP, Ruotsalainen Kansanpuolue, Swedish People’s Party in Finland; 6 = LKP,
1979
1975
1972
1970
I966
0.
W
N
;4
m
r
r
0
T
L
U
rn
n
P
z
z
>
270
FORMING COALITIONS A N D MEASURING VOTING POWER
7 . THE PUBLIC GOOD INDEX
When calculating the voting power, we should consider each minimum
winning coalition only once, since by (5) of our story every winning decisive set
corresponds with one specific outcome. Having abstracted from introducing
specific preferences for the players of our voting game, we will give equal values
u ( S ) to the various outcomes of the coalitions S E M ( u ) . Due to the public good
character of the coalition outcome, this value u ( S ) will be valid for each
member of the specific coalition S . If we assume that every MWC being
element of M ( u ) occurs with equal probability, the following measure for the
(relative) a priori voting power of player i results:
thereby
n
1 hi(U) = 1.
i= I
If we standardize the value of the minimum winning coalition ( S ) with
o( S) = 1 , h,(u) measures the number of times a player i is a member of a minimum winning coalition S, divided by the number of times the n players of the set
N are members of S E M(u). This index was used by Holler for measuring the
voting power in the Finnish Parliament.25 The author gave no theoretical
justification. He was doubtful about the validity of this measure, for it is not
necessarily monotonic with respect to the voting weights. This can easily be
verified by using the identical example for which the Deegan-Packel index has
also shown nonmonotonicity. Given the voting body u2 = (51; 35, 20, 15, 15,
15), it follows the power distribution
One could accept this result, hereby referring to the fact that for specific voting
bodies (like the above) a player with a smaller voting weight can be member of
more minimum winning coalitions and will therefore have a higher a priori
voting power than a player with higher voting weight. Indeed, the larger the
number of coalitions which player i can turn from winning into nonwinning by
changing his vote, the more likely it is that the resulting policy corresponds with
player i’s preferences. One could alternatively question the equiprobability of
minimum winning coalitions. For instance, since Leiserson’s bargaining concept
favours coalitions with small numbers of players,26 the coalition (35, 20)
becomes more likely than others. We may thus choose probability weights for the
considered minimum coalitions which increase the power measures for both the
35 per cent player and the 20 per cent player. The monotonicity of apriori voting
power and voting weights could then be re-established. It is, however, not the
concern of this paper to discuss this alternative in detail.
*’
’’
M. J . Holler, ‘ A Priori Party Power and Government Formation’, Munich Social Science
Reoirw, 4 (1978), pp. 32 ff.
Leiserson. ‘Factions and Coalitions in One-Party Japan’.
MANFRED J. HOLLER
27 I
8. SOME EMPIRICAL RESULTS
Table 1 shows some empirical results from calculating a priori voting power
for the parties in the Finnish Parliament in the period of 1948-79.’’ In the first
row you will find figures attached to the eleven parties which have been
represented in the Finnish Parliament since the election of 1948. In the second
row the voting weights (Lee, the relative numbers of seats) are listed in
decreasing magnitude. In the third row you will find the values of the index h,
and in the fourth row there are the (standardized) values of the Banzhaf
index.
For smaller parties (party 5 to 11), with exception of party 6 (LKP) in the
years 1948 and 1970, the values of the index h are larger than the values of the
Banzhaf index and the relative seat shares. For larger parties (party 1 to 4) the
values of the index h tend to be smaller than both of these measures. For the
periods 1951, 1954, 1966,1970, 1972, and 1975 the power weights as measured
by the index h are not monotonic with the seat shares. Thus, by this empirical
result we cannot say that the ‘paradox of nonmonotonicity’ is an exception. It
seems to occur rather frequently. However, from the discussion above we can
see that there is nothing paradoxical about this phenomenon if we accept the
notion of power and the hypothesis of coalition formation underlying the
index h.
’’ Holler, ‘ A Priori Party Power and Government Formation’, pp. 38 ff.