Sociology Mind
2013. Vol.3, No.2, 149-155
Published Online April 2013 in SciRes (
Copyright © 2013 SciRes. 149
Ex Ante and ex Post Voting Power: A Method for Calculating
Party Power in Party Government
Jan-Erik Lane, Alexander M. Preker
Department of Political Science, University of Freiburg, Freiburg im Breisgau, Germany
Received December 30th, 2012; revised February 3rd, 2013; accepted February 22nd, 2013
Copyright © 2013 Jan-Erik Lane, Alexander M. Preker. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
As political power tends to be wielded in the form of voting power in the national assembly, especially
under the institutions of party government, one needs a method to calculate the voting power of political
parties, both longitudinally and for a cross-section of European democracies. This paper suggests such a
method, derived from the power index approach in cooperative game theory. The application of the
method on the history of democracy in the two German nations results in party scores—mandates, ex ante
and ex post voting power—that are much in congruence with the standard interpretations of the interwar
period and the post-war politics in these two countries.
Keywords: Party Government; Government Coalitions and Coalitions in the Legislature; Ex Ante and ex
Post Voting Power; Penrose-Banzhaf-Coleman Framework; Banzhaf Numbers or Scores;
Double Banzhaf Games; Party Government in the Two German Speaking Nations
Political power under a regime with party government in a
multi-party system derives from the capacity of parties to enter
into winning coalitions, either ad hoc in relation to the voting
on decision issues in Parliament, or as a permanent member of
a coalition in government. The Penrose-Banzhaf-Coleman
model of coordination in N-person gamescoalitionsoffers a
method to estimate this form of political power. In short: The
cooperative game theory model measures the capacity of a
party to contribute to the success of a coalition.
Here we wish to show how the power index method can be
employed for the systematic and comparative analysis of the
power of political parties in the typical regime of party gov-
ernment. We give as example a small study of party govern-
ment in GermanyWeimar and FRGas well as Austria1st
and 2nd Republic (1919-1933 and 1945-today). The research
question can be posed as such: Have the various political par-
ties been able to exercise voting power in proportion to their
electoral support, as measured by the seats of the party in the
nation’s representative assembly?
Logic of Party Government: Double Banzhaf
In order to estimate the power of political parties under the
regime of party government one needs to calculate Banzhaf
numbers for the players involved in the games of party gov-
ernment, i.e. the political parties. In addition, one has to calcu-
late the Banzhaf power for governments, whether majority or
minority ones. All voting power derives from the capacity to
form winning coalitions in the national assembly.
Banzhaf Voting Power
The Banzhaf voting power index (Banzhaf, 1965; Coleman,
1971; Felsenthal & Machover, 1988, 2002, 2004, 2005) models
the marginal contribution of a player to the success of a coali-
tion, given a social choice mechanism. Looking here at the so-
called Penrose-Banzhaf-Coleman framework for solving one
kind of N-person games, namely so-called simple games, one
starts from the characteristic function of a game. Basically, one
considers the set 2N of all possible coalitions S employing the
assumption of equally likely coalitions. From these 2N coali-
tions, the Banzhaf score of player i is calculated as the number
of coalitions in which i is critical according to:
that is, the number of coalitions that player i is able to swing.
The Penrose-Banzhaf power index β of player i is defined by
the ratio of swings η of i to the number of coalitions not con-
taining the player i. Thus, we have:
(2) 1
The employment of the Banzhaf formula results in voting
power scores, either absolute or relative ones (normalised).
Since we compare these scores with the percentage number of
seats or mandates in Parliament, we use relative Banzhaf scores
below. All data contained in the Tables below have been calcu-
lated with the help of a computer algorithm for voting power
analysis by David Leech. Online: [last re-
view: 29 January 2013].
Below political parties will be looked upon as organised
players making coalitions involving a perfect degree of indi-
vidual party member discipline.
Ex Ante and ex Pos t Voting Power
The representative assembly makes a huge number of deci-
sions concerning for instance government formation and sup-
port, the budget and taxation as well as regulation in the form
of legislation and policies. Additional functions of parliament
include controlling government and articulating relevant prob-
lems as well as electing government. Under party government,
the actors in collective decisions are the political parties that
tend to vote with one voice, based on their differential number
of mandates. Both the constitutional rules of decision-making
and the strategy of coalition formation are decisive for the out-
comes of parliamentary decision-making.
Most decisions in Parliament require simple majorities,
which is why a minimum winning coalition is enough. Some-
times certain decisions of constitutional relevance are singled
out requiring a heavier support or qualified majority. To prevail,
a winning coalition would then have to be oversized. In the
normal business of government, simple majority suffices in
most parliamentary countries.
A coalition that is minimum winning does not need to be a
minimum sized coalition. Actually, coalitions may be of many
kinds, but the essential thing is that they win the voting se-
quences. Coalitions may be temporary, focusing on one voting
sequence, or they may be permanent over an election period,
such as for instance 4 years. Government coalitions are formed
for the purpose of exercising power over a longer period, but
government coalitions may break up prematurely, leading either
to a new government or to new elections.
A political party maximizes its voting power by actively par-
ticipating in the formation of coalitions. If it enters a govern-
ment coalition, then it shares the voting power of the govern-
ment with its governing partners. It may also exercise voting
power by participating in temporary coalitions that achieve the
minimum-winning format. Political power is mainly exercised
through the capacity to influence voting in the national assem-
bly. Typical for continental European democracies is the multi-
party system, meaning that the parliamentary arena is com-
prised of three or more political parties, where no party has a
majority position.
One may enquire after the differences in seats and voting
power for the major political parties under the regime of party
government by calculating two power index scores: ex ante
power measures based upon the election outcomes before gov-
ernment formation, and ex post power measures based upon the
pattern of government formation: minority, simple majority,
oversized, etc., after government formation. The following dis-
tinctions can be made in this new method for the calculation the
power of parties under party government:
a) The calculation of ex ante power scores is the straightfor-
ward solution of the Banzhaf game for parties in the legislature
before the formation of a government or government coalition.
b) The calculation of the ex post power scores involves a
double Banzhaf game for the parties after government forma-
tion, where the parties in a government coalition share whatever
power government has in the legislature, on the basis of a una-
nimity game between the government coalition partners.
One may in principle distinguish between two parliamentary
situations: a majority coalition and a minority coalition. The
voting power score of the government coalition will here be
assumed to be equally shared among the coalition parties on the
basis of an imagined second Banzhaf game within the govern-
ment—a unanimity game among equal coalition partners. Thus,
we have:
1) Banzhaf number or score for a simple majority govern-
ment = the party(ies) in government constitute the winning
coalition in legislative voting (Banzhaf score = 1/number of
parties in the government coalition).
2) Banzhaf number or score for a minority government = the
party(ies) in government enter some of the winning coalitions
in the legislature (Banzhaf score 0 < 1/number of coalition par-
ties). When the coalition government does not control the vot-
ing in the legislature, it is forced to make ad hoc coalitions with
the other parties—see Appendix for an example.
When there is a minority coalition government, other parlia-
mentary actors like the opposition parties will receive Banzhaf
scores as well, depending on whether they are decisive in coali-
c) One may further calculate the parties’ Banzhaf power in-
dex numbers for both minority and majority coalitions for every
legislative period and weighted for every single year, thus ar-
riving at aggregated scores for legislative periods, and longer
time spans.
d) By relating the ex post Banzhaf scores to the relative size
of the parties’ representation in the legislative assembly (seats
or mandates), it can be shown how the parties’ voting power
differ from their sizes in Parliament, because some parties tend
to dominate the political game, whereas others do not play a big
role despite their relative strength.
Let us proceed to a couple of examples from the spectacular
political history of democracy in Germany and Austria.
Example: Party Government in the Two
German Speaking Nations
Proportional election formulas have been utilized in Ger-
many to establish a legislative assembly. In compliance with
parliamentarism, main power lies with the Kanzler. Two kinds
of governments prevail: simple majority governments (FRG)
and minority governments/coalitions (Weimar). There have
only been a few consociational governments. In the Weimar
Republic, the disproportion between the size of representation
of the left-wing (seats, mandates) and its ex post Banzhaf vot-
ing power is striking. In the FRG, an effort was made to elimi-
nate this effect, but has it been successful?
The Weimar Republic
In the Weimar Republic the smaller and right wing parties
often augmented their voting power in comparison to their
share of seats in the Reic hstag by joining government. This
discrepancy and disproportionality can be demonstrated by
subtracting the parties’ relative share of seats from its ex post
Banzhaf points: Especially DDP and BVP enlarged their influ-
ence and voting power in comparison to their share of seats
(DDP + 6.7 per cent, BVP + 5.3 per cent). But also larger con-
servative parties, that is Zentrum and DVP, increased their vot-
ing power by entering the mainly minority governments. Ap-
parently, party coalitions, first minority governments, then
Copyright © 2013 SciRes.
Copyright © 2013 SciRes. 151
shifting majority coalitions, were the key to exercise power in
the Weimar Republic, highly successfully accomplished by
DVP’s Gustav Stresemann between 1923-1929 (Table 1).
On the contrary, the Social Democratic party (SPD) and the
Table 1.
Banzhaf power distribution in the weimar republic 1919-1933.
Party Modified aggregation Deviations from proportionality (power – seats)
SPD seats (rel.) 0.254736842
ex ante 0.291411 0.0366742
ex post 0.179639259 0.0750976
U SPD seats (rel.) 0.154
(1919-1924) ex ante 0.139715 0.014285
ex post 0.1057972 0.0482028
DNVP seats (rel.) 0.138947368
ex ante 0.126354053 0.0125933
ex post 0.113363011 0.0255843
Zentrum seats (rel.) 0.139635857
ex ante 0.129780053 0.0098558
ex post 0.155244942 0.0156091
BVP seats (rel.) 0.037010234
ex ante 0.034639444 0.0023708
ex post 0.090371939 0.0533617
DVP seats (rel.) 0.084253906
ex ante 0.077607053 0.0066469
ex post 0.136149469 0.0518955
DDP seats (rel.) 0.058270387
ex ante 0.052078947 0.00619144
ex post 0.124839478 0.066569091
NSDAP seats (rel.) 0.132896772
(1924-1933) ex ante 0.148824857 0.015928085
ex post 0.116784334 0.016112438
Wirtschafts- seats (rel.) 0.024752883
partei ex ante 0.023595211 0.001157672
ex post 0.029837938 0.005085055
KPD seats (rel.) 0.096936858
ex ante 0.094663278 0.00227358
ex post 0.045465496 0.051471362
Dt.-Hannov. seats (rel.) 0.006994286
Partei ex ante 0.006280333 0.000713953
ex post 0.001587514 0.005406772
Christl.-nat. seats (rel.) 0.020770765
Landvolk (1924-1933) ex ante 0.018560071 0.002210694
ex post 0.012855946 0.007914819
CSVD seats (rel.) 0.015807965
(1930-1933) ex ante 0.014741333 0.001066632
ex post 0.013180917 0.002627048
Konservative seats (rel.) 0.001299827
Volkspartei ex ante 0.0034224 0.002122573
(1930-1932) ex post 0.013798129 0.012498302
Deutsche seats (rel.) 0.010404347
Bauernpartei ex ante 0.008974556 0.001429791
(1928-33) ex post 0.003109667 0.00729468
Others seats (rel.) 0.006491473
ex ante 0.00438 0.002111473
ex post 0.001472844 0.005018629
Abbr.: SPD = Sozialdemokratische Partei Deutschlands (social democratic); U SPD = Unabhängige Sozialdemokratische Partei Deutschlands (independent, socialist);
DNVP = Deutschnationale Volkspartei (nationalist, conservative); Zentrum = Center party (catholic); BVP = Bayerische Volkspartei (Bavarian, conservative); DVP =
Deutsche Volkspartei (liberal); DDP = Deutsche Demokratische Partei (social liberal); NSDAP = Nationalsozialistische Deutsche Arbeiterpartei; Wirtschaftspartei =
liberals; KPD = Kommunistische Partei Deutschlands (communists); Deutsch-Hannoversche Partei (local conservative party); Christl.-nat. Landvolk = Christlich-Nationale
Bauern- und Landvolkpartei (agrarian, conservative); CSVD = Christlich sozialer Volksdienst (protestant conservative); Konservative Volkspartei = small conservative
party; Deutsche Bauernpartei (agrarian).
Communist Party (KPD) respectively were often unable to turn
their mandate strength in parliament effectively into ex post
influence or voting power after the broad “Weimar coalition”.
Social democrats became frequently the largest party in parlia-
ment after elections. Additionally, their ex ante Banzhaf score
was greater than their relative mandate strength. But even so,
they seldom joined government. A loss of power can be stated
from the ex ante to the ex post stage of government formation,
which can be explained by a lack of coalition options from the
mid-1920s onwards.
The nationalist DNVP was a large right-wing opposition
party, which often supported one of the minority governments.
But, in contrast to SPD or KPD, the DNVP shows only a small
loss of power when comparing its relative proportion of seats
and its voting power. Thus, only a small difference can be
stated between the DNVP with its nearly equal ex ante and its
post Banzhaf scores and the government parties.
The Federal Republic (FRG)
At first sight, the results for the FRG seem to differ from the
findings for the Weimar republic: The Federal Republic devel-
oped a 2.5 party system (Poguntke, 1999), comprising two
“catch all” parties (Kirchheimer, 1965), namely CDU respec-
tively CSU and SPD, and the FDP as a 0.5 “pivotal” party
(Keman, 1994).
But when applying the Banzhaf-Coleman approach to the
distribution of power, we thus find a parallel between the Wei-
mar republic and the FRG: Again, the political left shows a
tendency of underrepresentation in governments on the federal
level, also when the party system changed since the 1980s and
As can be seen in Table 2, only the FDP has been able to in-
crease its voting power in comparison to its relative mandate
strength. The “Union”, consisting of CDU and CSU, shows
only small losses of power in the ex ante and ex post govern-
ment estimates. The FDP has played a powerful role by means
of its pivotal status. Taking the discrepancy between the rela-
tive share of seats and the ex post Banzhaf scores into account,
it can be noted that only the liberal FDP (+28 per cent) and to a
lesser degree the Greens (+4.6 per cent) are the parties whose
Banzhaf ex post power scores increased between 1949 and 2012,
compared with their share of the seats (mandates).
The Social Democrats have often been unable to transfer
their profound mandate strength into power through an ex post
government formation. Their voting power is small compared
with their representation. Thus, the loss of power in ex post
government formation is nearly twice as much (18.1 per cent)
as the Christian democratic loss (9.5 per cent). Yet, the lack of
power of the left-wing on the federal level is compensated by
participation in several Länder governments and their power in
the Bundesrat, although Die Linke is still regarded as “regi-
erungsunfähig” (incapable of governing) by the other four par-
ties at the federal level.
When looking at the First Republic in Austria, the right-wing
parties were heavily overrepresented in comparison to their
relative share of seats in the Nationalrat. On the other hand, in
Table 2.
Banzhaf power distribution in the FRG 1949-2012.
Party Modified aggregation Deviations from proportional ity (power – seats)
CDU/CSU seats (rel.) 0.52482205
ex ante 0.501040618 0.023781432
ex post 0.429696945 0.095125105
SPD seats (rel.) 0.435363492
ex ante 0.301324545 0.134038947
ex post 0.254545455 0.180818037
FDP seats (rel.) 0.111550297
ex ante 0.242562455 0.131012158
ex post 0.395151491 0.283601194
Grüne (since 1983) seats (rel.) 0.070312448
ex ante 0.153269967 0.082957519
ex post 0.116666667 0.046354219
PDS (since 1991)/ seats (rel.) 0.055674471
Linkspartei.PDS / ex ante 0.072640727 0.016966256
Die Linke ex post 0.000000000 0.055674471
Others seats (rel.) 0.01402697
(1949-1961) ex ante 0.01501596 0.00098899
ex post 0.01301044 0.00101653
Abbr.: CDU/CSU = Christdemokratische Partei Deutschlands/Christlichsoziale Partei Bayerns (Christian democratic); SPD = Sozialdemokratische Partei Deutschlands
(Social democratic); FDP = Freie Demokratische Partei (liberal); Grüne (ecological); PDS = Partei des demokratisches Sozialismus, Linkspartei. PDS, Die Linke (left
Copyright © 2013 SciRes.
the Second republic (1945-today) with its consociational gov-
ernments, an opposing finding can be made: Both the Austrian
social democrats (SPÖ) and the Christian democratic (ÖVP) are
able to augment voting power from the ex ante to the ex post
government formation. In contrast to the German system, the
smaller parties (FPÖ, Greens, BZÖ) play a minor role in par-
liament and government.
The First Republic (1919-1933)
Between 1919 and 1933 especially three political blocks
dominated the Austrian party system: a socialist (SDAP), a
Christian-catholic (Christlichsoziale) and a German nationalist
block, which included some smaller parties (Pelinka, 2004: p.
Neglecting the two years, in which SDAP and Christlich-
soziale formed a grand coalition before the constitution was
passed in 1920, it was the Christian social party and the smaller
German nationalist party group, which was participating in
government on a regular basis. As can be seen in the Banzhaf
power distribution in Table 3, the SDAP was heavily underrep-
resented in comparison to its relative share of seats (25.9 per
cent). The German nationalist party group, in contrast, is mas-
sively overrepresented with a plus of 31 per cent in relation to
the mandate strength won in elections.
The Second Republic (1945-Today)
Germany is often described as the “Grand coalition state”
(Schmidt, 1996). But, when comparing the number of grand
coalitions in the FRG (2) and Austria (10), it has to be empha-
sized that this concept would fit Austria better. This is reflected
in the Banzhaf numbers for the parties ex post government for-
mation: in Austria, it is only the two “catch all” parties that
augment their power by frequently forming government (SPÖ +
4.2 per cent, ÖVP + 2.7 per cent) while all other parties incur
For a long time, the Austrian party system was a 2.5 party
system like the German one, consisting of SPÖ, ÖVP and FPÖ.
But in contrast to the FRG, the Austrian FPÖ never gained the
“pivotal” status as the FDP could in Germany, because of the
major parties exercising power in an oversized coalition (Table
The Banzhaf-Coleman approach may be developed into a
tool for the systematic and comparative analysis of the power of
political parties under the institutions of party government.
Voting power is wielded over legislation, budgets and taxation,
policies and regulations. It derives from winning by simple
majority in the legislative assembly, either as a single party or
as a coalition partner in government, with majority or minority
Applying this new method to the political history of democ-
racy in the two German nations, we observe quite a remarkable
fit between the stylised images of the politics of these countries
and the power scores for the political parties. Thus, the interwar
period comprised highly skewed party government, with dis-
Table 3.
Banzhaf power distribution in Austria 1919-1933 (1st Republic).
Party Modified aggregation
Deviations from proportionali ty (power – seats)
modified a ggregati o n
SDAP seats (rel.) 0.326533333
ex ante 0.333333333 0.0068
ex post 0.066666673 0.259866666
Christlichsoziale seats (rel.) 0.462751467
ex ante 0.333333333 0.129418134
ex post 0.466666666 0.003915199
Deutschnationale seats (rel.) 0.13902
ex ante 0.3333333 0.1943133
ex post 0.449999967 0.310979967
Landbund seats (rel.) 0.14
ex ante 0 0.14
ex post 0 0.14
Heimatbund/Heimwehr seats (rel.) 0.04424
ex ante 0 0.04424
ex post 0.01777767 0.02646233
Others seats (rel.) 0.012
ex ante 0 0.012
ex post 0 0.012
Abbr.: SDAP = Sozialdemokratische Arbeiterpartei (social democratic); Christlichsoziale = Christlichsoziale Partei (christian-conservative); Deutschnationle = Deutsch-
Nationale Bewegung (nationalist); Landbund (agrarian); Heimatbund/Heimwehr (paramilitary, nationalist).
Copyright © 2013 SciRes. 153
Table 4.
Banzhaf power distribution in Austria after 1945 (2nd Republic).
Party Modified a g gregation De v ia tions from proportionality (p ower – seats)
KPÖ (1945-1959) seats (rel.) 0.024675318
ex ante 0.035714293 0.011038975
ex post 0 0.024675318
Greens (since 1986) seats (rel.) 0.074190836
ex ante 0.058608031 0.015582805
ex post 0 0.074190836
SPÖ seats (rel.) 0.433602479
ex ante 0.391257975 0.042344504
ex post 0.475124378 0.041521899
ÖVP seats (rel.) 0.418770529
ex ante 0.366382331 0.052388198
ex post 0.445273631 0.026503102
VdU/FPÖ (since 1949) seats (rel.) 0.104511154
ex ante 0.212018132 0.107506978
ex post 0.084656084 0.01985507
Liberales Forum (1994-1999) seats (rel.) 0.055737705
(1994-2002) ex ante 0 0.055737705
ex post 0 0.055737705
BZÖ (since 2006) seats (rel.) 0.089253188
ex ante 0.142857 0.053603812
ex post 0 0.089253188
Abbr: KPÖ = Kommunistische Partei Österreichs (communist), Greens = Grünen – Die grüne Alternative (ecological); SPÖ = Sozialdemokratische Partei Österreichs
(social democratic); ÖVP = Österreichische Volkspartei (Christian democratic); VdU/FPÖ = Verband der Unabhängigen (1945-1955)/since 1955: Freiheitliche Partei
Österreichs (right-wing-demagogic, populist); Liberales Forum (liberal); BZÖ = Bündnis Zukunft Österreichs (nationalist-right-wing demagogic, economically liberal).
proportionate power of the centre (Germany) and the right-wing
(Austria), indicating democratic instability. The post-war shows
considerably more of balance between the right and the left,
especially in Austria, where consociationalism provides the left
with more power than the simple majority governments in the
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Calculation of ex ante and ex post voting power in a 5 player legislature with parties of the following size: P1 = 35%, P2 = 25%, P3 = 18%, P4 = 12%
and P5 = 10%.
P1 P2 P3 P4 P5
Ex ante voting power: 0.38 0.23 0.23 0.077 0.077
Ex post voting power:
Majority gov P1 + P2: 1/2 1/2 0 0 0
Minority gov with P1 + P5: 0.5/2 0.17 0.17 0.17 0.5/2
Copyright © 2013 SciRes. 155